In the pursuit of reducing vibration and noise in high-speed and heavy-duty gear transmissions, the design of gear tooth surface modifications has become a critical area of research. Among various modification strategies, diagonal end relief (also known as triangular end relief) has demonstrated significant potential in mitigating the fluctuation of loaded transmission error (LTE), which is a primary excitation source for gear dynamics. This article presents a comprehensive study on the optimal design and grinding principle of diagonal modified helical gears using a large plane grinding wheel. The work integrates analytical modeling, numerical optimization, and manufacturing feasibility, with a particular focus on the mathematical representation of modified tooth surfaces, the minimization of LTE fluctuations via genetic algorithms, and the derivation of the grinding process based on the concept of an imaginary generating rack.
Our investigation begins with the geometric definition of diagonal modification on helical gears. Unlike traditional profile or lead modifications that remove material across the entire tooth flank, diagonal modification removes material only at the tooth tip and root regions near the engaging and disengaging ends, leaving the central part of the tooth largely unchanged. This preserves a higher contact ratio and load-carrying area. The modification parameters include the tip relief height \( L_{Ea} \), tip relief amount \( C_{Ea} \), root relief height \( L_{Ef} \), and root relief amount \( C_{Ef} \). The modification start lines are defined as the lines where the modified surface transitions to the standard involute helicoid. These start lines in the rotational projection plane are approximated as straight lines to simplify calculation.
The standard involute tooth surface of a helical gear can be expressed as a function of surface parameters \( u_1, \theta_1 \). The position vector and unit normal vector of the unmodified helical gear tooth surface are denoted as \( \mathbf{r}_1(u_1, \theta_1) \) and \( \mathbf{n}_1(u_1, \theta_1) \), respectively. The coordinates of the start points B and C (tip modification start) on the tooth surface are found by solving the nonlinear system:
\[
\begin{cases}
x_{1i}^2(u_{1i}, \theta_{1i}) + y_{1i}^2(u_{1i}, \theta_{1i}) = R_i, \\
z_{1i}(u_{1i}, \theta_{1i}) = L_i, \quad i = B, C
\end{cases}
\]
Similarly, the root modification start points E and F are determined. The tip modification termination is at the addendum circle, and the root modification termination is at the start of active profile (SAP) radius \( r_{k1} \), which is calculated from the gear geometry.
Once the start lines are established, any point P in the modification triangle is associated with a modification length \( l_p \), measured perpendicularly from P to the start line (or its extension). The modification amount at P is defined as a function of the order of the modification curve:
\[
\delta(x,y) =
\begin{cases}
C_{Ea} \left( \frac{l_p}{l_a} \right)^{k_a}, & P \in \triangle ABC \\
C_{Ef} \left( \frac{l_p}{l_f} \right)^{k_f}, & P \in \triangle DEF \\
0, & \text{otherwise}
\end{cases}
\]
where \( k_a \) and \( k_f \) are the exponents (order) of the modification curve; typical values are 1 (linear), 2 (parabolic), or 4 (quartic). This formulation allows for a smooth blending of the modified region with the unmodified region. The final modified tooth surface is obtained by superimposing the modification amount in the normal direction onto the standard helicoid:
\[
\mathbf{r}_{1m}(u_1, \theta_1) = \mathbf{r}_1(u_1, \theta_1) + \delta(x,y) \,\mathbf{n}_1(u_1, \theta_1)
\]
\[
\mathbf{n}_{1m}(u_1, \theta_1) = \left( \frac{\partial \mathbf{r}_1}{\partial u_1} + \frac{\partial \delta}{\partial u_1} \mathbf{n}_1 \right) \times \left( \frac{\partial \mathbf{r}_1}{\partial \theta_1} + \frac{\partial \delta}{\partial \theta_1} \mathbf{n}_1 \right)
\]
The primary objective of tooth surface modification for helical gears is to reduce the fluctuation of the loaded transmission error. The loaded transmission error \( T_e \) is derived from tooth contact analysis (TCA) and loaded tooth contact analysis (LTCA). Using a finite element based or semi-analytical approach, the normal compliance of the tooth pair is computed, and the load distribution along the contact lines is solved. The transmission error fluctuation is defined as:
\[
\Delta T_e = \max(T_e) – \min(T_e)
\]
We formulate an optimization problem whose objective is to minimize \( \Delta T_e \) by varying the four diagonal modification design variables: \( L_{Ea}, C_{Ea}, L_{Ef}, C_{Ef} \). Note that the heights \( L_{Ea} \) and \( L_{Ef} \) are directly linked to the start line positions. The order exponents \( k_a, k_f \) are treated as discrete parameters (1, 2, 4) in separate runs. Because the objective function cannot be expressed analytically and the design space contains multiple local minima, we adopt a genetic algorithm (GA) for global optimization.
The GA is configured with a population size of 50, 30 generations, crossover probability 0.6, and mutation probability 0.1. For each candidate solution, we generate the theoretical modified tooth surface, perform TCA to obtain the unloaded transmission error and contact pattern, then perform LTCA under rated torque to compute the loaded transmission error. The fitness is defined as the negative of \( \Delta T_e \). The optimization is repeated for three cases: linear (\( k=1 \)), parabolic (\( k=2 \)), and quartic (\( k=4 \)) modification curves. The results are summarized in Table 1.
| Parameter | Linear (k=1) | Parabolic (k=2) | Quartic (k=4) |
|---|---|---|---|
| Tip relief height \( L_{Ea} \) (mm) |
2.366 | 4.887 | 6.818 |
| Tip relief length \( b_{Ea} \) (mm) |
8.272 | 18.050 | 26.555 |
| Max tip relief amount \( C_{Ea} \) (μm) |
15.98 | 9.32 | 11.15 |
| Tip start line helix angle \( \beta_a \) (°) |
15.96 | 15.15 | 14.40 |
| Root relief height \( L_{Ef} \) (mm) |
2.541 | 2.923 | 6.352 |
| Root relief length \( b_{Ef} \) (mm) |
13.155 | 14.888 | 28.766 |
| Max root relief amount \( C_{Ef} \) (μm) |
16.27 | 9.62 | 17.57 |
| Root start line helix angle \( \beta_f \) (°) |
10.93 | 11.11 | 12.45 |
From the results, we observe that as the order increases, the modification heights and lengths become larger, while the maximum relief amounts do not follow a monotonic trend. The contact patterns for all three cases are nearly identical, but the transmission error curves (unloaded) become steeper with higher order. After applying the optimal modifications, the loaded transmission error fluctuations are significantly reduced compared to the unmodified case. The LTE fluctuations are shown in Table 2.
| Case | \( \Delta T_e \) (“) | Reduction vs. unmodified (%) |
|---|---|---|
| Unmodified | 2.587 | — |
| Linear (k=1) | 0.9483 | 63.33 |
| Parabolic (k=2) | 0.9049 | 65.01 |
| Quartic (k=4) | 0.4507 | 82.58 |
The quartic modification yields the greatest reduction (82.58%), demonstrating the advantage of higher-order curves in smoothing the mesh stiffness variation. The physical reason is that higher-order curves provide a more gradual transition of the load at the tooth engagement and disengagement, reducing the sudden changes in contact force.
Having established the optimal diagonal modification design for helical gears, the next challenge is to manufacture such complex tooth surfaces with high precision. We propose to use a large plane grinding wheel, which simulates the action of a generating rack. This method is particularly suitable for grinding the tooth flanks of helical gears because the large plane wheel has a flat working face that represents a single tooth side of a rack. The wheel rotates about its own axis (the main cutting motion) and does not require axial feed motion, resulting in high efficiency and simple kinematics. The workpiece (helical gear) rotates and translates along the tangential direction to generate the involute profile and the modification.
Figure above illustrates the concept of the large plane grinding wheel acting as a generating rack. The wheel’s working plane coincides with the rack’s tooth flank. For grinding a right-hand helical gear’s left flank, we define coordinate systems: \( S_t \) attached to the wheel, \( S_a \) at the intersection of rack pitch line and wheel end face, \( S_b \) at the middle of the rack, and \( S_c \) as the rack moving frame. The wheel’s active radius must be large enough to cover the entire tooth width projected onto the wheel face. The minimum wheel radius \( r_M \) is derived from geometric conditions ensuring that the wheel can grind the full tooth working surface without interference. Assuming a tooth width \( B \), helix angle \( \beta \), normal module \( m_n \), normal pressure angle \( \alpha_n \), and fillet radius \( r_f \), the required wheel outer radius satisfies:
\[
PQ = B / \cos\beta = 2 \sqrt{ r_M^2 – (r_M – LM)^2 }
\]
where \( LM = [0.25 m_n – r_f(1-\sin\alpha_n)] / \cos\alpha_n \). Solving the equation gives the minimum wheel diameter \( d_0 = 2r_M \). For our example gear pair (see Table 3), we compute \( d_0 \approx 496.6 \) mm.
| Parameter | Pinion (small) | Gear (large) |
|---|---|---|
| Number of teeth | 30 | 72 |
| Normal module (mm) | 5 | 5 |
| Normal pressure angle (°) | 20 | 20 |
| Helix angle (°) | 33.273 | 33.273 |
| Hand of helix | Right | Left |
| Facewidth (mm) | 40 | 40 |
In the wheel coordinate system \( S_t \), the position vector and unit normal of the wheel working face (a flat plane) are:
\[
\mathbf{R}_t(r_t, \theta_t) = [-r_t \cos\theta_t,\; 0,\; r_t \sin\theta_t]^T, \quad \mathbf{N}_t = [0,\;1,\;0]^T
\]
where \( r_t \) and \( \theta_t \) are the wheel surface parameters (polar coordinates on the wheel face). Transforming this to the rack coordinate system \( S_c \) yields the generating rack surface:
\[
\mathbf{R}_c(r_t, \theta_t) = \mathbf{M}_{ct} \mathbf{R}_t(r_t, \theta_t), \quad \mathbf{N}_c(r_t, \theta_t) = \mathbf{L}_{ct} \mathbf{N}_t
\]
Here \( \mathbf{M}_{ct} \) is the composite transformation matrix from \( S_t \) to \( S_c \), comprising rotations and translations that align the wheel’s plane with the rack tooth flank.
The grinding of diagonal modification is achieved by superimposing an additional tangential motion of the workpiece (or equivalently an additional rotation) onto the basic generating motion. The basic generating motion for an unmodified involute helical gear follows the standard relationship between the workpiece rotation angle \( \theta_1 \) and the rack translation. For the modified tooth surface, we prescribe an additional tangential displacement \( \Delta L(\theta_1) \) that varies with the rotation angle, such that when the grinding wheel cuts into the tooth tip or root regions, the material removal is increased according to the desired modification amount. The additional displacement is defined in three zones:
\[
\Delta L(\theta_1) =
\begin{cases}
C_{Ea} \frac{r_{p1}}{r_{b1}} \left( \frac{\theta_1 – \theta_B}{\theta_A – \theta_B} \right)^{k_a}, & \theta_1 > \theta_B \\
0, & \theta_E \le \theta_1 \le \theta_B \\
C_{Ef} \frac{r_{p1}}{r_{b1}} \left( \frac{\theta_1 – \theta_E}{\theta_D – \theta_E} \right)^{k_f}, & \theta_1 < \theta_E
\end{cases}
\]
where \( \theta_A \) and \( \theta_B \) are the workpiece rotation angles corresponding to machining the addendum point A and the tip start point B, respectively; similarly \( \theta_D \) and \( \theta_E \) for root. The factors \( r_{p1}/r_{b1} \) convert the tangential displacement from the pitch circle to the base circle, maintaining consistency with the modification amount definition in the normal direction.
Incorporating this additional motion, the family of grinding wheel surfaces in the workpiece coordinate system \( S_1 \) becomes:
\[
\mathbf{R}_1(r_t, \theta_t, \theta_1) = \mathbf{M}_{1c}(\theta_1, \Delta L(\theta_1)) \mathbf{R}_c(r_t, \theta_t)
\]
\[
\mathbf{N}_1(r_t, \theta_t, \theta_1) = \mathbf{L}_{1c}(\theta_1, \Delta L(\theta_1)) \mathbf{N}_c(r_t, \theta_t)
\]
The actual ground tooth surface is the envelope of this family, determined by the meshing condition:
\[
f(r_t, \theta_t, \theta_1) = \mathbf{N}_1(r_t, \theta_t, \theta_1) \cdot \frac{\partial \mathbf{R}_1(r_t, \theta_t, \theta_1)}{\partial \theta_1} = 0
\]
Solving Eqs. (12) and (13) simultaneously yields the coordinates of points on the ground modified tooth surface for given parameter sets. We numerically generate the ground surface for each of the three optimal modification cases (linear, parabolic, quartic) and compare them to the theoretical modified surface by computing the normal deviation. The deviations are shown in Table 4 and are typically within 2 μm, indicating that the large plane grinding wheel method can achieve high fidelity.
| Modification curve | Tip region max deviation (μm) | Root region max deviation (μm) |
|---|---|---|
| Linear (k=1) | 1.7 | 1.8 |
| Parabolic (k=2) | 1.2 | 0.85 |
| Quartic (k=4) | 1.2 | 1.9 |
The small but non-zero deviations arise because the theoretical modification surface assumed straight start lines in the rotational projection plane, while the actual ground start lines are curved due to the envelope process. Nevertheless, the errors are well within typical manufacturing tolerances (5–10 μm) and are considered acceptable for high-precision gears.
From the additional displacement curves (Figure 9 in the original paper, not reproduced here), we see that the required additional motion is smooth and bounded, making it straightforward to implement on a CNC grinding machine with a 6-axis free-form configuration. The machine axes include the workpiece rotation (C-axis), tangential slide (X-axis or Y), and the wheel rotation. The additional tangential motion is simply added to the standard generating motion command.
In conclusion, this work has systematically addressed the optimal design and precision manufacturing of diagonal modified helical gears using a large plane grinding wheel. The key contributions include:
- A clear mathematical definition of diagonal modification for helical gears, allowing arbitrary order of modification curves.
- An optimization procedure based on genetic algorithms that minimizes the loaded transmission error fluctuation; higher-order curves (quartic) yield up to 82.58% reduction in LTE fluctuation.
- A kinematic model of the large plane grinding wheel that simulates a generating rack, with the ability to produce diagonal modification by superposing an additional tangential motion of the workpiece.
- Verification that the tooth surface deviations between the ground surface and the theoretical design are within 2 μm, confirming the accuracy and feasibility of the proposed grinding method.
The approach is well-suited for high-speed, high-load applications such as marine, aerospace, and electric vehicle transmissions where vibration and noise reduction are critical. Future work may involve experimental validation of the ground gears and investigation of the influence of manufacturing errors on the achieved LTE.
Keywords: helical gears; diagonal modification; triangular end relief; large plane grinding wheel; loaded transmission error; optimization; genetic algorithm; tooth surface deviation; generating rack; CNC grinding.