In my research, I have focused on the use of concave cutters to machine straight bevel gears on domestically produced CNC spiral bevel gear milling machines. The motivation for this work arises from the fact that straight bevel gears are critical components for intersecting-axis transmissions, and they are widely applied in aviation, military, and other demanding fields. Traditional manufacturing methods for straight bevel gears include gear shaping, dual-cutter milling, and fixed-setting hob generation. A well-known industrial approach is the Coniflex method, which uses concave cutters on Phoenix-type milling machines, but its core technology remains proprietary. Several authors have studied practical aspects of concave-cutter machining and contact pattern control, while others have proposed modified virtual generating gears or tooth contact analysis for shaped straight bevel gears. However, published work specifically addressing the mathematical modeling and tooth contact analysis of straight bevel gears machined by concave cutters is still limited. Therefore, I established a tooth surface mathematical model and a corresponding tooth contact analysis method for straight bevel gears produced with concave cutters. I also verified the model and the analysis method through actual cutting experiments and contact pattern inspections. The outcome provides a theoretical basis for determining reasonable machining parameters and cutter parameters for straight bevel gears.
The cutting principle of a concave cutter for straight bevel gears can be described through the concept of a virtual generating gear. The rotation of the cutter forms the tooth surface of a virtual generating wheel, and the cutting process is equivalent to the meshing transmission between this generating wheel and the straight bevel gear being machined. Concave cutters used for straight bevel gears are generally classified into two types: integral cutters and cutters equipped with carbide blades. The essential feature of a concave cutter is that its main cutting edge is inclined inward by a certain angle relative to the end face of the cutter head. When the cutter rotates, it forms a conical surface with an internal cone angle. Milling with such a concave cutter produces a crowning tooth trace. The amount of crowning is measured by the distance $\Delta S$ between the vector radius line at the middle of the tooth trace and the vector radius lines at the two ends of the tooth trace.
The crowning amount is calculated as
$$ \Delta S = \Delta H \left[ \tan(\alpha_f + \delta) – \tan\alpha_0 \right] $$
where $\alpha_f$ is the pressure angle of the gear being machined, $\delta$ is the concave angle of the main cutting edge, and $\alpha_0$ is the cutter pressure angle. In addition, the crowning amount can be expressed as
$$ \Delta S = \frac{b^2 \cos\alpha_f \tan\delta}{4D_e} $$
From this relationship, I can see that the crowning amount is directly proportional to the tooth width $b$ of the gear being machined and to the concave angle $\delta$ of the cutter, while it is inversely proportional to the pressure angle $\alpha_f$ and the cutter diameter $D_e$. Because the cutter does not move along the tooth width direction during machining, the root line of the machined gear appears as an arc, and its depth value is denoted by $\Delta H$. The cutter diameter $D_e$ is related to the tooth width $b$ and the pressure angle $\alpha_f$ by
$$ D_e = \frac{b^2 \cos\alpha_f}{4\Delta H} $$
The top width of the cutter depends on the bottom slot width at the large end and the bottom slot width at the small end of the gear being machined. The top width must simultaneously satisfy the condition of being larger than half of the large-end bottom slot width and smaller than the small-end bottom slot width. These geometric relationships are fundamental for designing concave cutters for straight bevel gears.
To build the tooth surface model, I first defined a coordinate system for the virtual generating wheel. The origin is placed at the apex of the face cone of the generating wheel. In this coordinate system, the position vector $r_c$ and the unit normal vector $n_c$ of the generating wheel tooth surface are
$$ r_c(u,v) = \begin{bmatrix} v \cos\lambda_0 \\ \pm(v \sin\lambda_0 + u \sin\alpha_0) \\ u \cos\alpha_0 \end{bmatrix} $$
$$ n_c(u,v) = \frac{r_{cu} \times r_{cv}}{\left| r_{cu} \times r_{cv} \right|} $$
where $u$ and $v$ are parameters along the tooth height direction and tooth width direction, respectively. The two angles $\alpha_0$ and $\lambda_0$ can be calculated from the basic parameters of the gear as
$$ \lambda_0 = \frac{s_e/2 – h_a \tan\alpha_f}{R_e / \cos\theta_f} $$
$$ \alpha_0 = \arctan\left( \tan\alpha_f \cos\lambda_0 \right) $$
where $s_e$ is the large-end tooth thickness, $\theta_f$ is the root angle, $h_a$ is the addendum, and $R_e$ is the outer cone distance.
For the mathematical model of the straight bevel gear tooth surface, I established a cutter coordinate system whose origin is the intersection of the cutter rotation axis and the tip plane. The position vector of an arbitrary point $m$ on the cutting edge can be expressed as a function of the distance $s$ from the tip and the rotation angle $\theta$. The cutting surface equation of the cutter is
$$ r_t = \begin{bmatrix} \sin\theta (r – s\cos\delta) \\ \cos\theta (r – s\cos\delta) \\ s\sin\delta \\ 1 \end{bmatrix} $$
where $r$ is the cutter radius and $\delta$ is the concave angle. Taking partial derivatives with respect to $s$ and $\theta$ gives
$$ r_{ts} = \begin{bmatrix} \cos\delta \sin\theta \\ -\cos\theta \cos\delta \\ \sin\delta \end{bmatrix} $$
$$ r_{t\theta} = (r – s\cos\delta) \begin{bmatrix} -\cos\theta \\ -\sin\theta \\ 0 \end{bmatrix} $$
Thus, the unit normal vector on the cutting surface is
$$ n_t = \frac{r_{t\theta} \times r_{ts}}{\left| r_{t\theta} \times r_{ts} \right|} = \begin{bmatrix} -\sin\theta \sin\delta \\ \cos\theta \sin\delta \\ \cos\delta \end{bmatrix} $$
During cutting of the workpiece tooth surface, I used the apex of the workpiece pitch cone as the origin of the machine coordinate system $S_m$. The virtual generating wheel coordinate system $S_c$ shares its origin with $S_m$, and its $Z_c$ axis coincides with the $Z_m$ axis. The generating wheel rotates about the $Z_c$ axis with angular velocity $\omega_1$, while the workpiece coordinate system $S_g$ rotates about the $Z_g$ axis with angular velocity $\omega_2$. Auxiliary coordinate systems $S_1$ and $S_2$ are used to determine the relative position between the cutter and the machine tool, and an auxiliary coordinate system $S_3$ is used to determine the relative position between the machine tool and the workpiece.
The relative position relationship among the generating wheel, the machine tool, and the workpiece involves several parameters: $E$ is the distance from the cutter center to the $X_m O_m Z_m$ plane; $D$ is the distance from the cutter center to the $X_m O_m Y_m$ plane; $L$ is the distance from the cutter center to the $Y_m O_m Z_m$ plane; $\lambda$ is the angle between the tip motion trajectory and the tooth slot bisector passing through the machine center, i.e., the tooth space angle; $\delta_f$ is the root cone angle; $\alpha$ is the cutter tilt angle; $\varphi_1$ is the virtual generating wheel rotation angle; and $\psi_1$ is the workpiece rotation angle. The coordinate transformations required for the cutting motion analysis are listed below.
The rotation matrix from $S_t$ to $S_1$ is
$$ M_{1t} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & \cos\alpha & \sin\alpha & 0 \\ 0 & -\sin\alpha & \cos\alpha & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} $$
The rotation matrix from $S_1$ to $S_2$ is
$$ M_{21} = \begin{bmatrix} \cos\lambda & \sin\lambda & 0 & 0 \\ -\sin\lambda & \cos\lambda & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} $$
The translation matrix from $S_2$ to $S_c$ is
$$ M_{c2} = \begin{bmatrix} 1 & 0 & 0 & L \\ 0 & 1 & 0 & -E \\ 0 & 0 & 1 & D \\ 0 & 0 & 0 & 1 \end{bmatrix} $$
The rotation matrix from $S_c$ to $S_m$ is
$$ M_{mc} = \begin{bmatrix} \cos\varphi_1 & -\sin\varphi_1 & 0 & 0 \\ \sin\varphi_1 & \cos\varphi_1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} $$
The rotation matrix from $S_m$ to $S_3$ is
$$ M_{3m} = \begin{bmatrix} \sin\delta_f & 0 & \cos\delta_f & 0 \\ 0 & 1 & 0 & 0 \\ -\cos\delta_f & 0 & \sin\delta_f & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} $$
The rotation matrix from $S_3$ to $S_g$ is
$$ M_{g3} = \begin{bmatrix} \cos\psi_1 & \sin\psi_1 & 0 & 0 \\ -\sin\psi_1 & \cos\psi_1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} $$
Given a reference point $p$ on the generating wheel tooth surface, its position vector is obtained from the generating wheel equation, and its unit normal vector is obtained from the corresponding normal equation. By moving the position vector and unit normal vector of point $m$ on the cutter to coincide with the reference point $p$ on the generating wheel through coordinate transformations, I obtain
$$ r_{cp} = M_{c2} M_{21} M_{1t} r_{tm} $$
$$ n_{cp} = L_{c2} L_{21} L_{1t} n_{tm} $$
where each $L$ matrix is formed by removing the last row and last column from the corresponding $M$ matrix. Solving these two equations yields the five unknowns $\lambda$, $\alpha$, $L$, $E$, and $D$.
In the milling process, the rolling ratio $I_{12}$, defined as the ratio of the virtual generating wheel rotation angle to the workpiece rotation angle, is constant and is given by
$$ I_{12} = \frac{\cos\theta_f}{\sin\delta_d} $$
where $\delta_d$ is the pitch cone angle of the workpiece. The relationship between the virtual generating wheel rotation angle $\varphi_1$ and the workpiece rotation angle $\psi_1$ is
$$ \psi_1 = I_{12} \varphi_1 $$
Using the transformation matrices above, the cutter cutting surface equation is transformed to the workpiece as
$$ r_g = M_{g3} M_{3m} M_{mc} M_{c2} M_{21} M_{1t} r_t $$
$$ n_g = L_{g3} L_{3m} L_{mc} L_{c2} L_{21} L_{1t} n_t $$
The position vector and unit normal vector in the machine coordinate system $S_m$ are
$$ r_m = M_{mc} M_{c2} M_{21} M_{1t} r_t $$
$$ n_m = L_{mc} L_{c2} L_{21} L_{1t} n_t $$
I set the angular velocity of the generating wheel during cutting as
$$ \omega_1 = \begin{bmatrix} 0 & 0 & 1 & 1 \end{bmatrix}^T $$
Then the angular velocity of the workpiece is
$$ \omega_2 = I_{12} \begin{bmatrix} -\cos\delta_f & 0 & \sin\delta_f & 1 \end{bmatrix}^T $$
In the machine coordinate system, the relative velocity is
$$ v_{12} = (\omega_1 – \omega_2) \times r_m $$
According to the meshing equation of the gear pair,
$$ v_{12} \cdot n_m = 0 $$
I can solve for the parameter $s$ as a function of $\varphi_1$ and $\theta$. Substituting $s$ into the workpiece tooth surface equation gives the contact line on the tooth surface parameterized by $\theta$. As $\varphi_1$ varies continuously, different contact lines are obtained, and the collection of these contact lines forms the tooth surface of the machined straight bevel gear.
Based on the method described above, I obtained the tooth surface mathematical models of both the pinion and the gear. For tooth contact analysis, the tooth surface equations and unit normal vectors of the pinion and the gear must be represented in the same coordinate system. I established a meshing coordinate system $S_f$ that coincides with the pinion coordinate system $S_p$. It is only necessary to transform the tooth surface equation and unit normal vector of the gear from its coordinate system $S_g$ into $S_f$. In the meshing coordinate system, the position vectors of a point on the pinion and gear tooth surfaces are
$$ r_{fp} = M_{f1} r_1 $$
$$ r_{fg} = M_{\Sigma} M_{f2} r_2 $$
where
$$ M_{f1} = \begin{bmatrix} \cos\eta_1 & -\sin\eta_1 & 0 & 0 \\ \sin\eta_1 & \cos\eta_1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} $$
$$ M_{f2} = \begin{bmatrix} \cos\eta_2 & \sin\eta_2 & 0 & 0 \\ -\sin\eta_2 & \cos\eta_2 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} $$
$$ M_{\Sigma} = \begin{bmatrix} \cos\Sigma & 0 & -\sin\Sigma & 0 \\ 0 & 1 & 0 & 0 \\ \sin\Sigma & 0 & \cos\Sigma & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} $$
Here, $\eta_1$ and $\eta_2$ are the rotation angles of the pinion and the gear from their initial positions, and $\Sigma$ is the shaft angle. In the meshing coordinate system $S_f$, the unit normal vectors of the pinion and the gear at a point are
$$ n_{fp} = L_{f1} n_1 $$
$$ n_{fg} = L_{\Sigma} L_{f2} n_2 $$
When the pinion and the gear mesh, the contact point on the tooth surfaces satisfies
$$ \begin{cases} r_{fp} = r_{fg} \\ n_{fp} = n_{fg} \end{cases} $$
To analyze the sensitivity of the straight bevel gear pair to installation errors during meshing transmission, I usually specify three initial calculation points on the gear tooth surface: one near the heel, one at the midpoint, and one near the toe. Their distances from the heel are $0.25b$, $0.5b$, and $0.75b$, respectively, and they are located at the middle of the tooth height at those positions. Then I calculate the contact pattern and transmission error curve for each of the three positions.
In actual meshing, the transmission error function is defined as
$$ \Delta\eta_2 = (\eta_2 – \eta_{20}) – \frac{z_1}{z_2} (\eta_1 – \eta_{10}) $$
I implemented the tooth contact analysis method in a program. Substituting the basic parameters and cutter parameters of the straight bevel gears, together with the machine tool adjustment parameters for cutting, into the program yields the contact patterns and transmission error curves for the heel, midpoint, and toe positions. The contact patterns and the corresponding values of $V$ and $H$ show that when the straight bevel gear pair contacts at different installation positions, the contact pattern along the tooth width direction occupies approximately one half of the tooth face width. The transmission error curve has a shape similar to a downward-opening parabola. This behavior is a result of modifying the tooth length curvature, i.e., producing a crowning tooth, by using the concave cutter to machine the straight bevel gears.
To validate the model and the analysis method, I prepared gear blanks and cutters and performed actual cutting on an H350C CNC milling machine. The basic parameters and cutter parameters used in the example are listed in the following table.
| Parameter | Pinion | Gear |
|---|---|---|
| Number of teeth | 35 | 35 |
| Module $m$ / mm | 2.5 | 2.5 |
| Cutter pressure angle $\alpha_0$ / deg | 20 | 20 |
| Tooth width $b$ / mm | 15 | 15 |
| Pitch cone angle $\delta_d$ / deg | 45 | 45 |
| Root cone angle $\delta_f$ / deg | 42.22 | 42.22 |
| Face cone angle $\delta_a$ / deg | 47.78 | 47.78 |
| Dedendum $h_f$ / mm | 2.5 | 2.5 |
| Addendum $h_a$ / mm | 3 | 3 |
| Cutter radius $r$ / mm | 72.78 | 72.78 |
| Concave angle $\delta$ / deg | 2 | 2 |
The machine tool adjustment parameters for cutting are summarized in the next table.
| Parameter | Pinion | Gear |
|---|---|---|
| Rolling ratio | 1.412554 | 1.412554 |
| Workpiece mounting angle $\delta_f$ / deg | 42.22 | 42.22 |
| $L$ / mm | 54.30 | 54.30 |
| $E$ / mm | 28.05 | 28.05 |
| $D$ / mm | 67.12 | 67.12 |
| Tooth space angle $\lambda$ / deg | 1.612 | 1.612 |
The theoretical contact patterns obtained from the tooth contact analysis program were compared with the actual contact patterns obtained after rolling inspection. The actual contact pattern was obtained by mounting a pair of machined straight bevel gears on a rolling inspection machine and adjusting the relative position of the gear pair. The comparisons for the toe, midpoint, and heel contact regions show that the actual contact pattern agrees with the theoretical contact pattern predicted by the tooth contact analysis. In addition, the gears were mounted on a gear measuring center for tooth profile error inspection. The maximum tooth profile error was $11.9~\mu\text{m}$, which satisfies practical engineering requirements. The main sources of this error are cutter errors, machine tool machining errors, and gear blank errors. The experimental results verified the correctness of the established tooth surface mathematical model and the tooth contact analysis method for straight bevel gears machined with concave cutters.
Because the straight bevel gears produced by concave cutters have a crowning tooth trace, the contact pattern is localized and the transmission error is controlled. This is beneficial for reducing the sensitivity of the straight bevel gear pair to installation errors. In my analysis, the contact pattern along the tooth width direction occupied approximately half of the face width, and the transmission error curve was parabolic. These characteristics are typical for straight bevel gears with a crowning tooth trace. The crowning amount $\Delta S$ depends on the tooth width $b$, the pressure angle $\alpha_f$, the concave angle $\delta$, and the cutter diameter $D_e$. A larger tooth width or a larger concave angle increases the crowning amount, while a larger pressure angle or a larger cutter diameter decreases the crowning amount. Therefore, the cutter parameters must be selected carefully to achieve the desired contact pattern and transmission error for straight bevel gears.
For a more systematic view of the influence of the main parameters on the crowning amount and on the tooth contact behavior of straight bevel gears, I summarized the trends in the following table.
| Parameter | Increase effect on crowning amount $\Delta S$ | Expected influence on contact pattern |
|---|---|---|
| Tooth width $b$ | Increases proportionally to $b^2$ | Contact pattern becomes more localized |
| Pressure angle $\alpha_f$ | Decreases | Contact pattern tends to become wider |
| Concave angle $\delta$ | Increases | Stronger crowning and more localized contact |
| Cutter diameter $D_e$ | Decreases | Weaker crowning and contact pattern may widen |
| Root depth parameter $\Delta H$ | Increases directly | Affects tooth root geometry and contact position |
The tooth contact analysis algorithm that I implemented consists of the following steps. First, I input the basic gear data, cutter parameters, and machine tool adjustment parameters. Second, I calculate the virtual generating wheel tooth surface and its unit normal vector. Third, I transform the cutter cutting surface to the workpiece coordinate system and derive the contact lines using the meshing equation. Fourth, I obtain the complete tooth surface of the pinion and the gear as the envelope of the contact lines. Fifth, I establish the meshing coordinate system and impose the equality of position vectors and unit normal vectors at the contact point. Sixth, I solve the nonlinear contact equations for the unknown rotation angles at the selected initial points. Seventh, I compute the contact pattern and the transmission error curve. Eighth, I compare the theoretical results with experimental measurements. This procedure is general and can be applied to straight bevel gears with different parameters.
The meshing equation $v_{12} \cdot n_m = 0$ is the core of the tooth surface generation. For the concave cutter, the normal vector $n_t$ on the cutting surface is not constant because of the concave angle $\delta$ and the rotation angle $\theta$. The relative velocity $v_{12}$ depends on the angular velocities $\omega_1$ and $\omega_2$, which in turn depend on the rolling ratio $I_{12}$ and the root cone angle $\delta_f$. Solving the meshing equation gives the parameter $s$ along the cutting edge for each pair of $\varphi_1$ and $\theta$. This means that the contact line on the workpiece tooth surface is a curve. When $\varphi_1$ increases, the contact line sweeps across the tooth surface and generates the tooth surface of the straight bevel gear. The resulting tooth surface is smooth and has a crowning shape in the lengthwise direction, which is essential for avoiding edge contact in straight bevel gears.
In the tooth contact analysis, I specified three initial points on the gear tooth surface: one at $0.25b$ from the heel, one at $0.5b$ from the heel, and one at $0.75b$ from the heel. These points are at the middle of the tooth height. For each point, I solved the contact equations and obtained the contact pattern and the transmission error curve. The results showed that the contact pattern is located near the middle of the tooth width when the gear pair is mounted at the nominal position. When the mounting position is changed, the contact pattern moves along the tooth width direction, but it remains localized and does not spread to the edges. The transmission error curve is nearly parabolic, with a small amplitude. This indicates that the straight bevel gears machined with concave cutters have good meshing performance and are not overly sensitive to installation errors.
The experimental verification included actual cutting, rolling inspection, and tooth profile error measurement. The cutting was performed on a CNC milling machine with the adjustment parameters listed earlier. After cutting, the pinion and gear were mounted on a rolling inspection machine. The contact pattern was inspected by applying a marking compound to the gear teeth and rotating the pair under a light load. The observed contact patterns at the toe, midpoint, and heel were consistent with the theoretical predictions. The tooth profile error was measured on a gear measuring center. The maximum error was $11.9~\mu\text{m}$, which is within the acceptable range for many industrial applications. The error sources include cutter manufacturing errors, machine tool geometric errors, thermal deformation, and gear blank errors. Even with these errors, the agreement between theory and experiment confirms the validity of the mathematical model and the tooth contact analysis method.
One important aspect of my study is the use of the concave cutter to produce straight bevel gears with a crowning tooth trace. The concave angle $\delta$ is the key parameter that controls the crowning amount. If $\delta$ is too small, the crowning amount is insufficient, and the contact pattern may extend to the tooth edges, causing stress concentration. If $\delta$ is too large, the contact pattern becomes too small, and the load capacity of the straight bevel gears decreases. Therefore, the concave angle must be optimized together with the tooth width, pressure angle, and cutter diameter. The formula for the crowning amount shows that $\Delta S$ is proportional to $b^2$ and $\tan\delta$, and inversely proportional to $\cos\alpha_f$ and $D_e$. This provides a clear guideline for cutter design and process planning for straight bevel gears.
Another important parameter is the cutter diameter $D_e$. It affects not only the crowning amount but also the depth of the root line arc. A smaller cutter diameter increases the curvature of the root line and can lead to a larger crowning amount for a given concave angle. However, a very small cutter diameter may cause interference or insufficient clearance. Therefore, the cutter diameter must be selected to satisfy the geometric constraints of the straight bevel gear pair. The formula $D_e = b^2 \cos\alpha_f / (4\Delta H)$ relates the cutter diameter to the tooth width, pressure angle, and root line depth. In practice, I use this formula as a first estimate and then refine the cutter design through tooth contact analysis.
The tooth contact analysis method I developed is not limited to the specific example presented here. It can be applied to straight bevel gears with different numbers of teeth, modules, pressure angles, and cone angles. The only requirement is that the basic parameters and cutter parameters are known. The method can also be extended to analyze the influence of installation errors, such as offset, shaft angle error, and axial displacement. By changing the corresponding parameters in the meshing coordinate system, I can predict how the contact pattern and transmission error change. This is useful for tolerance allocation and quality control in the manufacturing of straight bevel gears.
In terms of computational implementation, I used a numerical solver to solve the nonlinear contact equations. The unknowns are the rotation angles $\eta_1$ and $\eta_2$ and the surface parameters of the pinion and gear. The equations are the equality of position vectors and unit normal vectors. Because the tooth surfaces are represented analytically, the Jacobian matrix can be derived analytically or computed numerically. I used a Newton-Raphson iteration with a tolerance of $10^{-8}$ for convergence. The contact pattern was obtained by projecting the contact points onto the gear tooth surface and plotting the contact ellipse. The transmission error was computed as a function of the pinion rotation angle. The entire procedure is efficient and can be run on a personal computer.
For the straight bevel gears machined with concave cutters, the contact pattern and transmission error are strongly influenced by the machine tool adjustment parameters. The rolling ratio $I_{12}$ determines the relative motion between the generating wheel and the workpiece. If the rolling ratio is not correct, the tooth thickness and the contact pattern will be incorrect. The workpiece mounting angle $\delta_f$ determines the orientation of the tooth root. The parameters $L$, $E$, and $D$ determine the position of the cutter relative to the machine center. The tooth space angle $\lambda$ determines the angular position of the cutter. All these parameters must be set accurately to produce the desired straight bevel gears. In my experiments, I used the values listed in the table and obtained good results.
The measurement of the tooth profile error showed a maximum value of $11.9~\mu\text{m}$. This error is small enough for many applications, but it can be further reduced by improving the accuracy of the cutter and the machine tool. The error topology showed that the error is distributed along the tooth profile and tooth trace. The crowning shape was clearly visible in the topology, confirming that the concave cutter produced the intended modification. The contact pattern inspection also confirmed that the contact was localized and did not extend to the edges. This is a strong indication that the straight bevel gears machined with concave cutters have good meshing characteristics.
In conclusion, I established a tooth surface mathematical model for straight bevel gears machined with concave cutters. The model is based on the cutting principle of the concave cutter and the concept of a virtual generating wheel. I derived the equations for the cutter cutting surface, the unit normal vector, and the coordinate transformations. I also established a tooth contact analysis method for the straight bevel gear pair. The method solves the contact equations in a common meshing coordinate system and yields the contact pattern and transmission error curve. I implemented the method in a computer program and applied it to a numerical example. The theoretical results were verified by actual cutting experiments and contact pattern inspections. The maximum tooth profile error was $11.9~\mu\text{m}$, which is acceptable. The research provides a theoretical basis for determining reasonable machining parameters and cutter parameters for straight bevel gears. It also supports the use of concave cutters on domestic CNC spiral bevel gear milling machines for manufacturing straight bevel gears.
The main contributions of my work can be summarized as follows. First, I established a complete mathematical model for the tooth surface of straight bevel gears machined by concave cutters, including the concave angle and the crowning effect. Second, I developed a tooth contact analysis method that can predict the contact pattern and transmission error of straight bevel gears. Third, I verified the model and the method through actual cutting and inspection. Fourth, I provided formulas and tables that relate the cutter parameters to the crowning amount and contact behavior. These contributions are useful for engineers and researchers working with straight bevel gears. The method can be extended to other types of bevel gears and to the optimization of cutting parameters.
Future work may include the optimization of the concave angle and cutter diameter for different load conditions, the analysis of the influence of installation errors on the transmission error, and the development of a more efficient numerical algorithm for tooth contact analysis. I also plan to apply the method to straight bevel gears with different modules and tooth numbers to further validate its generality. In addition, the effect of cutter wear on the tooth surface and contact pattern can be studied. These extensions will make the method more robust and more useful for industrial applications of straight bevel gears.
Overall, my study demonstrates that concave cutter machining is a viable method for producing high-quality straight bevel gears. The mathematical model and tooth contact analysis method provide a solid foundation for understanding the meshing behavior of straight bevel gears. The experimental results confirm that the theoretical predictions are accurate and that the manufactured gears meet practical requirements. I believe that this work will contribute to the wider adoption of concave cutter machining for straight bevel gears in domestic CNC gear milling machines.