Internal gears are critical components in mechanical transmission systems, offering advantages such as high reliability, compact design, and precision. As an internal gear manufacturer, producing high-quality internal gears efficiently is essential for applications in aerospace, defense, energy, and transportation. Shaping is a primary method for machining internal gears, and optimizing cutting parameters like chip thickness, cutting speed, and feed rates can significantly enhance productivity and tool longevity. This article details the analytical approach for calculating chip thickness during internal gear shaping, provides guidelines for selecting cutting parameters, and introduces methods for predicting tool wear. By implementing these strategies, internal gear manufacturers can achieve better process control and improved outcomes for internal gears.
The shaping process for internal gears involves a generating motion where the cutter and workpiece rotate in synchronization. During this process, the chip thickness varies across different parts of the tooth profile, including the entry side, exit side, and tooth top. Understanding these variations is crucial for internal gear manufacturers to prevent tool damage and ensure smooth machining. The chip thickness can be categorized into three segments: entry-side chip ($\delta_j$), tooth-top chip ($\delta_d$), and exit-side chip ($\delta_o$). An analytical method, derived from graphical principles, is used to compute these thicknesses accurately. This method considers the instantaneous engagement point and the geometry of the cutter and internal gear, providing a practical tool for internal gear production.
To calculate chip thickness, the fundamental principle relies on the relationship between the rotation angle and the perpendicular distance from the instantaneous center to the normal at any point on the cutting edge. For a point M on the cutter edge, the chip thickness $\delta_M$ is approximated as $\delta_M = \theta_M \cdot PQ$, where $\theta_M$ is the angular difference between successive cuts and PQ is the distance from the instantaneous center P to the normal at M. This approximation is valid for small circular feed values, typically less than 1 mm/r, and offers sufficient accuracy for practical applications in internal gear manufacturing.
For the exit side of the cutter, consider point A on the tooth profile. The chip thickness $\delta_A$ is determined using the geometry of the engagement. The formula is given by:
$$\delta_A = \frac{f_c (Z_2 – Z_0)}{r_{j2} Z_2} (r_b + r_{j0}) \cos(\alpha_A + \theta_Z – \varphi_0)$$
where $f_c$ is the circular feed per revolution (mm/r), $Z_0$ and $Z_2$ are the number of teeth on the cutter and internal gear, respectively, $r_{j2}$ is the pitch radius of the internal gear (mm), $r_b$ is the base radius (mm), $r_{j0}$ is the pitch radius of the cutter (mm), $\alpha_A$ is the pressure angle at point A (degrees), $\theta_Z$ is the tooth spacing angle of the cutter (degrees), and $\varphi_0$ is the semi-angle of the cutter tip arc (degrees). This equation shows that the exit-side chip thickness decreases as the cutter engages deeper into the workpiece, with maximum thickness at the start of cut and minimum near zero. The pressure angle influences the thickness, being largest at the tooth tip and smallest at the base circle.
On the entry side, for point B on the cutter profile, the chip thickness $\delta_B$ is calculated similarly. The expression is:
$$\delta_B = \frac{f_c (Z_2 – Z_0)}{r_{j2} Z_2} \left( r_b \cos(\theta_Z + \varphi_B – \alpha_B) + r_{j0} \right)$$
Here, $\varphi_B$ is the arc semi-angle at point B (degrees) and $\alpha_B$ is the pressure angle at point B (degrees). The entry-side chip thickness decreases as the cutting progresses, reaching zero when the normal to the profile passes through the instantaneous pole P. The maximum thickness occurs when the distance PK2 is at its peak, specifically when $PK_2 = r_{j0} – r_b$. This behavior is opposite to that of the exit side, highlighting the need for internal gear manufacturers to balance cutting forces.
For the tooth top, point E on the cutter tip, the chip thickness $\delta_dE$ is derived from the geometry where the normal passes through the cutter center. The formula is:
$$\delta_dE = r_{j2} \theta \sin(\theta_E + \theta_Z)$$
where $\theta_E$ is the angle between the line from the cutter center to point E and the line to the instantaneous center (degrees), and $\theta$ is the angular difference given by $\theta = \frac{f_c (Z_2 – Z_0)}{r_{j2} Z_2}$. The tooth-top chip thickness is generally thicker and requires careful monitoring to avoid excessive tool wear during internal gear shaping.
The cutting edge radius of the shaper cutter also affects chip formation and minimum chip thickness. The edge radius $r$ can be estimated using the formula:
$$r \approx 0.00075 \left[ 35 – 0.55 (\lambda + \gamma) \right]$$
where $\lambda$ is the relief angle (degrees) and $\gamma$ is the rake angle (degrees). For standard cutters with a rake angle of 5° and relief angle of 6°, the edge radius is approximately 0.22 mm. In practice, optical measurements show values between 0.015 mm and 0.025 mm, which align with this estimation. Internal gear manufacturers should consider this when setting parameters to prevent issues like built-up edge or poor surface finish on internal gears.
Selecting appropriate cutting parameters is vital for efficient internal gear shaping. Key parameters include the number of cuts, stroke length, cutting speed, circular feed, radial feed, and machining time. The number of cuts depends on the machine capability and cutting forces. For internal gears with tooth depths up to 5 mm, a single cut may suffice, but deeper teeth often require roughing and finishing cuts, typically 2-3 roughing cuts and 1-2 finishing cuts. The stroke length $L_c$ ensures full tooth engagement and is calculated as:
$$L_c = B + 2.2 + 1.1 \pi m_t \sin \beta$$
where $B$ is the face width of the gear (mm), $m_t$ is the transverse module (mm), and $\beta$ is the helix angle (degrees). This accounts for over-travel at both ends to complete the tooth profile.
Cutting speed in shaping is limited by the intermittent nature of the process. The cutting velocity $v_c$ in meters per minute is related to the stroke rate and material hardness. The formula for cutting speed is:
$$v_c = \frac{\varepsilon}{HB^{0.2}} \cos \beta$$
where $\varepsilon$ is a material constant (e.g., 76.2 for high-speed steel), HB is the Brindle hardness of the workpiece, and $\beta$ is the helix angle. Increasing workpiece hardness by 10% can raise tool wear by 40%, and a 30% increase may lead to 150% more wear, so internal gear manufacturers must adjust speeds accordingly for internal gears. The stroke rate $n$ (strokes per minute) is derived from the cutting speed and stroke length:
$$n = \frac{1000 v_c}{2 L_c}$$
This conversion facilitates machine setup.
Circular feed $f_c$ and radial feed $f_r$ are critical for productivity. The circular feed per revolution (mm/r) influences chip load and power requirements. A recommended value for $f_c$ is calculated as:
$$f_c = K_2 \frac{160}{HB^{0.2}} m_t^{0.5}$$
where $K_2$ is a coefficient ranging from 0.7 to 0.9. For conventional machines, $f_c$ typically falls between 0.1 mm/r and 0.6 mm/r, while CNC machines may use 0.05 mm/r to 0.5 mm/r. Radial feed $f_r$ per stroke (mm/stroke) is often set as a fraction of circular feed:
$$f_r = (0.2 \text{ to } 0.5) \frac{f_c}{Z_2}$$
with common values from 0.02 mm/r to 0.1 mm/r for conventional machines and 0.001 mm/r to 0.02 mm/r for CNC. Roughing feeds are usually 2-4 times higher than finishing feeds to optimize material removal rates for internal gears.
Machining time $t_m$ in minutes is the sum of radial and circular feed phases, given by:
$$t_m = k_c \left( \frac{\pi m_t Z_2}{n f_c} + \frac{H}{n f_r} \right)$$
where $k_c$ is the number of cuts, $H$ is the total tooth depth (mm), and other symbols are as defined. This helps internal gear manufacturers plan production schedules and estimate costs for internal gear batches.
Tool wear prediction is essential for maintaining quality. The flank wear $\Delta H_b$ on the cutter (mm) can be estimated using the empirical formula:
$$\Delta H_b = \frac{HB^{0.2} Z_2 N}{100 Z_0 \varepsilon}$$
where $N$ is the number of workpieces machined. In practice, wear limits of 0.1 mm to 0.3 mm are common, with smaller values for finer-pitch cutters. Monitoring wear allows timely regrinding, preventing catastrophic failure and ensuring consistent quality in internal gear manufacturing.
To illustrate, consider an internal gear with module $m_t = 5$ mm, pressure angle 20°, tooth count $Z_2 = 24$, profile shift coefficient 0.0717, face width 20 mm, and workpiece hardness 220 HB. The circular feed $f_c$ is computed as 0.6 mm/r, radial feed $f_r$ as 0.008 mm/r, stroke length $L_c$ as 23.5 mm, cutting speed $v_c$ as 15.74 m/min, stroke rate $n$ as 335 strokes/min, and machining time $t_m$ as 8.3 minutes. The chip thicknesses are: entry-side $\delta_j = 0.4951$ mm, exit-side $\delta_o = 0.0347$ mm, and tooth-top $\delta_d = 0.4731$ mm. Internal gear manufacturers can use such calculations to adjust parameters based on observed chip conditions, avoiding tool wear and enhancing efficiency for internal gears.
The following table summarizes key symbols and formulas for chip thickness calculation, aiding internal gear manufacturers in quick reference:
| Symbol | Description | Formula or Value |
|---|---|---|
| $f_c$ | Circular feed per revolution | 0.1–0.6 mm/r (conventional), 0.05–0.5 mm/r (CNC) |
| $f_r$ | Radial feed per stroke | 0.02–0.1 mm/r (conventional), 0.001–0.02 mm/r (CNC) |
| $Z_0$ | Cutter teeth number | Depends on design |
| $Z_2$ | Internal gear teeth number | Specified by application |
| $r_{j2}$ | Internal gear pitch radius | $m_t Z_2 / 2$ (mm) |
| $\delta_A$ | Exit-side chip thickness | $\frac{f_c (Z_2 – Z_0)}{r_{j2} Z_2} (r_b + r_{j0}) \cos(\alpha_A + \theta_Z – \varphi_0)$ |
| $\delta_B$ | Entry-side chip thickness | $\frac{f_c (Z_2 – Z_0)}{r_{j2} Z_2} \left( r_b \cos(\theta_Z + \varphi_B – \alpha_B) + r_{j0} \right)$ |
| $\delta_dE$ | Tooth-top chip thickness | $r_{j2} \theta \sin(\theta_E + \theta_Z)$ |
| $r$ | Cutting edge radius | $0.00075 \left[ 35 – 0.55 (\lambda + \gamma) \right]$ |
Another table outlines cutting parameter selection guidelines for internal gear shaping:
| Parameter | Recommendation | Notes |
|---|---|---|
| Number of Cuts | 1 for depth ≤5 mm; 2–3 roughing + 1–2 finishing for deeper teeth | Adjust based on machine power |
| Stroke Length | $L_c = B + 2.2 + 1.1 \pi m_t \sin \beta$ | Ensure full tooth engagement |
| Cutting Speed | $v_c = \frac{\varepsilon}{HB^{0.2}} \cos \beta$ | Reduce for harder materials |
| Circular Feed | $f_c = K_2 \frac{160}{HB^{0.2}} m_t^{0.5}$ | Use higher values for roughing |
| Radial Feed | $f_r = (0.2 \text{ to } 0.5) f_c / Z_2$ | Fine-tune based on chip flow |
| Machining Time | $t_m = k_c \left( \frac{\pi m_t Z_2}{n f_c} + \frac{H}{n f_r} \right)$ | Optimize for batch production |
| Tool Wear Limit | 0.1–0.3 mm | Regrind when exceeded |
In conclusion, internal gear manufacturers can enhance shaping efficiency by applying these chip thickness calculations and cutting parameter selections. Regular monitoring of chip conditions allows for proactive adjustments, minimizing tool wear and maximizing productivity. For internal gears, this approach ensures high-quality outputs and cost-effective manufacturing, supporting industries that rely on precision gearing. By integrating these methods, internal gear manufacturers can achieve longer tool life and improved performance in producing internal gears.