I have focused my investigation on the cutting force and cutting deformation that arise during the gear cutting of cycloidal gears used in RV reducers. The cycloidal gear is a core functional component of an RV reducer. Its porous structure, which includes circular holes and trapezoidal holes, causes deformation under the influence of cutting forces during gear cutting. This deformation prevents the gear cutting process from meeting the required precision and ultimately affects the performance of the RV reducer. Therefore, predicting the cutting force and cutting deformation during gear cutting, and analyzing the deformation laws of cycloidal gears during gear cutting, are particularly important for improving the accuracy of cycloidal gear manufacturing. I derived the cycloidal gear tooth profile equation and the cycloidal gear hob blade equation based on the enveloping method and the gear meshing principle. I established geometric models, used the finite element method to simulate the gear cutting process, studied the cutting force during gear cutting, and performed a finite element analysis of the gear cutting deformation caused by the cutting force. My main research work and findings are summarized in the following sections.
Geometric Modeling of the Cycloidal Gear and Hob for Gear Cutting
I established the cycloidal gear tooth profile by using the enveloping method. The gear pair curve is an arc-short epicycloid equidistant line. I fixed the cycloidal gear to a coordinate system \(O_1X_1Y_1Z_1\) and the pin gear to a coordinate system \(O_2X_2Y_2Z_2\). The base circle radius of the cycloidal gear is \(r_1\), the base circle radius of the pin gear is \(r_2\), the pin wheel radius is \(R_2\), the pin tooth radius is \(r_z\), and the center distance is \(e\). When the cycloidal gear base circle rolls inside the pin gear base circle, the moving point \(\mu\) generates a short epicycloid relative to the cycloidal gear center \(O_1\). Because the cycloidal gear and pin gear are always in mesh, I placed the moving point \(\mu\) at the pin tooth center. The pin gear tooth profile equation can be written as:
$$x_z = r_z \sin\gamma, \quad y_z = R_2 – r_z \cos\gamma$$
The angle relation is:
$$\tan\gamma = \frac{r_2 \sin\varphi_2}{R_2 – r_2 \cos\varphi_2}$$
Using homogeneous coordinate transformation and multi-system kinematic principles, I derived the cycloidal gear tooth profile matrix as:
$$\begin{bmatrix} x_1 \\ y_1 \\ z_1 \end{bmatrix} = \begin{bmatrix} \cos(\varphi_1 – \varphi_2) & \sin(\varphi_1 – \varphi_2) & 0 \\ -\sin(\varphi_1 – \varphi_2) & \cos(\varphi_1 – \varphi_2) & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} r_z \sin\gamma + e \sin\varphi_1 \\ R_2 – r_z \cos\gamma – e \cos\varphi_1 \\ 0 \end{bmatrix}$$
The relation between the rotation angles and the base circle radii is:
$$\frac{\varphi_1}{\varphi_2} = \frac{r_2 – e}{r_1} = \frac{r_2}{r_1}$$
Combining these equations and simplifying, I obtained the cycloidal gear tooth profile equation parameterized by \(\varphi_1\):
$$X_1 = R_2 \cos(\varphi_1 – \varphi_2) – e \cos\varphi_1 – r_z \cos(\varphi_1 – \varphi_2 – \gamma) + e \sin\varphi_1 \sin(\varphi_1 – \varphi_2)$$
$$Y_1 = R_2 \sin(\varphi_1 – \varphi_2) – e \sin\varphi_1 – r_z \sin(\varphi_1 – \varphi_2 – \gamma) – e \sin\varphi_1 \cos(\varphi_1 – \varphi_2)$$
I used the basic parameters listed in Table 1 to generate the cycloidal gear tooth profile curve. A point \(P\) on the curve is denoted as \(P(X_1,Y_1)\). I then built the complete three-dimensional cycloidal gear model according to the RV reducer structural design, which includes eight circular holes and trapezoidal holes.
| Parameter | Value |
|---|---|
| Pin wheel radius \(R_2\) (mm) | 82 |
| Center distance \(e\) (mm) | 1.5 |
| Cycloidal gear base circle radius \(r_1\) (mm) | 38.5 |
| Pin gear base circle radius \(r_2\) (mm) | 40 |
| Pin tooth radius \(r_z\) (mm) | 7 |
| Center distance rotation angle \(\varphi_1\) (°) | 10 to 360 |
Cycloidal Gear Hob Model for Gear Cutting
Because gear cutting with a hob is similar to worm and worm gear meshing, the hob tooth profile is the conjugate curve of the cycloidal gear tooth profile. I established a rack coordinate system \(O_CX_CY_C\) with its origin at the node \(P\), and a cycloidal gear coordinate system \(O_1X_1Y_1\). After a relative rotation angle \(\eta\), the rack tooth profile equation transformed by homogeneous coordinates is:
$$X_C’ = X_1 \cos\eta – Y_1 \sin\eta + r_1 \cos\eta – r_1 \eta \sin\eta$$
$$Y_C’ = X_1 \sin\eta + Y_1 \cos\eta + r_1 \sin\eta + r_1 \eta \cos\eta$$
The conjugation condition for the hob blade normal profile is:
$$\eta = \frac{\pi}{2} – \arccos\left(\frac{X_1 \cos\gamma + Y_1 \sin\gamma}{r_1}\right)$$
I then calculated the hob blade normal tooth profile as a transcendental curve with a period of \(2\pi X_C’\). The curve point \(P_C(X_C’,Y_C’)\) represents the hob blade normal profile. The hob design parameters include blade tooth height, normal pitch, module, pitch circle diameter, lead angle, and axial pitch. These are summarized in Table 2.
| Parameter type | Symbol | Relation |
|---|---|---|
| Cycloidal gear tooth number | \(Z_1\) | Primary parameter |
| Short amplitude coefficient | \(K_1\) | Primary parameter |
| Pin wheel radius | \(r_z\) | Primary parameter |
| Eccentricity | \(e\) | Primary parameter |
| Cycloidal gear width | \(B\) | Primary parameter |
| Pin gear tooth number | \(Z_2\) | \(Z_2 = Z_1 + 1\) |
| Cycloidal gear pitch radius | \(r_1\) | \(r_1 = e Z_1\) |
| Pin gear pitch radius | \(r_2\) | \(r_2 = e (Z_1 + 1)\) |
| Pin tooth distribution radius | \(R_2\) | \(R_2 = e (Z_1 + 1) / K_1\) |
| Cycloidal gear root radius | \(r_f\) | \(r_f = e (Z_1 + 1) / K_1 – e – r_z\) |
| Cycloidal gear tip radius | \(r_a\) | \(r_a = e (Z_1 + 1) / K_1 + e – r_z\) |
| Cycloidal gear tooth height | \(h_0\) | \(h_0 = 2e\) |
| Hob outer diameter | \(d\) | Primary parameter |
| Hob number of starts | \(Z_0\) | Primary parameter |
| Hob total length | \(L\) | Secondary parameter |
| Hob bore diameter | \(D\) | Secondary parameter |
| Hob flute number | \(KZ\) | Secondary parameter |
| Hob blade height | \(h_0\) | \(h_0 = 2(r_a – r_f) = 2e\) |
| Normal pitch | \(p_n\) | \(p_n = 2\pi r_a / Z_1\) |
| Module | \(m\) | \(m = p_n / \pi\) |
| Pitch circle diameter | \(d_0\) | \(d_0 = d – 2h_0 = d – 4e\) |
| Pitch circle lead angle | \(\gamma_z\) | \(\sin\gamma_z = m Z_0 / d_0\) |
| Axial pitch | \(p_x\) | \(p_x = p_n / \cos\gamma_x\) |
I selected a module of approximately 2.05 and built the three-dimensional hob model. The hob structure includes total length \(L\), outer diameter \(d\), bore diameter \(D\), shoulder width \(a\), flute number \(KZ\), and number of starts \(Z_0\). The design constraints are:
$$Z_1 / KZ \text{ should be an integer}, \quad KZ \leq 4$$
For finishing gear cutting, I chose a single-start hob. Table 3 lists the basic dimensions I used for the hob.
| Hob module | Hob length (mm) | Hob outer diameter (mm) | Bore diameter (mm) | Shoulder width (mm) | Flute number |
|---|---|---|---|---|---|
| 1 | 60 | 55 | 27 | 5 | 12 |
| 2 | 90 | 65 | 32 | 5 | 10 |
| 4 | 5 | 100 | 75 | 32 | 12 |
| 6 | 120 | 85 | 32 | 7 | 9 |
Gear Cutting Kinematics and Undeformed Chip Geometry
Gear cutting with a hob is a typical non-free interrupted cutting process. It has multi-edge periodic cutting and variable chip thickness. The hob blades distributed along the helical line perform strict generating motion with the workpiece and axial feed motion. I analyzed the relative motion and position relationship between the workpiece and the hob during gear cutting. I fixed a coordinate system \(O_1x_1y_1z_1\) to the gear blank, with its origin at the upper end face center and the \(z_1\) axis coinciding with the blank axis. The coordinate system \(O_2x_2y_2z_2\) is fixed in space, with its \(z_2\) axis coinciding with the workpiece table rotation center. The coordinate system \(O_3x_3y_3z_3\) is fixed to the column, with the \(z_3\) axis parallel to the blank rotation axis and the \(y_3\) axis parallel to the hob radial feed direction. The coordinate system \(O_4x_4y_4z_4\) is fixed to the axial feed ram, with axes parallel to \(O_3x_3y_3z_3\). The axial position of the hob relative to the blank is determined by \(a_z\) and \(h_0\), and the axial feed speed is \(v_a\). The coordinate system \(O_5x_5y_5z_5\) is fixed to the tool holder rotary table, with \(x_z\) representing the hob tangential feed position and \(v_t\) the tangential feed speed. The coordinate system \(O_6x_6y_6z_6\) is fixed to the hob, with its origin at the hob geometric center, the \(x_6\) axis coinciding with the hob rotation axis, and \(w_1\) the hob rotation speed. When the hob rotation angle \(\phi_1\) is zero, it coincides with \(O_5x_5y_5z_5\).
I established the homogeneous coordinate transformation matrices between these coordinate systems. The transformation from the hob coordinate system to the workpiece coordinate system is:
$$M_{61} = M_{65} M_{54} M_{43} M_{32} M_{21}$$
where each matrix represents a specific motion:
$$M_{65} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & \cos\phi_1 & \sin\phi_1 & 0 \\ 0 & -\sin\phi_1 & \cos\phi_1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}$$
$$M_{54} = \begin{bmatrix} \cos\gamma_{an} & 0 & \sin\gamma_{an} & x_z \cos\gamma_{an} \\ 0 & 1 & 0 & 0 \\ -\sin\gamma_{an} & 0 & \cos\gamma_{an} & -x_z \sin\gamma_{an} \\ 0 & 0 & 0 & 1 \end{bmatrix}$$
$$M_{43} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & a_z \\ 0 & 0 & 0 & 1 \end{bmatrix}$$
$$M_{32} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & -h_0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}$$
$$M_{21} = \begin{bmatrix} \cos\phi_2 & \sin\phi_2 & 0 & 0 \\ -\sin\phi_2 & \cos\phi_2 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}$$
I then derived the relative velocity vector between the hob and the workpiece at a contact point \(P\) in the fixed coordinate system:
$$v_{phw}^{(2)} = v_{ph}^{(2)} – v_{pw}^{(2)}$$
Expanding this vector gives the components along the three axes. I used these to define the spatial surface cluster of the hob blade edges. By numbering the hob teeth along the helical line, I expressed the position vector of any point on the \(i\)-th blade in the workpiece coordinate system as:
$$r_{pw}^{(i)} = M_{61} M_{hi} r_i^{(hi)}$$
where \(M_{hi}\) is the transformation matrix from the \(i\)-th blade coordinate system to the hob coordinate system. The spatial trajectory surface cluster is:
$$G_i(\phi) = R_x(\phi) T_z(\phi) R_z(\phi) r_i^{(hi)}$$
I simplified the motion matrices for my specific gear cutting case, which does not require tangential feed or radial feed, and the hob installation angle is zero. The hob rotation matrix is:
$$R_x(\phi) = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & \cos\phi & \sin\phi & 0 \\ 0 & -\sin\phi & \cos\phi & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}$$
The axial feed matrix is:
$$T_z(\phi) = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & \xi(\phi) \\ 0 & 0 & 0 & 1 \end{bmatrix}, \quad \xi(\phi) = \frac{n f \phi}{2\pi z}$$
The workpiece rotation matrix is:
$$R_z(\phi) = \begin{bmatrix} \cos[\psi(\phi)] & \sin[\psi(\phi)] & 0 & 0 \\ -\sin[\psi(\phi)] & \cos[\psi(\phi)] & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}, \quad \psi(\phi) = \frac{z_0}{z} \phi$$
I divided the axial feed process into three stages: the cut-in stage, the full cutting stage, and the cut-out stage. The cut-in and cut-out stages are relatively short compared to the full cutting stage, which occupies most of the gear cutting process. I established the gear slot model for each stage by Boolean operations between the hob blade spatial trajectory surface cluster and the gear blank. For the full cutting stage, I analyzed the undeformed chip geometry in one cutting cycle of a single tooth slot. The chip thickness is calculated as:
$$h_i = \frac{1}{|\frac{\partial G_i}{\partial i} \times \frac{\partial G_i}{\partial \phi}|} \left| \left[ \frac{\partial G_{i+1}}{\partial (i+1)} \times \frac{\partial G_{i+1}}{\partial \phi} \right] – \left[ \frac{\partial G_i}{\partial i} \times \frac{\partial G_i}{\partial \phi} \right] \right|$$
The chip volume is:
$$V_i = \int \int h_i \left| \frac{\partial G_i}{\partial i} \times \frac{\partial G_i}{\partial \phi} \right| di d\phi$$
The average chip thickness is:
$$h_{avg} = \frac{V_i}{A_i}$$
where the projected area is:
$$A_i = \int \int \left| \frac{\partial G_i}{\partial i} \times \frac{\partial G_i}{\partial \phi} \right| di d\phi$$
I analyzed the undeformed chip morphology for one tooth slot in the full cutting stage. The chip length varied by about 10% within one cutting cycle, with a maximum of about 13 mm and a minimum of about 11.5 mm. The chip width was widest for blades 1, 2, and 8, and narrower for blades 3, 4, 5, 6, and 7. The maximum chip thickness for each blade is shown in Table 4.
| Blade number | Maximum chip thickness (mm) |
|---|---|
| 1 | 0.34 |
| 2 | 0.35 |
| 3 | 0.20 |
| 4 | 0.18 |
| 5 | 0.16 |
| 6 | 0.15 |
| 7 | 0.22 |
| 8 | 0.35 |
Finite Element Simulation of Gear Cutting Force
I simplified the full-featured gear cutting process into a single-tooth and single-slot model to reduce computational cost while preserving the essential cutting mechanics. I used the Johnson-Cook constitutive model for the workpiece material 25CrMo4:
$$\sigma = (A + B\varepsilon^n)\left(1 + C\ln\frac{\dot{\varepsilon}}{\dot{\varepsilon}_0}\right)(1 – T^{*m})$$
where \(T^* = (T – T_{room})/(T_{melt} – T_{room})\). The material parameters are listed in Table 5. The hob material was M35 with a Poisson’s ratio of 0.23, thermal conductivity of 30 W/(m K), specific heat of 15 J/(kg K), and density of \(8.14 \times 10^{-6}\) kg/mm\(^3\).
| Temperature (°C) | Young’s modulus (MPa) | Thermal expansion coefficient (10\(^{-5}\)/K) | Thermal conductivity (W/(m K)) | Specific heat (J/(kg K)) |
|---|---|---|---|---|
| 20 | 212000 | 1.19 | 41.7 | 6.61885 |
| 100 | 207000 | 1.25 | 43.4 | 3.8936 |
| 200 | 199000 | 1.31 | 41.4 | 4.18405 |
| 300 | 192000 | 1.36 | 39.1 | 4.4588 |
| 400 | 184000 | 1.41 | 36.7 | 4.79635 |
| 500 | 175000 | 1.49 | 34.1 | 5.31445 |
| 600 | 164000 | 1.49 | 34.1 | 6.1073 |
| 1500 | 69440 | 1.49 | 34.1 | 6.1073 |
The Johnson-Cook parameters for 25CrMo4 are given in Table 6. I used the C-L physical separation criterion for material fracture:
$$\int_0^{\bar{\varepsilon}_f} \frac{d\bar{\varepsilon}}{\sigma^*} = C_4$$
where \(\sigma^* = \sigma_1 / \sigma_{max}\) for \(\sigma_1 \geq 0\). I set the critical stress for node separation to 0.1 MPa. For the friction model, I adopted Zorev’s sticking-sliding friction law:
$$\tau_f = \begin{cases} k_{chip}, & \mu\sigma_n \geq k_{chip} \\ \mu\sigma_n, & \mu\sigma_n < k_{chip} \end{cases}$$
| Parameter | \(A\) | \(B\) | \(C\) | \(D_0\) | \(E\) | \(n\) | \(m\) | \(\dot{\varepsilon}_0\) | \(T_{melt}\) | \(T_{room}\) |
|---|---|---|---|---|---|---|---|---|---|---|
| Value | 1200 | 891 | 0.02 | 1 | 1 | 0.2 | 0.64 | 1 | 1800 | 293 |
I meshed the model with tetrahedral elements. The total node number was 75264, surface polyhedron number was 48000, element number was 67088, and mesh size ratio was 6. The minimum non-refined mesh size was 0.1 mm. I refined the cutting zone to 0.02 mm. I fixed the workpiece bottom in the axial direction and applied symmetric constraints on the cut faces. The cooling condition was oil-based with a heat convection coefficient of 2125 W/(m K).
I validated the finite element model against experimental data from gear cutting of a 25CrMo steel blank. The hob had an outer diameter of 75 mm, one start, 14 flutes, a helix angle of 4.917°, a pressure angle of 20°, a rake angle of 0°, and a module of 4 mm. The cutting parameters were a hob speed of 300 r/min and 900 r/min with an axial feed of 1.667 mm/r. The comparison between simulated and experimental cutting forces is shown in Table 7. The maximum error for axial cutting force was 9.28%, and for radial cutting force it was 10.5%. These errors are within a reasonable range, confirming the validity of my gear cutting simulation model.
| Experiment number | Hob speed (r/min) | Axial feed (mm/r) | Simulated axial force (N) | Experimental axial force (N) | Error (%) | Simulated radial force (N) | Experimental radial force (N) | Error (%) |
|---|---|---|---|---|---|---|---|---|
| 1 | 300 | 1.667 | 556.4 | 609.1 | 8.64 | 76.05 | 84.04 | 10.5 |
| 2 | 900 | 1.667 | 571.7 | 624.8 | 9.28 | 81.6 | 89.9 | 6.79 |
I then analyzed the cutting force variation with processing time. For the 3rd, 6th, and 8th blade positions in a single tooth slot, I found that the axial cutting force curves rise rapidly in the first 2 ms, remain stable from about 2 to 5 ms, and then decrease from 5 to 7 ms. The radial cutting force peaks earlier at about 1.6 ms and then decreases. The tangential cutting force shows a brief drop after initial contact due to tool yielding, then rises to a peak at 4 to 6 ms. The 8th blade produced the most stable cutting force curve with the highest peak, so I selected it for detailed analysis. The cutting force data for different machining parameters are shown in Table 8. I found that increasing the hob speed from 300 to 900 r/min increased the axial cutting force from 791 N to 834 N. Increasing the axial feed from 0.25 to 1.0 mm/r increased the axial cutting force from 803 N to 914 N, with a sharp rise after 0.75 mm/r. The radial cutting force increased from 324 N to 554 N, and the tangential cutting force increased from 50.9 N to 81.9 N over the same feed range. The influence of axial feed on cutting force is greater than that of hob speed.
| Number | Cutting stage | Hob speed (r/min) | Axial feed (mm/r) | Tangential force (N) | Radial force (N) | Axial force (N) |
|---|---|---|---|---|---|---|
| 1 | Cut-in | 600 | 0.5 | 15.3 | 46.13 | 212.7 |
| 2-1 | Full cutting 1 | 300 | 0.5 | 26 | 179.4 | 791.1 |
| 2-2 | Full cutting 1 | 450 | 0.5 | 34 | 294.1 | 799.4 |
| 2-3 | Full cutting 1 | 600 | 0.25 | 51 | 324.4 | 803.4 |
| 2-4 | Full cutting 1 | 600 | 0.5 | 52 | 358.1 | 811.1 |
| 2-5 | Full cutting 1 | 600 | 0.75 | 54.7 | 408 | 823.5 |
| 2-6 | Full cutting 1 | 600 | 1 | 82 | 553.5 | 914.4 |
| 2-7 | Full cutting 1 | 750 | 0.5 | 55 | 366.2 | 823.6 |
| 3 | Full cutting 2 | 600 | 0.5 | 79 | 420.5 | 897.7 |
| 4 | Cut-out | 600 | 0.5 | 5.3 | 13.4 | 81.3 |
Finite Element Analysis of Gear Cutting Deformation
I analyzed the deformation direction and location of the cycloidal gear during gear cutting. The gear has three circular holes and three trapezoidal holes. The trapezoidal hole region has the lowest stiffness, the circular hole region has intermediate stiffness, and the region between the trapezoidal and circular holes has the highest stiffness. I defined these as the weak rigidity zone, medium rigidity zone, and strong rigidity zone. The main deformation directions are radial and tangential because of the clamping method. Axial deformation does not affect the cutting parameters significantly, so I focused on radial and tangential deformation.
I established the finite element model for gear cutting deformation. I assumed that only the cutting zone undergoes plastic deformation, while the rest of the gear deforms elastically. I neglected cutting heat because the wet cutting process effectively removes heat. I also neglected machine tool and fixture deformation because the gear’s own weak rigidity dominates. I used a static structural simulation with tetrahedral elements, refining the mesh near the outer region. I applied boundary conditions according to the actual clamping: radial and tangential constraints on the center hole, and axial constraints on the bottom of the gear rim. I applied the cutting force as a concentrated force at the cutting position using a point-surface coupling. The cutting force data were taken from the simulation results in Table 8.
I selected four models from the three cutting stages: one from the cut-in stage, two from the full cutting stage, and one from the cut-out stage. For the weak rigidity zone, the maximum deformation in the same tooth slot at different cutting stages is shown in Table 9. The deformation increased from 8.19 \(\mu\)m in the cut-in stage to 10.47 \(\mu\)m at the end of the full cutting stage, then decreased to 9.00 \(\mu\)m in the cut-out stage. The deformation follows a downward-opening parabolic trend. For different tooth slots in the same full cutting stage, the maximum deformation ranged from 10.48 to 11.29 \(\mu\)m, with the highest at slot 3. The deformation is always concentrated in the weak rigidity zone.
| Cutting stage | Maximum deformation (\(\mu\)m) |
|---|---|
| Cut-in | 8.19 |
| Full cutting 1 | 9.71 |
| Full cutting 2 | 10.47 |
| Cut-out | 9.00 |
| Tooth slot number | Maximum deformation (\(\mu\)m) |
|---|---|
| 1 | 10.79 |
| 2 | 11.22 |
| 3 | 11.29 |
| 4 | 11.05 |
| 5 | 10.87 |
| 6 | 10.48 |
For the medium rigidity zone, the maximum deformation in the same tooth slot at different cutting stages is shown in Table 10. The maximum deformation was 10.20 \(\mu\)m, but the majority of the medium rigidity zone deformed between 4 and 7 \(\mu\)m. The maximum deformation region remained in the weak rigidity zone. For different tooth slots in the same full cutting stage, the maximum deformation ranged from 10.15 to 10.66 \(\mu\)m, again concentrated in the weak rigidity zone. The deformation trend for the medium rigidity zone is similar to that of the weak rigidity zone but with smaller magnitudes.
| Cutting stage | Maximum deformation (\(\mu\)m) |
|---|---|
| Cut-in | 8.55 |
| Full cutting 1 | 9.82 |
| Full cutting 2 | 10.20 |
| Cut-out | 9.10 |
| Tooth slot number | Maximum deformation (\(\mu\)m) |
|---|---|
| 1 | 10.15 |
| 2 | 10.36 |
| 3 | 10.47 |
| 4 | 10.58 |
| 5 | 10.66 |
| 6 | 10.63 |
I analyzed the influence of gear cutting parameters on deformation. For the weak rigidity zone, increasing the hob speed from 300 to 750 r/min increased the maximum deformation from 10.69 to 11.29 \(\mu\)m. Increasing the axial feed from 0.25 to 1.0 mm/r increased the maximum deformation from 11.03 to 12.74 \(\mu\)m. For the medium rigidity zone, increasing the hob speed from 300 to 750 r/min increased the maximum deformation from 10.11 to 10.66 \(\mu\)m, while increasing the axial feed from 0.25 to 1.0 mm/r increased the maximum deformation from 10.33 to 11.44 \(\mu\)m. In all cases, the axial feed has a greater influence on deformation than the hob speed. This is because the axial feed directly determines the chip thickness and volume, which in turn affects the cutting force.
Radial and Circumferential Deformation Laws
I analyzed the radial and circumferential deformation of the cycloidal gear tooth profile after gear cutting. I selected a path on the upper surface of the full cutting stage model along the edge of an already machined tooth slot. I placed 15 points evenly along this path. The radial deformation of a point \(P\) is calculated as:
$$\Delta R = \sqrt{X’^2 + Y’^2} – \sqrt{X^2 + Y^2}$$
where \((X,Y)\) are the original coordinates and \((X’,Y’)\) are the deformed coordinates. The circumferential deformation is:
$$\Delta l = R \Delta \theta$$
where \(\Delta \theta\) is the angular displacement. The results for different hob speeds and axial feeds are summarized in Table 11 and Table 12.
| Hob speed (r/min) | Maximum radial deformation (\(\mu\)m) | Maximum circumferential deformation (\(\mu\)m) |
|---|---|---|
| 300 | 5.12 | 1.45 |
| 450 | 6.35 | 2.12 |
| 600 | 7.01 | 2.98 |
| 750 | 7.56 | 3.71 |
| Axial feed (mm/r) | Maximum radial deformation (\(\mu\)m) | Maximum circumferential deformation (\(\mu\)m) |
|---|---|---|
| 0.25 | 6.02 | 2.85 |
| 0.5 | 7.01 | 3.98 |
| 0.75 | 7.52 | 4.76 |
| 1.0 | 8.19 | 5.67 |
The radial deformation increases with both hob speed and axial feed. The maximum radial deformation was 8.19 \(\mu\)m at a hob speed of 600 r/min and an axial feed of 1.0 mm/r. The circumferential deformation also increased, reaching a maximum of 5.67 \(\mu\)m under the same conditions. The deformation trend along the tooth profile path showed a slight decrease from point 1 to point 15, which corresponds to the stiffness variation within the tooth slot. The influence of axial feed on both radial and circumferential deformation is greater than that of hob speed.
Conclusions
I derived the cycloidal gear tooth profile equation and the cycloidal gear hob blade equation based on the enveloping method and the gear meshing principle. I established a gear cutting kinematic model and a spatial surface cluster equation for the hob. I constructed geometric models for the gear slot and undeformed chip at each cutting stage. The undeformed chip length varied by about 10% within one cutting cycle, with a maximum of 13 mm and a minimum of 11.5 mm. The maximum chip thickness was 0.35 mm for blades 1, 2, and 8, and 0.15 mm for blade 6. I established a finite element model for gear cutting force analysis and validated it against experimental data. The axial cutting force error was 9.28% and the radial cutting force error was 10.5%. I found that higher hob speeds and larger axial feeds increase the cutting force, with the axial feed having a greater influence. The threshold for a sharp increase in cutting force was an axial feed of 0.75 mm/r. I analyzed the deformation direction and location and divided the gear into weak, medium, and strong rigidity zones. The maximum deformation occurred in the weak rigidity zone during the full cutting stage. The maximum deformation in the weak rigidity zone was 10.47 \(\mu\)m in the same tooth slot and 11.29 \(\mu\)m for different tooth slots. In the medium rigidity zone, the maximum deformation was 10.20 \(\mu\)m in the same tooth slot and 10.66 \(\mu\)m for different tooth slots, but the maximum deformation region remained the weak rigidity zone. The axial feed had a greater influence on deformation than the hob speed. The maximum radial deformation was 8.19 \(\mu\)m and the maximum circumferential deformation was 5.67 \(\mu\)m under the tested conditions. These findings provide a theoretical basis for optimizing gear cutting parameters and improving the precision of cycloidal gears in RV reducers.
Future Work
In my gear cutting force modeling, I did not consider the effect of tool wear on cutting force. Future work could investigate the influence of tool wear on gear cutting forces to further improve the prediction accuracy. In my deformation prediction, I only considered the effect of cutting force on gear deformation. Future studies could incorporate more variables such as machine tool fixtures and cutting temperature to provide a more comprehensive understanding of gear cutting deformation.