In modern industrial applications, internal gears play a critical role in transmission systems for sectors such as wind energy and electric vehicles, where high precision and efficiency are paramount. As an internal gear manufacturer, optimizing the broaching process is essential to meet the growing demand for high-quality internal gears. Broaching offers significant advantages over other gear cutting methods, including higher productivity and superior accuracy, making it the preferred technique for internal gear production. However, accurately predicting cutting forces during broaching remains a challenge due to the prolonged tool-chip contact interface, which influences energy dissipation and tool wear. This study addresses this issue by developing a cutting force model based on the Zorev stress distribution at the tool-chip interface, specifically tailored for internal gear broaching. The model enables real-time monitoring of machine tool motor load rates, providing a practical approach for internal gear manufacturers to enhance process control and improve the machining performance of internal gears.
The accurate prediction of cutting forces is fundamental for evaluating material deformation, optimizing process parameters, and ensuring the stability of machining systems. Traditional cutting force models, such as those derived from empirical coefficients or material shear strength, have been extensively studied for processes like turning and milling. However, broaching involves unique characteristics, such as low cutting speeds (typically below 15 m/min) and long chip-tool contact lengths, which lead to continuous chip formation and increased frictional energy losses. Existing models often overlook the impact of varying chip thickness and extended friction zones in broaching. To bridge this gap, we propose a novel analytical model that integrates the Zorev stress model with broaching dynamics, focusing on the internal gear broaching process. This approach allows internal gear manufacturers to better understand the cutting mechanics and optimize parameters for internal gears, ultimately reducing costs and improving product quality.
The Zorev stress model provides a realistic representation of the normal and shear stress distributions along the tool-chip interface, accounting for both sticking and sliding regions. The contact length between the tool and chip, denoted as \( l_c \), is expressed as:
$$ l_c = h r_i^{1.5} $$
where \( h \) is the undeformed chip thickness (equivalent to the broaching feed per tooth), and \( r_i \) is the chip compression ratio, given by \( r_i = h_c / h \), with \( h_c \) representing the chip thickness. This relationship highlights the influence of chip formation on the contact mechanics, which is crucial for internal gear broaching where chip control affects surface integrity. The shear angle \( \phi \) is derived from tool geometry and chip compression as:
$$ \phi = \arctan \left( \frac{\cos \gamma}{r_i – \sin \gamma} \right) $$
where \( \gamma \) is the tool rake angle. For internal gears, the tool geometry must be precisely designed to accommodate the gear profile, and internal gear manufacturers often customize broaching tools based on these calculations.
Based on the Zorev model, the normal stress \( \sigma \) and shear stress \( \tau \) at any point \( l_x \) from the tool tip are defined as:
$$ \sigma = \sigma_{\text{max}} \left( \frac{l_c – l_x}{l_c} \right)^{\kappa} $$
$$ \tau =
\begin{cases}
\mu_s \sigma_{\text{max}} \left( \frac{l_c – l_x}{l_c} \right)^{\kappa_s}, & l_1 \leq l_x \leq l_c \\
\tau_s, & 0 \leq l_x \leq l_1
\end{cases} $$
Here, \( \sigma_{\text{max}} \) is the maximum normal stress, \( \kappa \) and \( \kappa_s \) are material-dependent coefficients, \( \mu_s \) is the friction coefficient, and \( \tau_s \) is the material shear yield strength. The lengths \( l_1 \) and \( l_c \) demarcate the sticking and sliding regions, with the sticking region typically accounting for 90% of the total shear force, as per prior research. Integrating these stresses over the contact area yields the normal force \( F_\sigma \) and tangential force \( F_\tau \) on the tool face:
$$ F_\sigma = \int_0^{l_c} \sigma \, dA = \int_0^{l_c} \sigma_{\text{max}} \left( \frac{l_x}{l_c} \right)^{\kappa} b \, dl_x $$
$$ F_\tau = \int_0^{l_1} \tau_s b \, dl_x + \int_{l_1}^{l_c} \mu_s \sigma_{\text{max}} \left( \frac{l_x}{l_c} \right)^{\kappa} b \, dl_x $$
where \( b \) is the cutting width. The friction coefficient \( \mu_s \) and friction angle \( \beta \) are then calculated as \( \mu_s = F_\tau / F_\sigma \) and \( \beta = \arctan(\mu_s) \). These parameters are vital for internal gear broaching, as they influence tool life and surface finish of internal gears.
To model the cutting forces in the speed and feed directions, we apply fundamental cutting mechanics:
$$ F_t = F_\tau \sin \gamma + F_\sigma \cos \gamma $$
$$ F_f = F_\tau \cos \gamma – F_\sigma \sin \gamma $$
where \( F_t \) is the cutting force in the speed direction, and \( F_f \) is the force in the feed direction. In broaching, the process involves multiple teeth engaging sequentially, leading to dynamic force variations. The number of active teeth \( n(t) \) at time \( t \) is modeled as:
$$ n(t) =
\begin{cases}
\frac{vt}{\Delta s}, & 0 \leq t \leq \frac{\kappa}{v} \\
N – 1, & \frac{\kappa + \Delta s k}{v} \leq t \leq \frac{(N + k) \Delta s}{v} \\
N, & \frac{\Delta s (N + k)}{v} \leq t \leq \frac{\kappa + (k + 1) \Delta s}{v} \\
z – \frac{vt – \kappa}{\Delta s}, & \frac{\kappa + \Delta s (z – N)}{v} \leq t \leq \frac{\kappa + (z – 1) \Delta s}{v}
\end{cases} $$
Here, \( v \) is the broaching speed, \( z \) is the total number of teeth on the broach, \( \Delta s \) is the tooth spacing, \( \kappa \) is the workpiece thickness, \( N \) is the maximum number of engaged teeth, and \( k \) is an index. The total cutting forces during broaching are then expressed as \( F_t(t) = M n(t) F_t \) and \( F_f(t) = M n(t) F_f \), where \( M \) is the number of tooth rows. This dynamic model helps internal gear manufacturers predict forces under varying conditions, ensuring stable machining of internal gears.
To validate the model, we conducted broaching experiments focused on internal gear manufacturing. The experiments comprised two parts: orthogonal cutting tests to determine material-specific parameters and broaching tests to measure cutting forces and motor load rates. The orthogonal cutting tests used tools and workpiece materials identical to those in broaching, with parameters summarized in Table 1. The broaching tests involved machining internal gears from ZL107 aluminum alloy, which has a yield strength of approximately 107 MPa, using a broach with specified geometry. The broaching parameters are listed in Table 2, and the internal gear workpiece details are in Table 3.
| Parameter | Value |
|---|---|
| Tool Rake Angle (°) | 8 |
| Cutting Speed (m/min) | 10 |
| Undeformed Chip Thickness (mm) | 0.1, 0.2, 0.3, 0.4 |
| Cutting Width (mm) | 1 |
| Parameter | Value |
|---|---|
| Broaching Speed (m/min) | 8 |
| Feed per Tooth (mm) | 0.2 |
| Number of Tooth Rows, M | 28 |
| Tooth Spacing, Δs (mm) | 5 |
| Cutting Edge Length, b (mm) | 5.54 |
| Parameter | Value |
|---|---|
| Module | 2 |
| Number of Teeth | 28 |
| Pressure Angle (°) | 30 |
| Major Diameter (mm) | 59 |
| Minor Diameter (mm) | 54.16 |
| Workpiece Thickness (mm) | 30 |
The experimental setup involved a broaching machine where the workpiece was mounted on a table moving upward to perform internal gear broaching. During broaching, the main drive motor’s load rate was monitored to assess the additional load due to cutting forces. The motor load rate without broaching was recorded as 13%, while during broaching, it increased to 17–18%. The broaching-induced load rate \( \Delta \eta \) was calculated using the torque relationship:
$$ T_L = F_t r $$
$$ \eta_1 = \frac{T_i}{T_e} $$
$$ \eta_2 = \frac{1}{T_e} (T_L + T_i) $$
$$ \Delta \eta = \eta_2 – \eta_1 = \frac{T_L}{T_e} $$
where \( T_L \) is the motor torque due to cutting force, \( r \) is the screw radius driving the table, \( T_i \) is the motor’s rated load, \( T_e \) is the额定负载, and \( \eta_1 \) and \( \eta_2 \) are the load rates without and with broaching, respectively. This method provides a straightforward way for internal gear manufacturers to monitor process conditions in real-time.
The results from the broaching experiments were compared with theoretical predictions from the Zorev-based model. For stable cutting conditions, the model predicted a broaching-induced load rate of 3.73–3.84%, which aligns closely with the experimental range of 4–5%. The minimum deviation was 4%, indicating high model accuracy suitable for monitoring internal gear broaching operations. However, the average deviation was around 16%, primarily due to dynamic effects such as acceleration and deceleration of the worktable during broaching, which cause torque fluctuations. Additionally, variations in the actual torque radius during machining contributed to measurement discrepancies. These factors highlight areas for future refinement in monitoring systems for internal gear manufacturers.
To further analyze the model’s performance, we examined the sensitivity of cutting forces to key parameters like chip thickness and tool geometry. For instance, increasing the undeformed chip thickness \( h \) amplifies the contact length \( l_c \), leading to higher frictional forces and motor load. This relationship is crucial for internal gear manufacturers when selecting broaching parameters to avoid excessive tool wear or deflection in internal gears. The model’s ability to incorporate material properties—such as the shear yield strength \( \tau_s \) and friction coefficients—ensures its applicability to different alloys used in internal gear production.
In discussion, the Zorev stress model offers a physics-based approach that accounts for the extended tool-chip interface in broaching, a feature often neglected in conventional models. By integrating this with broaching dynamics, the model provides a comprehensive framework for predicting cutting forces and motor loads in internal gear broaching. Internal gear manufacturers can leverage this to optimize tooth geometry, cutting speeds, and feed rates, thereby enhancing the efficiency and accuracy of internal gears. For example, reducing the chip compression ratio \( r_i \) through tool design can decrease contact length and cutting forces, prolonging tool life. Moreover, the load rate model enables predictive maintenance by detecting abnormal conditions early, reducing downtime in production lines for internal gears.
Despite the model’s effectiveness, limitations such as ignoring thermal effects and tool wear in long-duration broaching warrant further investigation. Future work could incorporate thermomechanical coupling to address temperature-induced stress changes, which are relevant for high-strength materials used in advanced internal gears. Additionally, extending the model to include multi-tooth interactions and chip breaking phenomena would improve its robustness for internal gear manufacturers dealing with complex gear profiles.
In conclusion, this study presents a cutting force model for internal gear broaching based on the Zorev stress distribution at the tool-chip interface. The model accurately predicts broaching forces and motor load rates, as validated through experiments, providing a practical tool for internal gear manufacturers to monitor and optimize the broaching process. By focusing on the unique characteristics of internal gears, such as prolonged chip-tool contact, the model enhances process control and machining performance. Internal gear manufacturers can adopt this approach to reduce costs, improve product quality, and advance the production of internal gears for critical applications in industries like renewable energy and transportation.