Straight bevel gears are fundamental machine elements used for transmitting motion and power between intersecting axes. They are widely employed in automobiles, tractors, engines, agricultural machinery, construction equipment, printing machinery, and many other power transmission applications. The demand for high-quality straight bevel gears has been continuously increasing due to the rapid growth of the automotive and machinery industries. However, the conventional manufacturing methods of straight bevel gears, such as planing, circular broaching, double-cutter milling, forming milling, and grinding, generally suffer from low productivity, limited flexibility, or high tooling costs. Therefore, there is a strong need for a manufacturing method that can simultaneously achieve high efficiency, high precision, and low cost.
In recent years, the German company Klingelnberg introduced a new face-hobbing process for manufacturing straight bevel gears, based on the interesting mathematical property that a hypocycloid degenerates into a straight line when the radius of the rolling circle is exactly half of the radius of the base circle. Their patented Hypoflex process is performed on a dedicated six-axis CNC bevel gear cutting machine. The process uses continuous indexing and double-flank cutting, which offers a much higher productivity and better gear tooth contact behavior than traditional indexed cutting methods. Nevertheless, the dedicated six-axis gear cutting machine is very expensive, and its proprietary control system is not open for user extensions. These limitations prevent this attractive technology from being widely adopted in ordinary gear manufacturing workshops.
To overcome those limitations, our research group proposed an alternative implementation: performing face-hobbing of straight bevel gears on a general five-axis machining center equipped with an open-architecture CNC system. A general five-axis machining center has three linear axes and two rotary axes, which provides high structural stiffness and precision at a much lower cost than a dedicated gear cutting machine. In addition, the open CNC system allows us to develop custom modules for automatic NC program generation, electronic gearbox control, and on-machine gear metrology. This paper presents the complete methodology, including the mathematical model, the development of the CNC machining module, the on-machine measurement module, and the simulation verification through practical machining examples.
1. Manufacturing Principle
1.1 Face-Hobbing for Epicycloid Bevel Gears
In the classical Oerlikon face-hobbing process, the cutter head rotates about its own axis and simultaneously the work gear rotates about its axis. The cutter traces an extended epicycloid on an imaginary plane gear, commonly called the crown gear. The blades on the cutter head represent the teeth of the crown gear. Because the cutter revolves continuously and the work gear rotates continuously, the indexing is continuous rather than intermittent. The tooth flank is generated by the envelope of the cutting edges. The motion can be described by the rolling of a rolling circle of radius \(R_t\) on the outside of a base circle of radius \(R_b\). If a point lies on the extension of the rolling circle radius, the trajectory becomes an extended epicycloid, which is the ideal tooth line for Oerlikon spiral bevel gears.
1.2 Hypocycloid Straight-Line Property
For a point on the circumference of a rolling circle of radius \(r\) rolling inside a fixed base circle of radius \(R\), the trajectory is a hypocycloid. With the usual coordinate system, the parametric equations are:
$$
x = (R – r) \cos\theta + r\cos\left(\frac{R-r}{r}\theta\right),
$$
$$
y = (R – r) \sin\theta – r\sin\left(\frac{R-r}{r}\theta\right).
$$
In these equations, \(\theta\) is the rotation angle of the line connecting the centers. If we impose the condition
$$
R = 2r,
$$
then the equations simplify to
$$
x = 2r\cos\theta = R\cos\theta,
$$
$$
y = 0.
$$
Thus, the trajectory degenerates into a straight line passing through the center of the base circle. This is the key geometric principle that enables the face-hobbing of straight bevel gears. Instead of using the extended epicycloid motion, we use the internal rolling motion with the cutter radius equal to one half of the imaginary crown gear base circle radius. The cutter blades then move along truly straight lines on the imaginary crown gear.
1.3 Face-Hobbing of Straight Bevel Gears
The face-hobbing process for straight bevel gears uses a cutter head with multiple groups of inner and outer blades. Let the cutter radius be \(R_t\), and let the base circle radius of the imaginary crown gear be \(R_b\). The condition required for generating straight tooth lines is:
$$
R_b = 2R_t.
$$
The cutter center is offset from the crown gear center by the blade distance \(R_{mt}\), which satisfies
$$
R_{mt} = R_b – R_t = R_t.
$$
In other words, the cutter head radius is equal to the radial distance between the cutter center and the crown gear center. Because of this special relationship, each group of blades cuts a straight tooth flank on the imaginary crown gear. The imaginary crown gear then rolls with the workpiece gear according to the gear ratio:
$$
\theta_g = \frac{Z_t}{Z_g}\beta_t,
$$
where \(\theta_g\) is the workpiece rotation angle, \(\beta_t\) is the cutter head rotation angle, \(Z_t\) is the number of cutter blade groups, and \(Z_g\) is the number of teeth of the workpiece gear. Continuous generating motion is thus obtained while both the cutter and the work gear rotate at constant speeds. The process is a double-flank cutting method because each group of blades cuts both the convex and concave flanks of the tooth slot simultaneously.
2. Mathematical Model of Face-Hobbing Straight Bevel Gears
2.1 Model on a Cradle-Type Bevel Gear Generator
To derive the mathematical model, we first consider a mechanical cradle-type bevel gear generator. The cutter head is mounted on the cradle, and its rotation axis is parallel to the cradle axis. The imaginary crown gear is generated by the combination of cutter rotation and cradle rotation. The workpiece is mounted on a work spindle whose axis is inclined by the machine root angle \(\delta_1\).
We define the following coordinate systems:
- \(S_t\) fixed to the cutter head, with its \(z_t\)-axis along the cutter axis.
- \(S_1\) auxiliary coordinate system after cutter head rotation.
- \(S_m\) fixed to the machine frame / imaginary crown gear, with \(z_m\)-axis along the cradle axis.
- \(S_2\) auxiliary coordinate system after the cradle rotation.
- \(S_g\) fixed to the workpiece gear, with \(z_g\)-axis along the workpiece axis.
The position vector of a cutting point in the cutter coordinate system is denoted by \(\mathbf{r}_t\). After applying the successive coordinate transformations, the position vector of the generated tooth surface in the workpiece coordinate system is expressed as
$$
\mathbf{r}_g = \mathbf{M}_{g\theta_g}\mathbf{M}_{m\delta_1}\mathbf{M}_{m\xi_m}\mathbf{M}_{mt}\mathbf{M}_{t\beta_t}\mathbf{r}_t,
$$
where \(\mathbf{M}_{t\beta_t}\) is the cutter head rotation matrix, \(\mathbf{M}_{mt}\) is the translation matrix by \(R_{mt}\), \(\mathbf{M}_{m\xi_m}\) is the combined cradle rotation matrix, \(\mathbf{M}_{m\delta_1}\) is the inclination matrix by the machine root angle, and \(\mathbf{M}_{g\theta_g}\) is the workpiece rotation matrix.
The combined cradle angle \(\xi_m\) consists of the indexing motion \(\xi_{m1}\) and the generating motion \(\xi_{m2}\):
$$
\xi_m = \xi_{m0} + \xi_{m1} + \xi_{m2},
$$
where \(\xi_{m0}\) is the initial cradle angle. In the pure face-hobbing process, the indexing motion is locked to the cutter rotation by the relation:
$$
\xi_{m1} = 0.5\,\beta_t.
$$
Because of the hypocycloid straight-line condition, the ratio between the cutter rotation and the cradle indexing rotation is fixed at 0.5. The generating motion is related to the workpiece rotation by the gear ratio:
$$
\xi_{m2} = i_{mg}\theta_g,
$$
with
$$
i_{mg} = \frac{Z_g}{Z_m},
$$
where \(Z_m\) is the number of teeth of the imaginary crown gear.
2.2 Transformation to the General Five-Axis Machining Center
The coordinate system of the general five-axis machining center is shown in the following conceptual development. The machine has three linear axes \(X\), \(Y\), \(Z\) and two rotary axes \(A\) and \(C\). The cutter spindle rotation is denoted by \(W\). The workpiece is mounted on a rotary table that provides the \(A\) and \(C\) rotations. The reference point of the workpiece coordinate system is placed at the pitch apex of the gear blank.
The coordinate transformation chain for the five-axis machining center is:
$$
\mathbf{r}_{g}^{(5)} = \mathbf{M}_{g\theta_g}\mathbf{M}_{qC}\mathbf{M}_{rqZ}\mathbf{M}_{rA}\mathbf{M}_{pqY}\mathbf{M}_{pqX}\mathbf{M}_{pqZ}\mathbf{M}_{pW}\mathbf{r}_t.
$$
Equating this transformation with the cradle-type model, we obtain the inverse kinematic expressions that are used to generate the NC program. The final expressions are summarized as:
$$
A = \delta_1,
$$
$$
Z = -Z_{rq}\cos\delta_1,
$$
$$
X = R_{mt}\sin(\xi_0 + \xi_m),
$$
$$
Y = Z_{rq}\sin\delta_1 + R_{mt}\cos(\xi_0 + \xi_m).
$$
In the above equations, \(Z_{rq}\) is the distance from the machine origin to the gear pitch apex along the \(Z\)-axis direction. The \(C\)-axis command is obtained by combining the indexing and generating rotations:
$$
\Delta C = i_{gm}\Delta\theta_g + i_{tm}\Delta\beta_t,
$$
where \(i_{gm}\) and \(i_{tm}\) are appropriate gear ratios. In the electronic gearbox implementation, the spindle \(W\) is set to a constant speed and the \(C\)-axis follows the spindle according to the gear ratio. Thus, the spindle angle increment assigned to the \(C\)-axis is:
$$
\Delta W’ = 0,
$$
and the original spindle motion is absorbed into the \(C\)-axis command:
$$
\Delta C’ = \Delta C – \frac{1}{i_{tm}}\Delta W.
$$
This formulation allows the five-axis machining center to perform the face-hobbing process without changing the physical spindle motion control.
3. Development of the CNC Machining System Module
3.1 Open-Architecture CNC Platform
Our laboratory has developed an open-architecture CNC system based on a PC plus a motion control card. The platform uses an industrial PC as the human-machine interface and a PMAC-type motion controller as the real-time control kernel. This structure provides high flexibility for developing custom machining functions because new software modules can be integrated with standard programming languages such as C++ and MATLAB. The open CNC system supports electronic gearing, value-added G-codes, and automatic NC program generation. These features are essential for implementing face-hobbing of straight bevel gears on a general five-axis machining center.
3.2 Reference Points and Workpiece Coordinate System
For the machining of straight bevel gears, the workpiece is mounted on a fixture that is fixed to the rotary table. The gear pitch apex is chosen as the origin of the workpiece coordinate system. The distance from the machine origin to the gear pitch apex is calculated as:
$$
Z_{rq} = Z_{r1} + Z_{r2} + Z_{r3},
$$
where \(Z_{r1}\) is the distance from the machine center to the table surface, \(Z_{r2}\) is the fixture height, and \(Z_{r3}\) is the distance from the gear mounting surface to the pitch apex. The correct setting of this reference distance is critical for obtaining the correct tooth depth and pressure angle.
3.3 Machining Parameter Calculation
The spindle speed is one of the most important cutting parameters. In our experiments, the cutter head and tool holder are much heavier than ordinary milling tools. Therefore, the PID servoparameters of the spindle have to be re-tuned to achieve good dynamic performance and to suppress vibration. Based on the tuning procedure and cutting tests, a suitable spindle speed for the EN4-55 cutter head was found to be in the range of 80 to 120 revolutions per minute.
The feed rate is determined by the chip load per blade and the number of blade groups. Let \(h\) denote the average chip thickness removed during one interval, \(Z_t\) the number of cutter blade groups, and \(S\) the spindle speed in revolutions per minute. Then the feed rate is:
$$
F = h \cdot Z_t \cdot S.
$$
For the gears used in our case study, the feed rate was set to approximately 10.83 mm/min for the large gear and 9.33 mm/min for the small gear.
To generate discrete NC points, the continuous cradle angle \(\xi_m\) is sampled with an interval \(\Delta\xi_m\). The total number of points \(n\) is:
$$
n = Z_t S t,
$$
where \(t\) is the total cutting time. The interval cradle angle is:
$$
\Delta\xi_m = \frac{\xi_{m,n} – \xi_{m,0}}{n}.
$$
Substituting \(\Delta\xi_m\) into the mathematical model yields all required axis positions.
3.4 Automatic NC Program Generation
The NC machining module was developed using a combination of MATLAB and C++ programming. The mathematical model described in Section 2 was coded in MATLAB, converted into C++ compatible source files using MATCOM, and then integrated into the open CNC system. The module provides a user interface where the operator can enter the gear blank parameters, cutter parameters, machine setup parameters, and cutting parameters. The module then automatically generates the complete NC program for face-hobbing of straight bevel gears.
The typical NC program structure is:
G92 X0 Y0 Z0 A0 C0 G95 T4 L20 M04 S100 G90 G01 X... Y... Z... A... C... F... ... M05 G96 M30
The generated NC codes are saved as standard G-code files and can be directly executed by the open CNC system.
3.5 Electronic Gearbox Function
During face-hobbing, the cutter spindle and the workpiece spindle must maintain a strict angular relationship at every instant. Any synchronization error will cause the cutter to cut into the previously formed flank. To solve this problem, we developed a master-slave electronic gearbox function based on the PMAC motion controller. The spindle encoder is treated as the master, and the workpiece \(C\)-axis encoder is treated as the slave. The required gear ratio is:
$$
i_{gc} = \frac{Z_t}{Z_g}.
$$
In the PMAC implementation, the master and slave encoder resolutions must be considered. The spindle encoder has 4096 counts per revolution after quadrature decoding, and the \(C\)-axis encoder has 65536 counts per revolution. If \(Ix07\) and \(Ix08\) are the electronic gear variables, the synchronization relation is:
$$
Ix07 \cdot \Delta MP_n \cdot 4096 = Ix08 \cdot \Delta CP_n \cdot 65536,
$$
where \(\Delta MP_n\) is the master spindle encoder count and \(\Delta CP_n\) is the slave \(C\)-axis encoder count. Therefore, the electronic gear variables are set according to:
$$
\frac{Ix08}{Ix07} = \frac{4096}{65536} \cdot \frac{Z_t}{Z_g}.
$$
This electronic gearbox function guarantees continuous indexing and precise synchronization throughout the cutting process.
4. On-Machine Measurement System Module
4.1 Need for On-Machine Measurement
After machining, straight bevel gears must be inspected for dimensional accuracy. The most important quality parameters are the tooth pitch deviation and the tooth surface error. In conventional practice, the workpiece is removed from the cutting machine and transferred to a dedicated gear measuring machine. This process is time-consuming and introduces additional setup errors. By developing an on-machine measurement module in the open CNC system, we can measure the gear immediately after machining using the same machine and a touch trigger probe.
4.2 Pitch Deviation Measurement
The single pitch deviation \(f_p\) is defined as the difference between the actual pitch and the nominal pitch on the reference circle. For a gear with nominal module \(m\), the nominal circular pitch is:
$$
P_t = \pi m.
$$
If the measurement radius is \(r_p\), the angular pitch measurement gives:
$$
\Delta\delta_i = \delta_i – \delta_{i-1},
$$
and the single pitch deviation is:
$$
f_{p,i} = r_p \Delta\delta_i – P_t.
$$
The cumulative pitch deviation \(F_p\) over \(n\) teeth is the algebraic sum of the single pitch deviations:
$$
F_p = \sum_{i=1}^{n} f_{p,i}.
$$
The on-machine measurement module measures the positions of adjacent tooth flanks on the same reference circle and automatically calculates the pitch errors.
4.3 Tooth Surface Error Measurement
To evaluate the accuracy of the generated tooth flank, the theoretical tooth surface is divided into a grid of measurement points. In our implementation, a \(5 \times 7\) grid is used. The theoretical coordinates \(\mathbf{R}_l^{(i,j)}\) and the unit normal vectors \(\mathbf{n}_l^{(i,j)}\) are obtained from the mathematical model. The measured coordinates \(\mathbf{R}_s^{(i,j)}\) are captured by the touch probe. The normal surface error at each point is calculated as:
$$
e^{(i,j)} = \left(\mathbf{R}_s^{(i,j)} – \mathbf{R}_l^{(i,j)}\right) \cdot \mathbf{n}_l^{(i,j)}.
$$
The grid is constructed by projecting the tooth surface onto the axial section. The relationship between a surface point \((X,Y,Z)\) and its projected coordinates \((Z,U)\) is:
$$
Z = r_g^{(3)},
$$
$$
U = \sqrt{\left(r_g^{(1)}\right)^2 + \left(r_g^{(2)}\right)^2}.
$$
By specifying the boundaries of the tooth surface, the projected grid points are obtained. Then a nonlinear equation solver, such as the fsolve function in MATLAB, is used to find the full spatial coordinates of each theoretical measurement point.
Table 1 shows a representative set of theoretical measurement points on one tooth flank of a straight bevel gear.
| Point ID | X (mm) | Y (mm) | Z (mm) |
|---|---|---|---|
| 1 | 30.16 | 394.52 | 18.71 |
| 2 | 30.52 | 394.35 | 18.54 |
| 3 | 30.88 | 394.17 | 18.36 |
| 4 | 31.24 | 393.99 | 18.18 |
| 5 | 31.60 | 393.81 | 18.00 |
| 10 | 33.40 | 392.88 | 17.07 |
| 15 | 35.20 | 391.91 | 16.10 |
| 20 | 37.00 | 390.89 | 15.08 |
| 25 | 38.80 | 389.83 | 14.02 |
| 30 | 40.60 | 388.73 | 12.92 |
| 35 | 42.04 | 387.81 | 12.00 |
4.4 Development of the Measurement Module
The on-machine measurement module follows the same software architecture as the machining module. The input parameters are the gear blank parameters and the measured gear specification. The module then automatically plans the measurement paths and generates a measurement NC program. The probe used in our system is a Renishaw OMP400 touch trigger probe, which transmits a signal to the CNC system through an infrared receiver. To protect the probe from excessive travel, a special skip command G31 was implemented in the open CNC system. When the probe touches the workpiece, the system reads the current coordinates, saves them, and jumps to the next block.
5. Case Study and Simulation Verification
5.1 Gear Blank Data
To validate the proposed methodology, a pair of straight bevel gears with a shaft angle of \(90^\circ\) was designed and planned. The main gear blank parameters are presented in Table 2.
| Parameter | Large Gear | Small Gear |
|---|---|---|
| Module | 3 | 3 |
| Number of teeth | 30 | 20 |
| Pressure angle (deg) | 20 | 20 |
| Pitch cone angle (deg) | 56.31 | 33.69 |
| Pitch diameter (mm) | 90 | 60 |
| Tip diameter (mm) | 93.33 | 64.99 |
| Addendum (mm) | 3.00 | 3.00 |
| Dedendum (mm) | 3.60 | 3.60 |
| Whole tooth height (mm) | 6.60 | 6.60 |
| Face width (mm) | 10 | 10 |
| Root cone angle (deg) | 52.50 | 29.88 |
5.2 Cutter Head Parameters
The cutter head used in this study is an Oerlikon EN4-55 cutter head. The relevant parameters are listed in Table 3.
| Parameter | Value |
|---|---|
| Number of blade groups | 4 |
| Nominal cutter radius (mm) | 55 |
| Normal module | 3 |
| Inner blade pressure angle (deg) | 20 |
| Outer blade pressure angle (deg) | 20 |
| Cutter blade length (mm) | 11 |
5.3 Imaginary Crown Gear Parameters
The imaginary crown gear is the virtual gear to which the cutter blades belong. The parameters of the imaginary crown gear are calculated according to the mesh relationship with the workpiece gear. The values are listed in Table 4.
| Parameter | Large Gear | Small Gear |
|---|---|---|
| Number of teeth | 8 | 8 |
| Face width (mm) | 10 | 10 |
| Addendum (mm) | 3.60 | 3.60 |
| Dedendum (mm) | 3.00 | 3.00 |
| Whole tooth height (mm) | 6.60 | 6.60 |
| Cutter center offset (mm) | 55 | 55 |
5.4 Machine Adjustment Parameters
From the mathematical model, the cradle-type machine adjustment parameters are obtained. These parameters are then transformed into the five-axis machining center commands. The major machine adjustment parameters for the large and small gears are listed in Table 5.
| Parameter | Large Gear | Small Gear |
|---|---|---|
| Initial cradle angle (deg) | 52.31 | 52.31 |
| Final cradle angle (deg) | 81.73 | 81.73 |
| Machine root angle (deg) | 52.50 | 29.88 |
| Indexing ratio | 0.5 | 0.5 |
| Generating ratio | 3.75 | 2.50 |
5.5 Axis Commands and NC Program
Using the developed module, the NC program for the large and small gears is generated automatically. Representative axis commands for the small gear are given in Table 6.
| Cradle angle (deg) | X (mm) | Y (mm) | Z (mm) | A (deg) | C (deg) |
|---|---|---|---|---|---|
| 52.31 | 43.5232 | 273.2571 | -359.4455 | 29.8819 | -31.8002 |
| 52.94 | 43.8910 | 272.7756 | -359.4455 | 29.8819 | -31.9283 |
| 53.57 | 44.2534 | 272.2900 | -359.4455 | 29.8819 | -32.0548 |
| 54.20 | 44.6105 | 271.8006 | -359.4455 | 29.8819 | -32.1796 |
| 54.83 | 44.9622 | 271.3072 | -359.4455 | 29.8819 | -32.3027 |
| 55.47 | 45.3084 | 270.8099 | -359.4455 | 29.8819 | -32.4240 |
| 56.10 | 45.6492 | 270.3089 | -359.4455 | 29.8819 | -32.5436 |
| 56.73 | 45.9844 | 269.8042 | -359.4455 | 29.8819 | -32.6614 |
| 57.36 | 46.3140 | 269.2957 | -359.4455 | 29.8819 | -32.7773 |
| 57.99 | 46.6380 | 268.7837 | -359.4455 | 29.8819 | -32.8913 |
The generated machining parameters are listed in Table 7.
| Parameter | Large Gear | Small Gear |
|---|---|---|
| Spindle speed (r/min) | 100 | 100 |
| Feed rate (mm/min) | 10.83 | 9.33 |
| Electronic gear I707 | 96 | 108 |
| Electronic gear I708 | 96 | 162 |
5.6 Simulation with VERICUT
The generated NC program was further verified using the VERICUT simulation software. A three-dimensional model of the general five-axis machining center was constructed in SolidWorks and imported into VERICUT. The cutter head was modeled as a simplified EN4-55 face-hobbing cutter with four groups of blades. The workpiece blank was mounted on the rotary table. The coordinate systems and tool paths were configured according to the kinematic model presented in Section 2. Because VERICUT does not directly simulate electronic gearbox synchronization, the workpiece rotary axis was implemented as an interpolated axis in the simulation. Despite this simplification, the simulation successfully demonstrated that the cutter head and the workpiece blank do not interfere with the fixture or the machine structure, and that the generated tooth slots exhibit the basic straight-bevel tooth geometry. The simulation results therefore confirm the correctness of the mathematical model and the feasibility of the developed CNC machining module.
6. Conclusion
In this paper, we have presented a complete methodology for the face-hobbing manufacturing of straight bevel gears on a general five-axis machining center with an open-architecture CNC system. The main conclusions and contributions are as follows:
- A mathematical model for face-hobbing of straight bevel gears was derived using the hypocycloid straight-line property. The model was first developed on a cradle-type bevel gear generator and then transformed into inverse kinematics for a general five-axis machining center.
- An automatic NC program generation module was developed on the open CNC platform. The module accepts gear blank parameters, cutter parameters, machine settings, and process parameters, and produces complete NC code for face- hobbing of straight bevel gears.
- An electronic gearbox function was implemented to guarantee the strict synchronization between the cutter spindle and the workpiece \(C\)-axis during continuous indexing. This function is essential for face-hobbing of straight bevel gears.
- An on-machine measurement module was developed for measuring tooth pitch deviation and tooth surface error. The module automatically plans measurement paths, generates measurement NC programs, and calculates the error report according to the established error evaluation formulas.
- The proposed methodology was verified through machining calculations and VERICUT simulation for a pair of straight bevel gears with 30 and 20 teeth. The simulation results show that the process is feasible and can produce the desired straight tooth geometry without interference.
Future work will focus on the design of dedicated cutter heads for face-hobbing of straight bevel gears, the development of a more accurate simulation module that includes electronic gearbox effects, and the application of tooth contact analysis (TCA) and loaded tooth contact analysis (LTCA) to further optimize the machining parameters and surface contact quality.