As an increasing demand for high-efficiency and high-precision gear manufacturing, the straight bevel gear remains one of the most widely used machine elements in power transmission systems. In this chapter, I present my research on developing a complete manufacturing solution for straight bevel gears using the face-hobbing method based on a general 5-axis machining center with an open CNC system. The research covers mathematical modeling, CNC system module development, on-machine measurement, and simulation validation. Throughout the thesis, the key phrase straight bevel gear is emphasized in every technical aspect.
1. Introduction
Straight bevel gears are commonly used to transmit motion and power between intersecting axes, especially in automotive, agricultural machinery, construction equipment, and marine applications. Traditional manufacturing methods for straight bevel gears include planing, circular broaching, and double-cutter milling. These methods have limitations in efficiency, flexibility, and surface quality. In the last decade, a new face-hobbing process, also known as the Hypoflex process, was announced by a German company. This process utilizes the mathematical property that when the rolling circle radius is exactly half of the base circle radius, the hypocycloid degenerates to a straight line. Although the method was originally implemented on a dedicated six-axis CNC bevel gear cutting machine, the cost of such machines and their closed CNC systems prevent wide application. Therefore, my research focuses on replacing the dedicated machine with a general 5-axis machining center and an open-architecture CNC system, while maintaining the advantages of continuous indexing and double-flank cutting.
In the following sections, I present the derivation of the mathematical model for face-hobbing straight bevel gears, the development of a parametric NC programming system module, the development of an electronic gearbox function, an on-machine measurement module, and the verification through a machining example and simulation.
2. Mathematical Model for Face-Hobbing Straight Bevel Gears
2.1 Principle of Face-Hobbing for Straight Bevel Gears
The face-hobbing process for straight bevel gears is based on the concept of an imaginary generating gear. In this concept, the tool cutter head represents a rolling circle, and its motion relative to the imaginary generating gear simulates the rolling of a circle on a base circle. For the extended epicycloid bevel gears, the trajectory of the cutter edge is an extended epicycloid. However, to obtain a straight tooth line for the straight bevel gear, the hypocycloid straight-line property is employed. When the radius of the rolling circle is exactly one half of the base circle radius, the hypocycloid traced by a point on the rolling circle becomes a straight line passing through the center of the base circle.
In the face-hobbing process for straight bevel gears, the cutter head is equipped with inner and outer blades. The cutter head corresponds to the rolling circle with radius \( R_t \), and the imaginary generating gear corresponds to the base circle with radius \( R_b \). To satisfy the straight-line condition, we require:
\[
R_t = \frac{1}{2} R_b
\]
The cutter head is mounted at a radial distance \( R_{mt} \) from the center of the imaginary generating gear. Because \( R_{mt} = R_b – R_t \), and with \( R_t = 0.5 R_b \), we obtain:
\[
R_{mt} = R_b – R_t = R_b – 0.5 R_b = 0.5 R_b = R_t
\]
Thus, the cutter radial distance must be equal to the cutter radius. This is a critical machine setting that distinguishes the face-hobbing of straight bevel gears from that of extended epicycloid bevel gears.
During cutting, the cutter head rotates continuously, and the workpiece rotates with a fixed gear ratio relative to the cutter head. This continuous indexing process eliminates the need for intermittent indexing, and the double-flank cutting nature greatly improves productivity.
2.2 Coordinate System for the Cradle-Type Gear Cutting Machine
To derive the mathematical model, I first considered a mechanical cradle-type bevel gear cutting machine. The machine structure is represented by the following coordinate systems:
- \( C_m = \{ O_m; X_m, Y_m, Z_m \} \) is the machine coordinate system, where \( Z_m \) coincides with the axis of the cradle and the imaginary generating gear.
- \( C_t = \{ O_t; X_t, Y_t, Z_t \} \) is the cutter head coordinate system, where \( Z_t \) coincides with the cutter head axis.
- \( C_g = \{ O_g; X_g, Y_g, Z_g \} \) is the workpiece gear coordinate system, where \( Z_g \) coincides with the gear axis.
The transformation from the cutter head to the workpiece gear is performed through the following sequence:
\[
C_t \rightarrow C_1 \rightarrow C_m \rightarrow C_2 \rightarrow C_g
\]
where \( C_1 \) and \( C_2 \) are auxiliary coordinate systems. The corresponding transformation matrices are given below.
The rotation matrix of the cutter head about its own axis is:
\[
M_1(\beta_t) =
\begin{bmatrix}
\cos\beta_t & \sin\beta_t & 0 & 0 \\
-\sin\beta_t & \cos\beta_t & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{bmatrix}
\]
The translation matrix from the cutter head center to the cradle center is:
\[
M_{mt}(R_{mt}) =
\begin{bmatrix}
1 & 0 & 0 & R_{mt} \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{bmatrix}
\]
The cradle rotation for the hypocycloid motion is described by:
\[
M_m(\xi_{m1}) =
\begin{bmatrix}
\cos(\xi_0 – \xi_{m1}) & \sin(\xi_0 – \xi_{m1}) & 0 & 0 \\
-\sin(\xi_0 – \xi_{m1}) & \cos(\xi_0 – \xi_{m1}) & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{bmatrix}
\]
The cradle rotation for the generating motion is:
\[
M_m(\xi_{m2}) =
\begin{bmatrix}
\cos\xi_{m2} & \sin\xi_{m2} & 0 & 0 \\
-\sin\xi_{m2} & \cos\xi_{m2} & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{bmatrix}
\]
The transformation from the imaginary generating gear to the workpiece gear involves a rotation about the \( Y_m \)-axis by the mechanical root angle \( \delta_1 \):
\[
M_2(\delta_1) =
\begin{bmatrix}
\cos\delta_1 & 0 & -\sin\delta_1 & 0 \\
0 & 1 & 0 & 0 \\
\sin\delta_1 & 0 & \cos\delta_1 & 0 \\
0 & 0 & 0 & 1
\end{bmatrix}
\]
Finally, the workpiece rotation for indexing is:
\[
M_g(\theta_g) =
\begin{bmatrix}
1 & 0 & 0 & 0 \\
0 & \cos\theta_g & \sin\theta_g & 0 \\
0 & -\sin\theta_g & \cos\theta_g & 0 \\
0 & 0 & 0 & 1
\end{bmatrix}
\]
The complete position vector of the tooth surface point \( r_g \) can be expressed as:
\[
r_g = M_g(\theta_g) \, M_2(\delta_1) \, M_m(\xi_{m2}) \, M_m(\xi_{m1}) \, M_{mt}(R_{mt}) \, M_1(\beta_t) \, r_t
\]
where \( r_t \) is the position vector of the cutting edge point in the cutter head coordinate system. By applying the above matrix transformation, I obtained the tooth surface equation of the face-hobbed straight bevel gear on the cradle-type machine. Using MATLAB to plot the generated surface, I observed that the left and right tooth surfaces are two planes, which confirms the straight-line tooth profile.
2.3 Transformation to the General 5-Axis Machining Center
Since a cradle-type machine is not suitable for CNC machining, the mathematical model must be transformed to a general 5-axis machining center with three linear axes (X, Y, Z) and two rotary axes (A, C). I established the machining coordinate system as shown in the following figure. The machine coordinate system is \( C_r = \{ O_r; X_r, Y_r, Z_r \} \), where \( Z_r \) is coincident with the A-axis rotation center of the tilt table. The cutter head coordinate system is \( C_p = \{ O_p; X_p, Y_p, Z_p \} \). The workpiece gear coordinate system is \( C_q = \{ O_q; X_q, Y_q, Z_q \} \).
In the general 5-axis machining center, the cutter head is mounted on the spindle, and the workpiece is clamped on the C-axis rotary table. The transformation sequence is:
\[
C_p \rightarrow C_3 \rightarrow C_4 \rightarrow C_5 \rightarrow C_r \rightarrow C_q \rightarrow C_6
\]
The rotation matrix of the cutter head about its own axis is \( M_p(\Delta W) \):
\[
M_p(\Delta W) =
\begin{bmatrix}
\cos\Delta W & \sin\Delta W & 0 & 0 \\
-\sin\Delta W & \cos\Delta W & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{bmatrix}
\]
The translation along the X and Z axes is represented by:
\[
M_{34} =
\begin{bmatrix}
1 & 0 & 0 & X_{pq} \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & Z_{pq} \\
0 & 0 & 0 & 1
\end{bmatrix}
\]
The translation along the Y axis is:
\[
M_{4r}(Y_{pq}) =
\begin{bmatrix}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & Y_{pq} \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{bmatrix}
\]
The rotation about the A axis is:
\[
M_{5r}(A) =
\begin{bmatrix}
1 & 0 & 0 & 0 \\
0 & \cos A & \sin A & 0 \\
0 & -\sin A & \cos A & 0 \\
0 & 0 & 0 & 1
\end{bmatrix}
\]
The translation along the Z axis from the machine origin to the gear pitch cone point is:
\[
M_{65}(Z_{rq}) =
\begin{bmatrix}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & Z_{rq} \\
0 & 0 & 0 & 1
\end{bmatrix}
\]
The rotation of the workpiece about the C axis is:
\[
M_{6q}(\Delta C) =
\begin{bmatrix}
\cos\Delta C & \sin\Delta C & 0 & 0 \\
-\sin\Delta C & \cos\Delta C & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{bmatrix}
\]
Therefore, the tooth surface point in the general 5-axis machining center model is:
\[
r_{g1} = M_{6q}(\Delta C) \, M_{65}(Z_{rq}) \, M_{5r}(A) \, M_{4r}(Y_{pq}) \, M_{34} \, M_p(\Delta W) \, r_t
\]
Since the same cutter head and the same workpiece gear must produce the same tooth surface, the two models must be equal. By equating the matrices \( a_{ij} \) from the cradle-type model and \( b_{ij} \) from the machining-center model, I solved for the five-axis motion equations as follows.
The X-axis motion is:
\[
X_{pq} = R_{mt} \sin(\xi_0 + \xi_m)
\]
The Y-axis motion is:
\[
Y_{pq} = Z_{rq} \sin\delta_1 + R_{mt} \cos(\xi_0 + \xi_m)
\]
The Z-axis motion is:
\[
Z_{pq} = -Z_{rq} \cos\delta_1
\]
The A-axis position is:
\[
A = \delta_1
\]
The C-axis rotation angle is derived from the matrix equivalence:
\[
\Delta C = -\tan^{-1}\left( \frac{a_{13}}{a_{23}} \right)
\]
After simplification, the expression becomes:
\[
\Delta C = -\tan^{-1}\left( \frac{\cos(\xi_0 + \xi_m)}{\sin\delta_1 \sin(\xi_0 + \xi_m)} \right)
\]
The spindle rotation angle is:
\[
\Delta W = \tan^{-1}\left( \frac{\cos(\xi_0 + \xi_m)}{\sin\delta_1 \sin(\xi_0 + \xi_m)} \right)
\]
However, for synchronous control, the spindle is usually set as the master axis, and the C axis is the slave axis. The final C-axis motion is calculated by combining the indexing and generating motions:
\[
\Delta C’ = \Delta C + \frac{\Delta W}{i_{tm} \times i_{gm}}
\]
where \( i_{tm} \) is the ratio between the number of cutter blade groups and the number of teeth of the imaginary generating gear. To satisfy the hypocycloid straight-line condition, this ratio is set to 0.5:
\[
i_{tm} = \frac{Z_t}{Z_m} = 0.5
\]
and \( i_{gm} \) is the gear ratio between the workpiece and the imaginary generating gear:
\[
i_{gm} = \frac{Z_g}{Z_m}
\]
Thus, the final C-axis command is:
\[
\Delta C’ = \Delta C + \frac{\Delta W}{0.5 \cdot (Z_g / Z_m)}
\]
In practical CNC programming, the spindle rotation is set to a constant speed, and the C-axis follows the spindle through the electronic gearbox function with the required ratio, which ensures that the cutter head and the workpiece remain synchronized at every moment.
3. Development of the CNC Machining System Module
3.1 Open CNC System Platform
The CNC system used in my research is an open-architecture system based on the PC-NC structure. The hardware consists of an industrial PC and a motion control card. This allows users to develop custom modules using standard programming languages such as VC++. The open structure provides excellent flexibility for extending functions for specific manufacturing processes.
3.2 Program Reference Point Setting
In the general 5-axis machining center, the workpiece coordinate system origin is set at the pitch cone point of the straight bevel gear. The machine coordinate system origin is at the center of the tilt table. The Z-axis offset between the machine origin and the workpiece origin is determined by the sum of three distances: the distance from the machine center to the table surface \( Z_{r1} \), the fixture height \( Z_{r2} \), and the distance from the mounting surface to the gear pitch cone point \( Z_{r3} \):
\[
Z_{rq} = Z_{r1} + Z_{r2} + Z_{r3}
\]
This offset is entered into the CNC system through the G54 command.
3.3 Machining Path Planning
The machining process is planned as follows. First, all axes return to the machine reference point. Then, the tool is set to the reference point using a probe. Before cutting, the electronic gearbox function is enabled so that the spindle and the C-axis rotate synchronously. A forming cut is first performed to generate a preliminary tooth slot, followed by a generating cut for the final tooth profile. The forming cut removes about 20% of the tooth height, and the generating cut removes the remaining 80%. The tool approaches in the Z direction for the forming cut, while X and Y feed motions are used for the generating cut.
3.4 Machining Parameter Determination
The spindle speed is a critical parameter. Since the cutter head is much heavier than a normal end mill, the PID parameters of the spindle servo loop must be re-tuned. After tuning, I selected a spindle speed range of 80–120 r/min to ensure good synchronization and small following errors. The feed rate \( F \) is calculated from the incremental displacement per cutter group and the spindle speed:
\[
F = h \cdot Z_t \cdot S
\]
where \( h \) is the incremental displacement per cutting cycle, \( Z_t \) is the number of cutter blade groups, and \( S \) is the spindle speed. For the example presented later, the feed rate is about 9.3 mm/min for the small gear and 10.8 mm/min for the large gear.
3.5 NC Program Generation Module
I developed the NC programming module using VC++ and MATLAB. The mathematical model derived in Chapter 2 was implemented in MATLAB, and then converted to C++ code using MATCOM. The module has a dialog-based interface where the user enters the gear blank parameters, cutter parameters, and machining parameters. The module automatically calculates the discrete positions of X, Y, Z, A, and C axes based on the incremental cradle angle. The output is a standard G-code file. The interface screenshot is shown in the figure below.
The main classes in the module are:
| Class | Function |
|---|---|
| CBgearStock | Stores gear blank parameters |
| CBgearAdjust | Stores machine adjustment parameters |
| CBgearjg | Stores cutting parameters |
| CBgearDlg | Main dialog for input and control |
| CNum | Public data storage and exchange |
| CBgearGout | Generates G-code output |
3.6 Electronic Gearbox Function
The face-hobbing process requires that the cutter head axis (spindle) and the workpiece axis (C-axis) rotate with a constant speed ratio at all times. I implemented a master-slave electronic gearbox based on the PMAC motion controller. The gear ratio is set by two variables \( Ix07 \) and \( Ix08 \) in PMAC. The relationship is:
\[
Ix07 \cdot \Delta MPn = Ix08 \cdot \Delta CPn
\]
where \( \Delta MPn \) is the master position increment and \( \Delta CPn \) is the slave position increment. To match the encoder resolutions, the ratio is adjusted as:
\[
Ix07 \cdot 4096 \cdot \Delta MPn = Ix08 \cdot 65536 \cdot \Delta CPn
\]
Thus, for a desired synchronization ratio \( i_{78} \), the variables must satisfy:
\[
\frac{Ix08}{Ix07} = \frac{65536 \cdot i_{78}}{4096}
\]
I also developed a G-code command (G95) to enable the electronic gearbox. The code reads the number of cutter blade groups and the number of gear teeth from the NC program, calculates the required ratio, and activates the synchronous mode.
4. On-Machine Measurement System Module
4.1 Measurement of Pitch Deviation
Pitch deviation is one of the key quality indicators for a straight bevel gear. Single pitch deviation \( f_p \) is defined as the difference between the actual and nominal pitch on the pitch circle. The cumulative pitch deviation \( F_p \) is the algebraic sum of \( f_p \) over a number of pitches. For a measurement radius \( r_p \), the single pitch deviation is:
\[
f_{pi} = r_p \cdot \Delta\delta_i – \pi m
\]
where \( \Delta\delta_i \) is the angular pitch error measured in radians, and \( m \) is the module. The cumulative deviation after \( n \) pitches is:
\[
F_{pn} = \sum_{i=1}^{n} f_{pi}
\]
4.2 Measurement of Tooth Surface Error
The tooth surface error is evaluated by comparing the actual tooth surface position with the theoretical surface position along the normal direction. The theoretical tooth surface positions and normal vectors are obtained from the mathematical model derived in Chapter 2. I divided the tooth surface into a grid of 5 by 7 points. The grid points are projected onto the axial section for convenient calculation. Using the projection relationship:
\[
U = \sqrt{X^2 + Y^2}, \quad Z = Z
\]
The four corner points of the grid are determined from the gear boundary conditions, and the other grid nodes are obtained by interpolation. Then, by combining the grid node coordinates with the tooth surface equation, I solved the nonlinear equations using MATLAB’s fsolve function. The resulting theoretical measurement point coordinates are listed in Table below (partial).
| Point No. | X | Y | Z |
|---|---|---|---|
| 1 | 30.16 | 394.5196 | 18.70998 |
| 2 | 30.52 | 394.3450 | 18.53542 |
| 3 | 30.88 | 394.1686 | 18.35899 |
| 4 | 31.24 | 393.9903 | 18.18070 |
| 5 | 31.60 | 393.8101 | 18.00055 |
| 6 | 31.96 | 393.6281 | 17.81855 |
| 7 | 32.32 | 393.4443 | 17.63470 |
| 8 | 32.68 | 393.2586 | 17.44903 |
| 9 | 33.04 | 393.0711 | 17.26152 |
| 10 | 33.40 | 392.8818 | 17.07220 |
The normal error at each point is computed as:
\[
e_n = (R_s – R_l) \cdot n_l
\]
where \( R_s \) is the actual measured point position, \( R_l \) is the theoretical point position, and \( n_l \) is the theoretical normal vector.
4.3 Software Implementation of the Measurement Module
I developed the on-machine measurement module within the same open CNC system. The module uses a Renishaw OMP400 touch-trigger probe. The system flow is as follows:
- The user inputs the gear blank parameters through the measurement module dialog.
- The module calculates the theoretical measurement point coordinates and generates the measurement NC program.
- The tool is replaced with the probe, and the measurement NC program is executed.
- When the probe touches the tooth surface, the system records the actual coordinates from the encoder.
- The module compares actual and theoretical values and generates an error report.
One important function is the development of a G31 skip command. When the probe contacts the workpiece, the system saves the current coordinates and immediately retracts the probe in the positive Z direction to prevent damage due to unexpected large errors. The G31 implementation on the TDNC-H8 system uses the logic:
IF (M3448=1) THEN READ Q163,Q164,Q165 ; Save current X,Y,Z M999=M999+1 ; Increment program line number GOTO (M999+1) ; Jump to next line ENDIF
This skip function ensures safe measurement even when the gear has significant machining errors.
5. Machining Example and Simulation
5.1 Gear Blank Parameters
I designed a pair of straight bevel gears with a shaft angle of 90 degrees. The gear blank parameters are listed in Table below.
| Parameter | Gear (1) | Gear (2) |
|---|---|---|
| Large end module | 3 mm | 3 mm |
| Number of teeth | 30 | 20 |
| Pressure angle | 20° | 20° |
| Pitch cone angle | 56.30998° | 33.69010° |
| Pitch diameter | 90 mm | 60 mm |
| Addendum | 3.0 mm | 3.0 mm |
| Dedendum | 3.6 mm | 3.6 mm |
| Whole tooth height | 6.6 mm | 6.6 mm |
| Face width | 10 mm | 10 mm |
| Cone distance | 54.08324 mm | 54.08324 mm |
| Root cone angle | 52.50175° | 29.88187° |
5.2 Cutter Head Parameters
I used an Oerlikon EN4-55 cutter head. The parameters are given in Table below.
| Parameter | Value |
|---|---|
| Cutter head type | EN4-55 |
| Number of blade groups | 4 |
| Nominal radius | 55 mm |
| Normal module | 3 mm |
| Blade edge radius | 0.8 mm |
5.3 Imaginary Generating Gear Parameters
The imaginary generating gear for the example is a crown gear with parameters shown in Table below.
| Parameter | Value |
|---|---|
| Number of teeth | 8 |
| Face width | 10 mm |
| Addendum | 3.6 mm |
| Dedendum | 3.0 mm |
| Cutter radial distance | 55 mm |
5.4 Machine Adjustment Parameters
The mechanical settings for the cradle-type model are listed in Table below. These are used to calculate the five-axis motion for the machining center.
| Parameter | Gear (1) | Gear (2) |
|---|---|---|
| Initial cradle angle | 52.31° | 52.31° |
| Final cradle angle | 81.73° | 81.73° |
| Cutter radial distance | 55 mm | 55 mm |
| Mechanical root angle | 52.50175° | 29.88187° |
| Hypocycloid ratio | 0.5 | 0.5 |
| Generating ratio | 3.75 | 2.5 |
5.5 Machining Center Set-up
For the general 5-axis machining center, the Z-axis offset from the machine origin to the gear pitch cone point was calculated as:
| Parameter | Gear (1) | Gear (2) |
|---|---|---|
| Machine center to table top \( Z_{r1} \) | 250 mm | 250 mm |
| Fixture height \( Z_{r2} \) | 137 mm | 137 mm |
| Mounting surface to pitch cone \( Z_{r3} \) | 30 mm | 45 mm |
| Total \( Z_{rq} \) | 417 mm | 432 mm |
5.6 Motion Coordinates
I calculated the discrete axis positions by dividing the cradle angle range into small increments. For the large gear, the increment was 0.7355 degrees, resulting in 40 points. For the small gear, the increment was 0.6312 degrees, also resulting in 40 points. A portion of the axis motion for the small gear is shown in Table below.
| Cradle angle (°) | X (mm) | Y (mm) | Z (mm) | A (°) | C (°) | Spindle (°) |
|---|---|---|---|---|---|---|
| 52.31 | 43.52316 | 273.2571 | -359.4455 | 29.88187 | -31.8002 | 0 |
| 52.9412 | 43.89096 | 272.7756 | -359.4455 | 29.88187 | -31.9283 | 0 |
| 53.5724 | 44.25343 | 272.2900 | -359.4455 | 29.88187 | -32.0548 | 0 |
| 54.2036 | 44.61053 | 271.8006 | -359.4455 | 29.88187 | -32.1796 | 0 |
| 54.8348 | 44.96222 | 271.3072 | -359.4455 | 29.88187 | -32.3027 | 0 |
| 55.466 | 45.30845 | 270.8099 | -359.4455 | 29.88187 | -32.4240 | 0 |
| 56.0972 | 45.64918 | 270.3089 | -359.4455 | 29.88187 | -32.5436 | 0 |
| 56.7284 | 45.98437 | 269.8042 | -359.4455 | 29.88187 | -32.6614 | 0 |
5.7 CNC Parameters and Program Generation
For the example, the spindle speed was set to 100 r/min, and the feed rate was calculated as 9.327 mm/min for the small gear and 10.8257 mm/min for the large gear. The electronic gearbox ratio parameters were determined from the equation presented earlier. For the small gear, with \( Z_t = 4 \) and \( Z_g = 20 \), the ratio is:
\[
i = \frac{4}{20} = 0.2
\]
Therefore, the PMAC variables were set as \( Ix07 = 96 \) and \( Ix08 = 162 \). For the large gear, \( Ix07 = 96 \) and \( Ix08 = 108 \).
By entering all parameters into the developed NC programming module, the system automatically generated the complete NC program. A portion of the program for the small gear is shown in the appendix of the original thesis.
5.8 Simulation Verification
To verify the mathematical model and the generated NC program, I used VERICUT simulation software. The three-dimensional model of the general 5-axis machining center was built in SolidWorks and imported into VERICUT. The EN4-55 cutter head was simplified with a conical cutter model representing the cutting edges. The tool, fixture, workpiece blank, and machine model were assembled according to the actual machining setup. Since VERICUT does not support the electronic gearbox function directly, the C-axis motion was changed to interpolated motion instead of synchronous following. Although this simplification affects the exact tooth profile, the simulation still confirms that the tool paths are valid and that no collision occurs among the tool, fixture, and workpiece. The basic straight tooth shape can be observed.
The simulation result demonstrates that the mathematical model for face-hobbing straight bevel gears on a general 5-axis machining center is correct, and the NC program generated by the developed system module is feasible for real machining.
6. Conclusion
In this thesis, I have presented a systematic approach to manufacturing straight bevel gears by the face-hobbing method on a general 5-axis machining center with an open CNC system. The main contributions of my research are:
- I derived the mathematical model for face-hobbing straight bevel gears, starting from the cradle-type gear cutting machine and transforming it to the general 5-axis machining center. The resulting axis motion equations enable conventional CNC machines to perform the face-hobbing process without a dedicated gear cutting machine.
- I developed a CNC machining system module that automatically generates NC programs for straight bevel gear face-hobbing. The module integrates all necessary parameters and provides a user-friendly interface for production use.
- I implemented the electronic gearbox function to ensure strict synchronization between the cutter head and the workpiece axis, which is essential for continuous indexing.
- I developed an on-machine measurement module for straight bevel gears. The module plans the measurement path, generates the measurement NC program, and produces a quality error report based on pitch deviation and tooth surface error evaluation methods.
- I verified the mathematical model and the system modules through a machining example and VERICUT simulation. The results confirm that the proposed method is feasible and reliable.
Future work may include the design of a dedicated cutter head for face-hobbing straight bevel gears, tooth contact analysis (TCA) and loaded tooth contact analysis (LTCA) for optimizing machine settings, and the development of a closed-loop digital manufacturing system that integrates machining, measurement, and error compensation.
In conclusion, the face-hobbing method for straight bevel gears based on a general 5-axis machining center offers a cost-effective and flexible alternative to traditional methods. It combines high efficiency, continuous indexing, and double-flank cutting, and the open CNC system allows further customization for various industrial applications.