In the field of metro vehicle manufacturing, the mechanical transmission system plays a critical role in ensuring operational efficiency and safety. Gear grinding is a vital process in producing high-precision gears for these transmissions, as it directly impacts the performance and durability of the components. However, the positioning of gear grinding fixtures faces significant challenges due to factors such as vehicle-induced vibrations, cutting forces, and fixture self-weight, which can lead to deviations in gear placement and the formation of grinding cracks. These issues necessitate robust positioning methods to minimize errors and enhance the quality of gear profile grinding. In this paper, we propose a comprehensive approach to designing a positioning method for gear grinding fixtures in metro vehicle mechanical transmissions. Our method focuses on analyzing positioning source deviations, constructing robust models, and optimizing the fixture layout to achieve high precision and stability. By integrating geometric modeling, robust design principles, and advanced optimization algorithms, we aim to address the limitations of existing methods, such as low accuracy and poor noise resistance. Throughout this work, we emphasize key aspects of gear grinding, including the mitigation of grinding cracks and the refinement of gear profile grinding processes, to ensure reliable and efficient manufacturing outcomes.
The foundation of our positioning method lies in the development of a detailed geometric model for the grinding fixture. This model enables the quantitative analysis of positioning source errors and gear position deviations, which are critical for understanding the dynamics of gear grinding. We begin by defining the gear’s positioning contact surface, represented as $f_e = (x_e, y_e, z_e)^T \times t_e$, where $x_e$, $y_e$, and $z_e$ denote parameters such as the friction angle between the gear spring collet and the taper sleeve, the friction angle between the spring collet and the workpiece, and the semi-taper angle of the spring collet, respectively, while $t_e$ is the coordinate matrix of any point on the gear. The tangent plane at the contact point on this surface is given by $\phi_e^{yo} = m_e^o \times (t_e – t_{ero}) \times f_e$, where $m_e^o$ is the unit normal vector and $t_{ero}$ is the contact point coordinate. To simplify the analysis, we transform the gear coordinates from a three-dimensional system to a two-dimensional system using the equation $t = Y(\lambda_e) T + t_e \times \phi_e^{yo}$, where $\lambda_e = (\beta_e, \chi_e, \omicron_e)$ represents the orientation parameters relative to the two-dimensional coordinate system, and $Y(\lambda_e)$ is the coordinate rotation transformation matrix. This transformation facilitates the derivation of the translation plane $\phi_p^o = Y(\lambda_e)^T \times w_e \times (\beta_e, \chi_e, \omicron_e) \times t_z^o$, where $w_e$ is the tangent plane point coordinate and $t_z^o$ is the coordinate vector. The distance between the numerical value $H_i^o$ of the gear positioning contact point tangent plane and the corresponding value $R_{jh}$ of the tool setting point is expressed as $Z_f = H_i^o \times R_{jh} \times H_{mn} \times \phi_p^o$, where $R_{jh}$ is the distance parameter and $H_{mn}$ is the workpiece diameter. This leads to the determination of the gear positioning structure, as illustrated in the following diagram, which highlights the critical aspects of gear profile grinding and the potential for grinding cracks if not properly managed.
Building on this geometric foundation, we derive the positioning point equation $F_{XYZ} = G_X \times G_Y \times G_Z \times Z_f$, where $G_X$, $G_Y$, and $G_Z$ are the coordinate vectors of the positioning point. This allows us to establish the position deviation vector of the positioning point and construct the fixture geometric model as $H_{JK} = K_M + \mu \times O_{6 \times 6} \times F_{XYZ}$, where $H_{JK}$ represents the fixture geometric model, $K_M$ is the generalized inverse matrix, $\mu$ is a constant, and $O_{6 \times 6}$ is a 6×6 order coordinate rotation transformation matrix. This model serves as the basis for analyzing deviations in gear grinding, particularly in relation to gear profile grinding and the prevention of grinding cracks. The geometric model accounts for various factors, such as the elastic deformation of the fixture and gear system, which can exacerbate positioning errors if not properly controlled. By quantifying these deviations, we can identify critical areas for improvement in the fixture design, ultimately enhancing the accuracy and reliability of the gear grinding process.
To address the challenges of positioning accuracy and stability, we develop a robust model for gear grinding fixture positioning. This model aims to achieve preliminary fixture positioning by considering both controllable design variables and uncontrollable noise factors. We define the function $G_o(X, Z)$ to describe the correct position of the gear in the grinding fixture, where $X$ and $Z$ represent the positioning dimensions. The fluctuation of this function due to external disturbances leads to positioning errors, which we mitigate using the average quality loss function as the criterion function $\phi_o(X, Z) = H_{JK} \times R \times [(G_o – \bar{G}_o)]^2 + [(\bar{G}_o – G’)]^2$, where $R$ is the positioning benchmark position, $G_o$ is the positioning distribution parameter, $\bar{G}_o$ is the mean value, and $G’$ is the expected value. To ensure that the deviation of positioning errors does not exceed the specified precision, we impose constraints on the positioning benchmark position parameters, leading to the robust model for gear grinding fixture positioning:
$$ \text{find } Y = [Y_1, Y_2, \ldots, Y_o] $$
$$ \min [\zeta^2 G_1(X, Z), G_2(X, Z), \ldots, G_n(X, Z)] $$
$$ \text{s.t. } \omega G_o \leq \phi_o(X, Z) \times \Delta_o^F \quad (o = 1, 2, \ldots, N) $$
$$ x_{Zk} \leq x_k \leq x_{ik} $$
$$ \Delta x_k \geq \Delta x’_k $$
In this model, $Y$ represents the tooth tip height, $\zeta$ is the elastic modulus, $\omega$ is the load value, $Y_1, Y_2, \ldots, Y_o$ are the axial cutting forces, $G_n(X, Z)$ is the elastic deformation coefficient, $\Delta_o^F$ is the precision, $x_{Zk}$ is the gear module, $x_k$ is the number of teeth, $x_{ik}$ is the tooth width, $\Delta x_k$ is the design variable, and $\Delta x’_k$ is the design variable deviation. This robust model allows us to achieve a more stable positioning solution, reducing the impact of uncertainties in the gear grinding process. By focusing on key aspects such as gear profile grinding and the minimization of grinding cracks, we ensure that the fixture positioning remains accurate under varying operational conditions. The model’s constraints help maintain the integrity of the gear geometry, which is essential for preventing defects like grinding cracks that can compromise the transmission’s performance.
Following the initial positioning, we optimize the fixture layout by designing the balance equation for the grinding fixture and gear elastic system. The balance equation is given by $P_H = L \times I \times G + Q \times \text{find } Y$, where $L$ is the unknown displacement vector, $I$ is the stiffness matrix, $G$ is the vector of other external forces, and $Q$ is the unknown spring contact force vector. We express the balance equation in terms of selected node $K$ and other nodes $P$ as $L_{KV} = I_P \times I_V \times P \times K \times P_H$, where $I_P$ is the gear shaving mandrel and $I_V$ is the elastic gear shaving mandrel. The spring contact force vector $S_T$ is derived as $S_T = V_M \times A_M \times L_{KV}$, where $V_M$ is the number of mandrels and $A_M$ is the number of spring sleeves. Since the gear only承受 unidirectional normal pressure from the positioning elements, we impose a unilateral contact force constraint $B_n^M \geq 0$ for any contact point $n$, leading to the expression for the contact point unilateral contact force constraint $B_o^M = S_T \times m_o \times I_{oV}$, where $m_o$ is the number of degrees of freedom and $I_{oV}$ is the contact point displacement vector. We solve the balance equation using the nonlinear Newton method, where the Newton step size $x$ is given by $x = K(c) \times (x_i, i_a) \times B_o^M$, with $K(c)$ being the Jacobian matrix of the gear grinding fixture and $x_i$, $i_a$ as the Newton step size coefficients. By replacing the deformation of the fixture and gear with the deformation of characteristic points, we obtain the optimized layout of the grinding fixture positioning as $\min g = E_d \times D_e \times B_a \times A_b \times x$, where $g$ is the positioning modulus of the grinding fixture, $E_d$ is the position error between the actual and theoretical positioning of the metro vehicle mechanical transmission gear, $D_e$ is the positioning Jacobian matrix, $B_a$ is the deviation coefficient, and $A_b$ is the unit normal vector matrix. We employ the Method of Moving Asymptotes (MMA) algorithm and sensitivity analysis to obtain and update the design variables, repeating this process until convergence conditions are met. This optimization step enhances the precision of gear grinding, particularly in gear profile grinding, by reducing the likelihood of grinding cracks and improving overall fixture stability.
To validate the effectiveness of our proposed positioning method, we conducted a series of experiments using a GPU server and the CFDS program for testing. The parameters of the mechanical transmission gear used in the experiments are summarized in the table below, which includes critical dimensions relevant to gear grinding and gear profile grinding processes.
| Parameter | Value |
|---|---|
| Cylindrical Gear Teeth Number | 25 |
| Face Gear Teeth Number | 77 |
| Nominal Pressure Angle (°) | 20 |
| Pinion Helix Angle (°) | 0 |
| Module (mm) | 3 |
| Shaft Angle (°) | 90 |
| Face Gear Inner Radius (mm) | 112 |
| Face Gear Outer Radius (mm) | 133 |
We compared our method with two existing approaches from the literature, focusing on positioning accuracy, comprehensive performance, and robustness. The positioning accuracy was evaluated by measuring the errors in the gear tooth profile direction and tooth width direction. The results demonstrated that our method achieved errors within approximately ±10 μm for both directions, significantly lower than the ±50 μm and ±30 μm errors observed in the comparative methods. This improvement is attributed to our robust model’s ability to control positioning benchmark parameters, thereby minimizing deviations in gear grinding. Additionally, we assessed comprehensive performance using the hypervolume indicator, where a higher value indicates better overall performance. Our method achieved a hypervolume value of 0.36, outperforming the values of 0.24 and 0.28 from the other methods, as shown in the following table. This enhancement stems from the iterative optimization process using the MMA algorithm, which refines design variables to achieve convergence.
| Method | Hypervolume Indicator |
|---|---|
| Proposed Method | 0.36 |
| Literature Method 1 | 0.24 |
| Literature Method 2 | 0.28 |
Robustness was evaluated based on the assembly success rate under varying error conditions, such as grinding wheel offset distance error and grinding wheel length measurement error. Our method maintained an assembly success rate of around 80% even with errors up to 0.10 mm, whereas the other methods showed significant declines. This stability is crucial for preventing grinding cracks and ensuring consistent gear profile grinding quality. The following equation summarizes the relationship between assembly success rate $A_s$ and error magnitude $\epsilon$ for our method: $A_s = 80 – 5 \times \epsilon^2$, which highlights the method’s resilience to perturbations. By integrating these experimental findings, we confirm that our positioning method offers superior accuracy, performance, and robustness, making it highly suitable for applications in metro vehicle transmission systems where gear grinding precision is paramount.
In conclusion, we have developed a comprehensive positioning method for gear grinding fixtures in metro vehicle mechanical transmissions, focusing on geometric modeling, robust design, and optimization. Our approach effectively addresses the challenges of positioning deviations, vibration-induced errors, and the formation of grinding cracks, which are critical in gear profile grinding processes. The geometric model provides a foundation for analyzing positioning source errors, while the robust model ensures stability under uncertainties. The optimization phase, utilizing the MMA algorithm and sensitivity analysis, further enhances positioning accuracy and convergence. Experimental results validate the method’s advantages, including reduced errors, improved comprehensive performance, and high robustness, as evidenced by consistent assembly success rates. This method not only enhances the quality of gear grinding but also contributes to the reliability and safety of metro vehicle operations. Future work could explore the integration of real-time monitoring systems to dynamically adjust fixture positioning, further advancing the prevention of grinding cracks and optimizing gear profile grinding outcomes.