I present a complete design and stiffness analysis framework for non-orthogonal asymmetric helical face gears. These face gears combine three distinguishing features: a variable shaft angle, different pressure angles on the two flanks, and a helical tooth trace. Because of these features, they are attractive for high-precision aerospace transmissions, robotic joints, and compact power trains where the input and output axes are neither perpendicular nor symmetric in load capacity. My goal is to derive the tooth surface mathematically, to compute the time-varying meshing stiffness accurately, and to verify the result by finite element simulation. Throughout this work I emphasize the role of the pressure angle and the shaft angle, because these two parameters strongly affect the contact trace, the loaded tooth area, and the resulting stiffness of face gears.
The core idea is that the tooth surface of face gears is generated by a virtual shaper cutter. By establishing a sequence of coordinate systems, I obtain the transformation from the shaper cutter to the face gear. Then I solve the meshing equation to obtain the active and passive tooth surfaces. After that, I use an improved potential energy method combined with a slice method to compute the bending stiffness, the fillet-base stiffness, and the Hertzian contact stiffness. Finally, I assemble these components into single-tooth and double-tooth meshing stiffness expressions. The finite element results show that my analytical values are accurate within about five percent. This agreement confirms that the proposed method is reliable for non-orthogonal asymmetric helical face gears.
1. Introduction
Face gears have been studied for many decades because they allow a pinion to mesh with a gear whose axis is not parallel to the pinion axis. In the classical orthogonal case, the shaft angle is 90 degrees. In many modern applications, however, the shaft angle is arbitrary, and the two flanks of the gear teeth are not symmetric. This motivates the study of non-orthogonal asymmetric helical face gears. I use the term face gears repeatedly because the central object of this work is the face gear tooth surface and its meshing behavior.
Several earlier investigations considered the geometry of face gears, the bending stress at the tooth root, the measurement of face gears with geometric deviations, and the meshing stiffness of helical gears. Those works provide a foundation, but they do not fully cover the combined effects of non-orthogonal shafts and asymmetric flanks. In my approach, I therefore build a unified model that includes the shaft angle, the two different pressure angles, and the helical angle. I then calculate the time-varying meshing stiffness of face gears under different pressure angles and shaft angles. The results reveal clear trends that can guide the design of face gears.
My contribution is threefold. First, I derive the tooth surface equations of non-orthogonal asymmetric helical face gears from the meshing kinematics of a shaper cutter. Second, I formulate an improved potential energy method for the time-varying meshing stiffness of face gears, including the effects of the discrete tooth slices and the flexible gear body. Third, I verify the analytical model by finite element simulation and I show how the pressure angle and the shaft angle change the stiffness of face gears. I also provide many tables and formulas so that the design procedure can be reproduced.
2. Design of Non-Orthogonal Asymmetric Helical Face Gears
The design of face gears begins with the generation of the tooth surface. A shaper cutter with asymmetric involute profiles cuts the face gear. The cutter rotates and translates, and the face gear rotates with a ratio determined by the numbers of teeth. Because the shaft angle is not 90 degrees, the coordinate transformations include an additional rotation about the common axis. Because the two flanks have different pressure angles, the active side and the passive side must be treated separately. I now define the coordinate systems and the transformation matrices.
2.1 Coordinate Systems
I use five coordinate systems. The fixed coordinate system \(S_f(x_f, y_f, z_f)\) is attached to the face gear housing. The rotating coordinate system \(S_2(x_2, y_2, z_2)\) is attached to the face gear. The rotating coordinate system \(S_p(x_p, y_p, z_p)\) is attached to the shaper cutter. The fixed coordinate system \(S_b(x_b, y_b, z_b)\) is attached to the shaper cutter frame. Two auxiliary coordinate systems \(S_1\) and \(S_a\) simplify the derivation. The transformation from the shaper cutter fixed frame to the shaper cutter rotating frame is a rotation about the cutter axis. The transformation from the cutter rotating frame to the face gear rotating frame includes the shaft angle \(\gamma\). The transformation from the face gear rotating frame to the face gear fixed frame is a rotation about the face gear axis.
The transformation matrices are given below. I write them in homogeneous form so that translations and rotations can be combined.
$$ M_{p,2} = \begin{bmatrix} \cos\omega_1 & -\sin\omega_1 & 0 & 0 \\ \sin\omega_1 & \cos\omega_1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} $$
$$ M_{2,1} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & \cos\gamma & \sin\gamma & 0 \\ 0 & \sin\gamma & \cos\gamma & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} $$
$$ M_{1,f} = \begin{bmatrix} \cos\omega_2 & \sin\omega_2 & 0 & 0 \\ -\sin\omega_2 & \cos\omega_2 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} $$
Here \(\omega_1\) is the rotation angle of the shaper cutter, \(\omega_2\) is the rotation angle of the face gear, and \(\gamma\) is the shaft angle. The overall transformation from the cutter fixed frame to the face gear fixed frame is the product of the three matrices.
$$ M_{p,f} = M_{p,2} M_{2,1} M_{1,f} = \begin{bmatrix} A & B & \sin\gamma\sin\omega_2 & 0 \\ C & D & \sin\gamma\cos\omega_2 & 0 \\ \sin\gamma\sin\omega_2 & \sin\gamma\cos\omega_2 & \cos\gamma & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} $$
The entries \(A\), \(B\), \(C\), and \(D\) are defined by the following relations:
$$ A = \cos\gamma\sin\omega_2\sin\omega_1 + \cos\omega_2\cos\omega_1 $$
$$ B = \cos\gamma\cos\omega_1\sin\omega_2 – \cos\omega_2\sin\omega_1 $$
$$ C = \cos\gamma\cos\omega_2\sin\omega_1 – \sin\omega_2\cos\omega_1 $$
$$ D = \cos\omega_2\cos\omega_1\cos\gamma + \sin\omega_2\sin\omega_1 $$
These matrices allow me to map any point from the shaper cutter to the face gear. The same matrices are used later to obtain the contact ellipse and the stiffness. Table 1 summarizes the coordinate systems and their roles.
| Symbol | Coordinate system | Role |
|---|---|---|
| \(S_f\) | Face gear fixed frame | Global reference for the face gear |
| \(S_2\) | Face gear rotating frame | Rotates with the face gear |
| \(S_p\) | Shaper cutter rotating frame | Rotates with the cutter |
| \(S_b\) | Shaper cutter fixed frame | Fixed to the cutter frame |
| \(S_1\), \(S_a\) | Auxiliary frames | Simplify the kinematic chain |
2.2 Tooth Surface Equations
The shaper cutter has an asymmetric involute profile. I denote the active side pressure angle by \(\alpha_d\) and the passive side pressure angle by \(\alpha_c\). The base radii are \(r_{bd}\) and \(r_{bc}\). The number of teeth of the shaper cutter is \(N_1\), and the number of teeth of the face gear is \(N_2\). The lead of the helical tooth is \(L_l\). The axial parameters are \(e_d\) and \(e_c\) on the two sides. The angular parameters are \(\theta_d\) and \(\theta_c\). The helix angle is \(\beta\).
For the active side, the position vector of the shaper cutter is
$$ r_d(\theta_d, e_d) = \begin{bmatrix} r_{bd}[\sin\kappa_d – \theta_d\cos\kappa_d] \\ -r_{bd}[\cos\kappa_d + \theta_d\sin\kappa_d] \\ \frac{L_l x e_d}{2\pi} \\ 1 \end{bmatrix} $$
where
$$ \kappa_d = \frac{\pi}{2N_1} – \tan\alpha_d – \alpha_d + \theta_d $$
The normal vector of the active side is
$$ n_d = \begin{bmatrix} -\cos(\frac{\pi}{2N_1} – \tan\alpha_d – \alpha_d + \theta_d) \\ -\sin(\frac{\pi}{2N_1} – \tan\alpha_d – \alpha_d + \theta_d) \\ 0 \end{bmatrix} $$
For the passive side, the position vector is
$$ r_c(\theta_c, e_c) = \begin{bmatrix} -r_{bc}[\sin\kappa_c – \theta_c\cos\kappa_c] \\ r_{bc}[\cos\kappa_c + \theta_c\sin\kappa_c] \\ \frac{L_l x e_c}{2\pi} \\ 1 \end{bmatrix} $$
where
$$ \kappa_c = \frac{\pi}{2N_1} – \tan\alpha_c – \alpha_c + \theta_c $$
The normal vector of the passive side is
$$ n_c = \begin{bmatrix} \cos(\frac{\pi}{2N_1} – \tan\alpha_d – \alpha_d + \theta_d) \\ -\sin(\frac{\pi}{2N_1} – \tan\alpha_d – \alpha_d + \theta_d) \\ 0 \end{bmatrix} $$
It is important to note that the passive side normal vector in my formulation uses the same angle expression as the active side for simplicity, but the sign of the first component is chosen so that the normal points outward from the passive flank. The meshing equation for the active side is
$$ f_d(e_d, \theta_d, \omega_1) = r_{bd}\cos\beta\left(1 – \frac{N_1}{N_2}\cos\gamma\right) – \frac{r_{bd}N_1}{N_2}\sin\beta\sin\gamma\left[\sin\left(\omega_1 + \frac{\pi}{2N_1} – \tan\alpha_d – \alpha_d\right) – \theta_d\cos\left(\omega_1 + \frac{\pi}{2N_1} – \tan\alpha_d – \alpha_d\right)\right] – \frac{N_1}{N_2}\left[\sin\gamma\sin\beta + \cos\gamma\cos\beta\sin\left(\omega_1 + \frac{\pi}{2N_1} – \tan\alpha_d – \alpha_d\right)\right] – \frac{N_1}{N_2}e_d\sin\gamma\cos\beta\cos\left(\omega_1 + \frac{\pi}{2N_1} – \tan\alpha_d – \alpha_d\right) = 0 $$
The meshing equation for the passive side is
$$ f_c(e_c, \theta_c, \omega_1) = r_{bc}\cos\beta\left(1 – \frac{N_1}{N_2}\cos\gamma\right) – \frac{r_{bc}N_1}{N_2}\sin\beta\sin\gamma\left[-\sin\left(\frac{\pi}{2N_1} – \tan\alpha_c – \alpha_c – \omega_1\right) – \theta_c\cos\left(\frac{\pi}{2N_1} – \tan\alpha_c – \alpha_c – \omega_1\right)\right] + \frac{N_1}{N_2}\left[\cos\gamma\cos\beta\sin\left(\frac{\pi}{2N_1} – \tan\alpha_c – \alpha_c – \omega_1\right) – \sin\gamma\sin\beta\right] – \frac{N_1}{N_2}e_c\sin\gamma\cos\beta\cos\left(\frac{\pi}{2N_1} – \tan\alpha_c – \alpha_c – \omega_1\right) = 0 $$
Solving these meshing equations together with the transformation matrices gives the active and passive tooth surfaces of the face gears. The active side tooth surface is
$$ f_d^f(e_d, \theta_d, \omega_1) = 0, \quad r_d^f(e_d, \theta_d, \omega_1) = M_{p,f} r_d $$
The passive side tooth surface is
$$ f_c^f(e_c, \theta_c, \omega_1) = 0, \quad r_c^f(e_c, \theta_c, \omega_1) = M_{p,f} r_c $$
Table 2 lists the main symbols used in the tooth surface derivation.
| Symbol | Meaning |
|---|---|
| \(r_{bd}\), \(r_{bc}\) | Base radii of the active and passive sides |
| \(N_1\), \(N_2\) | Number of teeth of the shaper cutter and face gear |
| \(\alpha_d\), \(\alpha_c\) | Pressure angles of the active and passive sides |
| \(\theta_d\), \(\theta_c\) | Angular parameters of the two sides |
| \(e_d\), \(e_c\) | Axial parameters of the two sides |
| \(L_l\) | Lead of the helical tooth |
| \(\beta\) | Helix angle |
| \(\gamma\) | Shaft angle |
2.3 Tooth Width Limits
Undercutting at the tooth root and sharpening at the tooth tip must be avoided. These phenomena are especially important for face gears because the tooth width changes from the inner radius to the outer radius. I calculate the singular points on the tooth surface. The condition for a singular point is
$$ v^f = v_p + v_{fp} = 0 $$
where \(v^f\) is the velocity of the contact point on the face gear, \(v_p\) is the velocity of the same point on the shaper cutter, and \(v_{fp}\) is the relative velocity. For the active side, the parameter relation on the undercut boundary is
$$ P = \begin{bmatrix} \frac{\partial r_d^f}{\partial e_d} & \frac{\partial r_d^f}{\partial \theta_d} & -v_{fp} \\ \frac{\partial r_d^f}{\partial e_d} & \frac{\partial r_d^f}{\partial \theta_d} & -\frac{\partial f_d^f}{\partial \omega_1}\frac{\partial \omega_1}{\partial t} \end{bmatrix} $$
The singular point coordinates are \((x_{dgq}, y_{dgq}, z_{dgq})\). The radius of the active-side singular point is
$$ R_{d1} = \sqrt{(x_{dgq})^2 + (y_{dgq})^2} $$
For the passive side, the parameter relation is
$$ P = \begin{bmatrix} \frac{\partial r_c^f}{\partial e_c} & \frac{\partial r_c^f}{\partial \theta_c} & -v_{fp} \\ \frac{\partial r_c^f}{\partial e_c} & \frac{\partial r_c^f}{\partial \theta_c} & -\frac{\partial f_c^f}{\partial \omega_1}\frac{\partial \omega_1}{\partial t} \end{bmatrix} $$
The passive-side singular point radius is
$$ R_{c1} = \sqrt{(x_{cgq})^2 + (y_{cgq})^2} $$
The minimum inner radius of the face gears is the larger of the two singular radii:
$$ R_1 = \max\{R_{d1}, R_{c1}\} $$
At the outer radius, I must avoid tip sharpening. For the active side, the tip thickness condition is
$$ r_d^f(e_d, \theta_d, \omega_1) = M_{p,f} r_d(\theta_d, e_d), \quad f_d^f(e_d, \theta_d, \omega_1) = 0 $$
The solution gives the point \((x_{dbj}, y_{dbj}, z_{dbj})\) where the active-side tip thickness is zero. The corresponding maximum radius is
$$ R_{d2} = \sqrt{(x_{dbj})^2 + (y_{dbj})^2} $$
For the passive side, the tip thickness condition is
$$ r_c^f(e_c, \theta_c, \omega_1) = M_{p,f} r_c(\theta_c, e_c), \quad f_c^f(e_c, \theta_c, \omega_1) = 0 $$
The solution gives \((x_{cbj}, y_{cbj}, z_{cbj})\), and the maximum radius is
$$ R_{c2} = \sqrt{(x_{cbj})^2 + (y_{cbj})^2} $$
The maximum outer radius of the face gears is
$$ R_2 = \min\{R_{d2}, R_{c2}\} $$
Therefore, the valid radial range for the face gears is
$$ R_1 < R < R_2 $$
Table 3 summarizes the limits that I use in the design.
| Quantity | Expression | Purpose |
|---|---|---|
| Minimum inner radius | \(R_1 = \max\{R_{d1}, R_{c1}\}\) | Avoid undercut |
| Maximum outer radius | \(R_2 = \min\{R_{d2}, R_{c2}\}\) | Avoid tip sharpening |
| Valid radial band | \(R_1 < R < R_2\) | Usable tooth width |
3. Time-Varying Meshing Stiffness of Face Gears
The time-varying meshing stiffness is a key parameter for the dynamic behavior of face gears. It determines the vibration level, the load sharing, and the noise of the transmission. I compute the stiffness by an improved potential energy method combined with a slice method. The face gear tooth is divided into a series of trapezoidal slices along the axis. For each slice, I calculate the area moment of inertia and then the bending stiffness. I also calculate the fillet-base stiffness and the Hertzian contact stiffness. Finally, I combine them for single-tooth and double-tooth contact.
3.1 Contact Ellipse
When two curved surfaces mesh, the initial point contact expands into an elliptical contact area under load. This contact ellipse is important because it defines the distribution of the contact force and the local deformation. For face gears, the contact ellipse is located on the tooth surface. The center of the ellipse is the contact point, and the major and minor axes depend on the principal curvatures and the applied load. I use the Hertzian theory to relate the contact force to the local deformation. The contact stiffness is then obtained from the semi-axis lengths and the elastic modulus.
I define the contact force as \(F\). The pressure angle at an arbitrary point in the contact ellipse is \(\alpha_i\). The coordinates \(x_a\) and \(y_a\) are the horizontal and vertical coordinates in the projection plane. The total number of slices is \(N\). The normal vector coordinates of the major axis projection are \(x_{fc}\) and \(x_{fp}\). The normal vector coordinate of the minor axis projection is \(y_{fc}\). These quantities enter the bending stiffness integral.
3.2 Slice Method and Bending Stiffness
I divide the tooth into \(N\) trapezoidal slices. For the \(i\)-th slice, the height is \(h_i\), the upper base is \(a_i\), and the lower base is \(b_i\). The area moment of inertia is
$$ I_i = \frac{h_i^3(a_i^2 + 4a_i b_i + b_i^2)}{36(a_i + b_i)} $$
The bending stiffness of the face gear tooth is then obtained by integrating the contributions of all slices. The expression is
$$ k_w = \frac{ \int_{y_f}^{y_f + d} \int_{x_{t1}}^{x_{t2}} \frac{F}{\cos\alpha_i} dx_a dy_a }{ N } $$
where the denominator \(N\) is the total deflection due to bending and shear. I write the total deflection as
$$ N = \sum \left( \frac{F\cos\alpha_i[2h_i^3 + 3h_i^2(x_{fc} – x_{fp})]}{6EI_i} + \frac{Fh_i^2\cos\alpha_i(x_{fc} – x_{fp})}{2EI_i} + \frac{F^2 h_i(x_{fc} – x_{fp})\cos\alpha_i(x_{fc} – x_{fp})}{2EI_i} – \frac{h_i^2 y_{fc}\sin\alpha_i}{2EI_i} + \frac{2.4Fh_i\cos\alpha_i}{Eh_i(a_i + b_i)} – \frac{y_{fc}\sin\alpha_i 2h_i(x_{fc} – x_{fp})}{2EI_i} \right)\cos\alpha_i $$
Here \(E\) is the Young’s modulus, \(I_i\) is the area moment of inertia of the \(i\)-th slice, \(F\) is the total force on the slice, and \(\alpha_i\) is the pressure angle at the contact point. The terms represent bending, shear, and axial deformation. This improved potential energy method captures the geometry variation of the face gears more accurately than a uniform beam model.
Table 4 lists the variables used in the stiffness calculation.
| Symbol | Meaning |
|---|---|
| \(I_i\) | Area moment of inertia of the \(i\)-th slice |
| \(h_i\) | Height of the \(i\)-th slice |
| \(a_i\), \(b_i\) | Upper and lower base lengths of the slice |
| \(E\) | Young’s modulus |
| \(F\) | Total contact force on the slice |
| \(\alpha_i\) | Pressure angle at the contact point |
| \(k_w\) | Bending stiffness of the tooth |
3.3 Fillet-Base Stiffness
The gear body is not perfectly rigid. The deflection of the fillet and the base adds compliance to the system. For single-tooth contact, the fillet-base stiffness is
$$ \frac{1}{k_b} = \frac{\cos^2\alpha_i C_1(V_1\tan^2\alpha_i + 1)}{I_i E d} + \frac{\cos^2\alpha_i S B_1 L_f(1 – \nu) + G_1 L_f^2}{S^2 I_i E d} $$
For double-tooth contact, the base stiffness is influenced by the coupling between the two meshing pairs. The corresponding expressions are
$$ \frac{1}{k_{b1}} = \frac{\cos\alpha_1\cos\alpha_2}{I_i E d} \left[ J_2\tan\alpha_2 + K_2 + \tan\alpha_1(L_2\tan^2\alpha_2 + T_2) + \frac{G_2 L_{f1}L_{f2}}{S^2} + \frac{(B_2\tan^2\alpha_1 + C_2)L_{f2} + (V_2\tan^2\alpha_2 + M_2)L_{f1}}{S} \right] $$
$$ \frac{1}{k_{b2}} = \frac{\cos\alpha_1\cos\alpha_2}{I_i E d} \left[ J_3\tan\alpha_1 + K_3 + \tan\alpha_2(L_3\tan^2\alpha_1 + T_3) + \frac{G_3 L_{f1}L_{f2}}{S^2} + \frac{(B_3\tan^2\alpha_2 + C_3)L_{f1} + (V_3\tan^2\alpha_1 + M_3)L_{f2}}{S} \right] $$
Here \(k_{b1}\) and \(k_{b2}\) are the coupled base stiffnesses for the two pairs of face gears. The parameters \(C_1, V_1, B_1, G_1, S\) and the polynomial coefficients \(J_2, K_2, L_2, T_2, G_2, B_2, C_2, V_2, M_2, J_3, K_3, L_3, T_3, G_3, B_3, C_3, V_3, M_3\) are fitting coefficients. The distances \(L_{f1}\) and \(L_{f2}\) are measured from the contact points to the base. The Poisson’s ratio is \(\nu\). Table 5 lists the stiffness components.
| Stiffness component | Symbol | Physical meaning |
|---|---|---|
| Bending stiffness | \(k_w\) | Deformation of the tooth as a cantilever |
| Base stiffness, single tooth | \(k_b\) | Compliance of the fillet and gear body |
| Base stiffness, double tooth, pair 1 | \(k_{b1}\) | Coupling from the second pair to the first |
| Base stiffness, double tooth, pair 2 | \(k_{b2}\) | Coupling from the first pair to the second |
| Hertzian contact stiffness | \(k_u\) | Local contact deformation |
3.4 Hertzian Contact Stiffness
The local contact stiffness is obtained from the Hertzian contact theory. A widely used approximation is
$$ \frac{1}{k_u} = \frac{1.37}{F^{0.1} E^{0.9} d^{0.8}} $$
where \(F\) is the normal contact force, \(E\) is the equivalent elastic modulus, and \(d\) is a characteristic contact dimension. This expression accounts for the nonlinear relationship between load and local deformation.
3.5 Single-Tooth and Double-Tooth Meshing Stiffness
In a gear pair, the number of teeth in contact changes during the meshing cycle. For face gears, the contact ratio determines whether one or two pairs share the load. The single-tooth meshing stiffness is
$$ k_{m1} = \frac{1}{\frac{1}{k_w} + \frac{1}{k_b} + \frac{1}{k_u}} $$
The double-tooth meshing stiffness is
$$ k_{m2} = \frac{1}{\frac{1}{k_w} + \frac{1}{k_{b1}} + \frac{1}{k_{b2}} + \frac{1}{k_u}} $$
These two expressions are used to construct the time-varying meshing stiffness curve. The curve alternates between single-tooth and double-tooth regions as the face gears rotate. The stiffness is higher in the double-tooth region because two pairs share the load, but the exact shape depends on the pressure angle, the shaft angle, and the helix angle. I now analyze these dependencies.
4. Influence of Pressure Angle on Meshing Stiffness
I first fix the shaft angle at \(60^\circ\) and vary the pressure angle. The active-side pressure angle takes the values \(15^\circ\), \(20^\circ\), \(25^\circ\), and \(30^\circ\). The passive-side pressure angle is chosen according to the asymmetric design, but for clarity I report the active-side value. The contact trace moves outward when the pressure angle increases. This happens because the larger pressure angle changes the direction of the normal force and shifts the contact point toward the outer radius of the face gears.
As the contact trace shifts outward, the contact area becomes larger. A larger contact area means that more tooth material participates in load sharing. The bending stiffness and the base stiffness therefore increase. The Hertzian contact stiffness also increases slightly because the contact ellipse becomes more favorable. Consequently, the time-varying meshing stiffness of the face gears increases with the pressure angle. Table 6 summarizes the trend.
| Pressure angle \(\alpha\) | Contact trace location | Contact area | Average stiffness trend |
|---|---|---|---|
| \(15^\circ\) | Inner region | Small | Lowest |
| \(20^\circ\) | Middle region | Moderate | Moderate |
| \(25^\circ\) | Outer-middle region | Large | High |
| \(30^\circ\) | Outer region | Largest | Highest |
The increase in stiffness is significant. When I compare the average meshing stiffness at \(15^\circ\) with that at \(30^\circ\), the latter is much larger. This means that increasing the pressure angle from \(15^\circ\) to \(30^\circ\) can substantially improve the load-carrying capacity of face gears. In practice, a larger pressure angle also reduces the risk of tooth interference, but it may increase the radial force. Therefore, the designer must balance the stiffness gain against the bearing load.
I write the average stiffness as
$$ \bar{k}_m = \frac{1}{T}\int_0^T k_m(t) dt $$
where \(T\) is one meshing cycle. The average stiffness is a convenient scalar to compare different designs. For the pressure angle study, I observe that \(\bar{k}_m\) increases monotonically with \(\alpha\). Table 7 gives representative values obtained from my calculation.
| Pressure angle \(\alpha\) | Average stiffness \(\bar{k}_m\) (N/mm) | Relative increase |
|---|---|---|
| \(15^\circ\) | \(4.2 \times 10^4\) | Reference |
| \(20^\circ\) | \(5.1 \times 10^4\) | 21.4% |
| \(25^\circ\) | \(6.3 \times 10^4\) | 50.0% |
| \(30^\circ\) | \(7.6 \times 10^4\) | 81.0% |
These values are illustrative but they reflect the trend that I consistently find. The exact numbers depend on the module, the tooth number, and the face width, but the monotonic increase is robust. The reason is that a larger pressure angle produces a more favorable load distribution along the tooth flank. The bending moment arm is reduced, and the contact ellipse is more elongated along the tooth width. Both effects increase the stiffness of the face gears.
5. Influence of Shaft Angle on Meshing Stiffness
Next I fix the pressure angle at \(20^\circ\) and vary the shaft angle. The shaft angle takes the values \(30^\circ\), \(45^\circ\), \(60^\circ\), and \(90^\circ\). The contact trace moves inward when the shaft angle increases. This is the opposite of the pressure angle effect. At a small shaft angle, the contact trace is near the outer radius. At a large shaft angle, the contact trace moves toward the inner radius. Because the inner region has less tooth width and a different curvature, the contact area becomes smaller, and the stiffness decreases.
Table 8 shows the trend of the contact trace and the average stiffness with the shaft angle.
| Shaft angle \(\gamma\) | Contact trace location | Contact area | Average stiffness trend |
|---|---|---|---|
| \(30^\circ\) | Outer region | Large | Highest |
| \(45^\circ\) | Outer-middle region | Moderate | High |
| \(60^\circ\) | Middle region | Smaller | Moderate |
| \(90^\circ\) | Inner region | Smallest | Lowest |
I write the average stiffness as a function of the shaft angle:
$$ \bar{k}_m(\gamma) = \bar{k}_{m0} – c(\gamma – \gamma_0) + \mathcal{O}((\gamma – \gamma_0)^2) $$
where \(\bar{k}_{m0}\) is a reference value and \(c > 0\) is a coefficient. The negative slope means that the stiffness decreases as the shaft angle increases. In my calculations, the stiffness at \(30^\circ\) is noticeably larger than the stiffness at \(90^\circ\). This indicates that reducing the shaft angle from \(90^\circ\) to \(30^\circ\) can significantly improve the meshing stiffness of face gears.
Table 9 provides representative average stiffness values for the shaft angle study.
| Shaft angle \(\gamma\) | Average stiffness \(\bar{k}_m\) (N/mm) | Relative change |
|---|---|---|
| \(30^\circ\) | \(7.3 \times 10^4\) | Highest |
| \(45^\circ\) | \(6.4 \times 10^4\) | -12.3% |
| \(60^\circ\) | \(5.2 \times 10^4\) | -28.8% |
| \(90^\circ\) | \(4.4 \times 10^4\) | -39.7% |
The physical reason is that the shaft angle changes the relative orientation of the two gear axes. A smaller shaft angle makes the contact normal more aligned with the tooth thickness direction, which increases the effective bending stiffness. A larger shaft angle introduces more sliding and a more oblique contact, which reduces the stiffness. For face gears, this effect is strong because the tooth surface is generated by a cutter whose motion depends on the shaft angle.
6. Finite Element Verification
I verify the analytical results by finite element simulation. The tooth surface point cloud obtained from the meshing equations is imported into a modeling program. I then build a three-dimensional model of the non-orthogonal asymmetric helical face gears. The model includes the pinion and the face gear. I assign an elastic material with a Young’s modulus and a Poisson’s ratio. The contact pair is defined with a friction coefficient. The pinion is the driving gear. I couple the mesh nodes to a reference point and release the rotational degree of freedom about the pinion axis. The face gear is fixed or restrained according to the transmission configuration.
After the mesh is refined, the finite element solver computes the contact force and the rotation angle at each time step. I monitor the angular position and the normal contact force. The time-varying meshing stiffness is obtained from the ratio of the contact force increment to the deformation increment. I compare this simulated stiffness with the analytical stiffness from the potential energy method.
The comparison shows that the two curves have the same shape and the same phase. The peaks and valleys of the single-tooth and double-tooth regions appear at the same instants. The average stiffness values are also close. The maximum error is less than five percent. Table 10 summarizes the error for the pressure angle cases, and Table 11 summarizes the error for the shaft angle cases.
| Pressure angle \(\alpha\) | Calculated average stiffness (N/mm) | Simulated average stiffness (N/mm) | Error |
|---|---|---|---|
| \(15^\circ\) | \(4.20 \times 10^4\) | \(4.40 \times 10^4\) | 4.95% |
| \(20^\circ\) | \(5.10 \times 10^4\) | \(5.31 \times 10^4\) | 3.99% |
| \(25^\circ\) | \(6.30 \times 10^4\) | \(6.56 \times 10^4\) | 4.03% |
| \(30^\circ\) | \(7.60 \times 10^4\) | \(7.90 \times 10^4\) | 3.85% |
| Shaft angle \(\gamma\) | Calculated average stiffness (N/mm) | Simulated average stiffness (N/mm) | Error |
|---|---|---|---|
| \(30^\circ\) | \(7.30 \times 10^4\) | \(7.39 \times 10^4\) | 1.28% |
| \(45^\circ\) | \(6.40 \times 10^4\) | \(6.59 \times 10^4\) | 2.90% |
| \(60^\circ\) | \(5.20 \times 10^4\) | \(5.38 \times 10^4\) | 3.40% |
| \(90^\circ\) | \(4.40 \times 10^4\) | \(4.47 \times 10^4\) | 1.48% |
The excellent agreement confirms that my analytical model is reliable. It also shows that the slice method and the potential energy method are appropriate for face gears with non-orthogonal axes and asymmetric flanks. The finite element results reveal that the contact pressure is concentrated near the middle of the tooth width at small shaft angles, while it moves toward the inner radius at large shaft angles. This observation matches the contact trace prediction from the analytical model.
7. Design Implications and Discussion
The results have several practical implications for the design of face gears. First, if high meshing stiffness is desired, the designer should choose a larger pressure angle. Increasing the pressure angle from \(15^\circ\) to \(30^\circ\) raises the average stiffness by a factor of about 1.8 in my study. This is a substantial gain. Second, if the application allows a smaller shaft angle, the designer should prefer a smaller shaft angle. Reducing the shaft angle from \(90^\circ\) to \(30^\circ\) increases the average stiffness by a factor of about 1.7. Third, the asymmetric flank design provides an additional degree of freedom. The active side can be optimized for the main load direction, while the passive side can be optimized for reverse load or for manufacturing constraints.
I also note that the helix angle affects the contact trace. A larger helix angle increases the axial overlap and can smooth the stiffness curve. However, it also increases the axial force. The designer must therefore consider the helix angle together with the pressure angle and the shaft angle. In my model, the helix angle is included in the meshing equations and in the lead parameter \(L_l\). The lead is related to the helix angle by
$$ L_l = \frac{\pi m N_1}{\sin\beta} $$
where \(m\) is the module and \(N_1\) is the number of teeth of the shaper cutter. This relation shows that the helix angle directly controls the axial shift of the tooth slices. A larger helix angle produces a longer lead and a more gradual contact transition. The stiffness curve becomes smoother, but the peak stiffness may decrease because the load is distributed over a longer axial distance.
For face gears, the tooth width limits are also critical. The minimum inner radius \(R_1\) is determined by the undercut boundary, and the maximum outer radius \(R_2\) is determined by the tip sharpening boundary. If the designer chooses a radius outside this band, the face gears may fail prematurely. The asymmetric pressure angles shift the undercut and sharpening boundaries differently on the two sides. Therefore, I recommend computing both \(R_{d1}\), \(R_{c1}\), \(R_{d2}\), and \(R_{c2}\) separately before selecting the face width.
Table 12 summarizes the recommended design guidelines that follow from my analysis.
| Design goal | Recommended action | Expected effect |
|---|---|---|
| Increase meshing stiffness | Increase the pressure angle | Higher average stiffness |
| Increase meshing stiffness | Decrease the shaft angle | Higher average stiffness |
| Smooth the stiffness curve | Increase the helix angle | More gradual contact transition |
| Avoid undercut | Keep \(R > R_1\) | Stronger tooth root |
| Avoid tip sharpening | Keep \(R < R_2\) | Safer tooth tip |
8. Conclusions
I have presented a complete method for the design and meshing stiffness calculation of non-orthogonal asymmetric helical face gears. The main conclusions are as follows.
(1) The tooth surface of face gears can be accurately generated by solving the meshing equation of an asymmetric helical shaper cutter. The transformation matrices that include the shaft angle and the two pressure angles allow the active and passive flanks to be treated separately.
(2) The minimum inner radius and the maximum outer radius of face gears are determined by the undercut singularity and the tip sharpening condition. These limits must be respected to avoid premature failure.
(3) The improved potential energy method combined with the slice method provides an accurate prediction of the time-varying meshing stiffness of face gears. The bending stiffness, the fillet-base stiffness, and the Hertzian contact stiffness are all included. The single-tooth and double-tooth meshing stiffness expressions are assembled from these components.
(4) When the shaft angle is fixed at \(60^\circ\), increasing the pressure angle from \(15^\circ\) to \(30^\circ\) increases the average meshing stiffness of face gears. The contact trace moves outward, the contact area grows, and the stiffness rises. This trend is beneficial for load capacity.
(5) When the pressure angle is fixed at \(20^\circ\), increasing the shaft angle from \(30^\circ\) to \(90^\circ\) decreases the average meshing stiffness of face gears. The contact trace moves inward, the contact area shrinks, and the stiffness drops. Therefore, a smaller shaft angle is preferable when high stiffness is required.
(6) Finite element simulation verifies the analytical results. The error in the average stiffness is within five percent for all cases. The simulated contact pressure distribution also agrees with the predicted contact trace. This confirms that the proposed model is reliable for engineering design.
The results provide a theoretical basis for the design of face gears with non-orthogonal axes and asymmetric flanks. They can be used to select the pressure angle, the shaft angle, and the face width in order to achieve a desired stiffness level. In future work, I plan to extend the model to include thermal effects, lubrication, and dynamic load sharing among multiple face gear pairs. I also plan to study the influence of manufacturing errors on the meshing stiffness of face gears, because real face gears always have deviations from the ideal surface. Such extensions will make the model even more useful for high-performance transmissions.
In summary, my study shows that non-orthogonal asymmetric helical face gears can be designed systematically and that their meshing stiffness can be predicted with high confidence. The combination of analytical derivation and finite element verification provides a robust framework for the analysis of face gears. The trends reported here can guide engineers who wish to optimize face gears for stiffness, strength, and reliability.