In high-precision applications such as aerospace and marine propulsion, herringbone gears are widely employed due to their high load capacity, smooth operation, and self-canceling axial thrust. However, under heavy loads and high speeds, gear deformation, manufacturing errors, and assembly misalignments induce significant vibration and noise. Tooth surface modification has proven to be an effective approach to mitigate these dynamic issues. This paper presents a comprehensive study on three-dimensional (3D) tooth surface modification for herringbone gears, integrating analytical modeling, multi-objective optimization, and experimental validation. The modification is implemented as a fourth-order parabolic profile combined with cubic B-spline fitting to represent the actual tooth geometry including errors. A 12-degree-of-freedom (DOF) bending–torsion–axial coupled dynamic model is established for a herringbone gear pair supported by rolling element bearings. The model incorporates transmission error excitation and mesh impact force excitation. The effects of modification on internal excitation factors such as loaded transmission error fluctuation and off-line mesh impact force are analyzed. Subsequently, a multi-objective dynamic optimization is performed using an improved adaptive genetic algorithm, targeting the minimization of the transmission error fluctuation amplitude, the off-line impact force amplitude, and the root-mean-square (RMS) value of the relative vibration acceleration along the line of action. Finally, a closed-power-flow test rig is constructed, and Heidenhain circular gratings are used to measure the vibration acceleration. The experimental results confirm the theoretical predictions, with a maximum deviation of less than 14.5%.
Introduction to Three-Dimensional Tooth Surface Modification
Tooth surface modification for herringbone gears can be classified into profile modification (along the tooth height) and lead modification (along the face width). Conventional approaches often apply these two types separately and then superimpose them numerically. However, this simple combination may not yield optimal dynamic performance, and in some cases can even degrade it. Therefore, we adopt a three-dimensional modification scheme using a fourth-order parabolic function applied to the pinion tooth surface. The modification amounts are defined by eight parameters: four for the profile direction (tip and root modification amounts and lengths) and four for the lead direction (both ends). The modification distribution is modeled using a cubic B-spline surface, which ensures high continuity and can accurately represent complex modified surfaces including manufacturing errors. A typical set of modification parameters is given in Table 1.
| Direction | Parameter | Value |
|---|---|---|
| Profile | y1 (tip modification amount) [μm] | 14 |
| y2 (tip modification length) [mm] | 1.8 | |
| y3 (root modification amount) [μm] | 15 | |
| y4 (root modification length) [mm] | 3.0 | |
| Parabola order | 4 | |
| Lead | y5 (left end modification amount) [μm] | 15 |
| y6 (left end modification length) [mm] | 11 | |
| y7 (right end modification amount) [μm] | 14 | |
| y8 (right end modification length) [mm] | 13 | |
| Parabola order | 4 |
The cubic B-spline surface allows the fitting of arbitrary deviation distributions with an accuracy of 1 μm. This capability is essential for subsequent tooth contact analysis (TCA) and loaded tooth contact analysis (LTCA) to evaluate the meshing performance of the modified herringbone gear pair.
Dynamic Model of the Herringbone Gear System
A 12-degree-of-freedom bending–torsion–axial coupled vibration model is developed for a herringbone gear pair under rolling bearing supports. The two helical sections of the herringbone gear are modeled separately, with axial coupling through the shaft and the gap. The generalized displacement vector is:
$$
\boldsymbol{\delta} = [y_{p1},\, z_{p1},\, \theta_{p1},\, y_{g1},\, z_{g1},\, \theta_{g1},\, y_{p2},\, z_{p2},\, \theta_{p2},\, y_{g2},\, z_{g2},\, \theta_{g2}]^{\mathrm{T}}
$$
where subscripts \(p\) and \(g\) denote pinion and gear, and 1 and 2 denote the left and right helical sections. \(y\), \(z\), and \(\theta\) represent lateral (bending), axial, and torsional displacements, respectively. The equations of motion for the left half of the pinion are given by:
$$
\begin{aligned}
m_{p1}\,\ddot{y}_{p1} + c_{p1y}\,\dot{y}_{p1} + k_{p1y}\,y_{p1} &= -F_{y1} + m_{p1}g \\
m_{p1}\,\ddot{z}_{p1} + c_{p12z}\,(\dot{z}_{p1} – \dot{z}_{p2}) + k_{p12z}\,(z_{p1} – z_{p2}) &= -F_{z1} \\
I_{p1}\,\ddot{\theta}_{p1} &= -F_{y1}R_p + T_{p1} – F_{s1}R_p
\end{aligned}
$$
Similar equations hold for the other three bodies. Here, \(m\), \(I\) are mass and moment of inertia; \(R_p\) and \(R_g\) are base radii; \(c\) and \(k\) are damping and stiffness coefficients for bearings and shafts; \(F_{y1}, F_{y2}\) are dynamic mesh forces along the transverse line of action; \(F_{z1}, F_{z2}\) are axial mesh forces; and \(F_{s1}, F_{s2}\) are off-line mesh impact forces. The system is solved using the Runge–Kutta method, discarding the transient part to obtain steady-state periodic solutions.
Effect of Modification on Internal Excitation Factors
To investigate how tooth surface modification alters the internal vibration excitations, we compute the loaded transmission error (TE) and the off-line mesh impact force for different modification schemes. The herringbone gear pair parameters used in the analysis are listed in Table 2.
| Parameter | Pinion | Gear |
|---|---|---|
| Normal module (mm) | 6 | 6 |
| Transverse pressure angle (°) | 20 | 20 |
| Helix angle (°) | 24.43 | −24.43 |
| Number of teeth | 17 | 44 |
| Face width (mm) | 55 | 55 |
| Density (g/cm³) | 7.85 | 7.85 |
| Input speed (r/min) | 2000 | – |
| Load torque (N·m) | – | 828 |
Transmission Error Fluctuation
The loaded transmission error \(\varepsilon\) is obtained from LTCA, considering the axial floating of the pinion. The fluctuation amplitude is a direct measure of the stiffness variation. Figure 1 in the original paper (not reproduced here) shows that a suitable modification (e.g., scheme A) reduces the TE fluctuation from 2.9″ to 2.2″. However, the waveform shape also matters: a modification that introduces fewer harmonic components in the TE spectrum yields better vibration attenuation over a wider speed range. Table 3 compares two modification schemes.
| Scheme | TE fluctuation amplitude (″) | Harmonic content |
|---|---|---|
| Unmodified | 2.9 | Fundamental only |
| Modification A | 2.2 | Fundamental + 2nd harmonic |
| Modification B | 2.3 | Fundamental only |
Although both modifications reduce the peak-to-peak amplitude, modification B is preferable because the absence of higher harmonics prevents resonance at sub-harmonic speeds (e.g., 1/2 and 1/3 of the mesh frequency). This highlights the need to consider not only the TE amplitude but also the overall vibration response.
Off-Line Mesh Impact Force
Off-line mesh impact occurs when a tooth pair enters contact before the theoretical line of action due to elastic deformations and profile errors. The impact velocity \(v_s\) at the actual mesh start point can be computed from TCA results:
$$
v_s = \frac{2\pi n}{60} \left( O_1M_1 \cdot i – O_2M_2 \right)
$$
where \(n\) is the pinion speed, \(i\) is the transmission ratio, and \(O_1M_1\), \(O_2M_2\) are the distances from the gear centers to the contact point projected onto the line of action. The impact force amplitude \(F_s\) is then derived from energy conservation:
$$
F_s = v_s \sqrt{ \frac{b J_1 J_2}{\left(J_1 (O_2M_2)^2 + J_2 (O_1M_1)^2\right) \left(q_s + \cos^2\theta \cdot q_p\right)} }
$$
where \(J_1, J_2\) are moments of inertia, \(b\) is face width, \(\theta\) is the angle between the instantaneous and theoretical contact lines, and \(q_s, q_p\) are tooth compliances. Table 4 lists two additional modification sets (C and D) used to study the impact force variation.
| Direction | Parameter | Modification C | Modification D |
|---|---|---|---|
| Profile | y1 (μm) | 8 | 16 |
| y2 (mm) | 1.8 | 1.9 | |
| y3 (μm) | 10 | 16 | |
| y4 (mm) | 2.8 | 2.9 | |
| Lead | y5 (μm) | 9 | 11 |
| y6 (mm) | 10 | 10 | |
| y7 (μm) | 9 | 13 | |
| y8 (mm) | 10 | 10 | |
| Parabola order | 4 | 4 | |
With the unmodified tooth surface, the load sharing coefficient at the mesh start point is 0.032 and the impact force is 1160 N. Reducing the load sharing coefficient to 0.016 (modification C) lowers the impact force to 786 N. Further increasing the modification (D) so that the tooth pair carries zero load at the start reduces the impact force to 435 N. However, excessive modification also reduces the effective contact ratio, which may degrade transmission smoothness. Therefore, a trade-off must be found through optimization.
Multi‑Objective Dynamic Optimization of Tooth Surface Modification
We formulate a multi‑objective optimization problem that minimizes three dynamic performance indices: (1) the peak‑to‑peak fluctuation of the loaded transmission error \(f_e\), (2) the amplitude of the off‑line mesh impact force \(f_I\), and (3) the RMS value of the relative vibration acceleration along the line of action \(f_a\). The optimization function is:
$$
f_C(\mathbf{y}) = \min \left[ w_1 f_e(\mathbf{y}) + w_2 f_I(\mathbf{y}) + w_3 f_a(\mathbf{y}) \right]
$$
subject to constraints on the eight modification parameters \(\mathbf{y} = [y_1, y_2, \dots, y_8]\). The weights \(w_1, w_2, w_3\) are chosen based on the contribution of each factor under multiple torque conditions (621, 828, and 1035 N·m) with operating frequency factors 0.2, 0.5, and 0.3, respectively. The search is performed using an improved adaptive genetic algorithm (IAGA). The optimized modification parameters obtained at 2000 r/min and the rated load of 828 N·m are listed in Table 5.
| Direction | Parameter | Value |
|---|---|---|
| Profile | y1 (μm) | 16 |
| y2 (mm) | 1.6 | |
| y3 (μm) | 18 | |
| y4 (mm) | 3.2 | |
| Lead | y5 (μm) | 14 |
| y6 (mm) | 11.2 | |
| y7 (μm) | 14 | |
| y8 (mm) | 11.2 | |
| Parabola order | 4 | |
Using these optimized parameters, the dynamic response of the herringbone gear system is recalculated. The RMS value of the relative vibration acceleration along the line of action decreases from 29.38 m/s² (unmodified) to 23.38 m/s² (modified), representing a reduction of 20.42%. The frequency spectrum shows a significant attenuation of high‑order harmonics, confirming the effectiveness of the modification in reducing mesh impact and vibration.
Experimental Validation
To verify the theoretical predictions, a closed‑power‑flow test rig for herringbone gears is constructed. The rig includes a DC motor, torque meter, high‑speed elastic shafts, and the test gearbox. Two Heidenhain ROD280 circular gratings (18,000 lines/revolution, resolution ±5″) are installed on the input and output shafts to measure the instantaneous angular positions of the pinion and gear. The measured signals are acquired using a high‑speed data acquisition card (PCI8502, 40 MHz sampling rate, 256 MB buffer).
The angular position at each sampling instant is computed by:
$$
\varphi(t_{i,j}) = \varphi(t_{i,j-1}) + \frac{360}{N} \cdot \frac{\theta(t_{i,j}) – \theta(t_{i,j-1})}{2\pi}
$$
where \(N = 18,000\) is the number of grating lines, and \(\theta(t_{i,j})\) is the arc‑sine of the sinusoidal signal amplitude. The dynamic transmission error in angular measure is:
$$
\varepsilon(t) = \left( \varphi_2 – \varphi_{20} \right) – \frac{z_1}{z_2} \left( \varphi_1 – \varphi_{10} \right)
$$
Converting to linear displacement along the line of action and taking the second derivative gives the relative vibration acceleration \(a_{12}(t)\). To improve the signal‑to‑noise ratio, time‑domain synchronous averaging is applied with \(N_{\text{avg}} = 100\) periods. The noise RMS is reduced by a factor of \(\sqrt{100}\).
Figure 2 in the original shows the measured vibration acceleration spectra for both unmodified and optimized herringbone gears at 2000 r/min and 828 N·m. The experimental RMS values drop from 32.13 m/s² (unmodified) to 25.13 m/s² (modified), a reduction of 21.8%. The dominant frequency component is the second harmonic of the mesh frequency, consistent with the theoretical analysis. The relative deviation between theory and experiment is within 14.5% across the tested torque range, as summarized in Table 6.
| Torque (N·m) | Unmodified (theo.) | Unmodified (exp.) | Optimized (theo.) | Optimized (exp.) | Deviation (%) |
|---|---|---|---|---|---|
| 414 | 18.5 | 20.1 | 19.2 | 20.8 | 7.7 |
| 621 | 24.1 | 26.3 | 20.8 | 22.5 | 7.6 |
| 828 | 29.4 | 32.1 | 23.4 | 25.1 | 6.8 |
| 1035 | 33.2 | 36.0 | 26.7 | 28.9 | 7.6 |
It should be noted that under the lightest load (414 N·m), the vibration of the modified herringbone gear is slightly higher than that of the unmodified one. This occurs because the optimization was performed for the rated load (828 N·m); the modification reduces the effective contact ratio at low loads, leading to increased excitation. Hence, the design modification must be tailored to the expected operating conditions.
Conclusions
This work presented a comprehensive investigation of three‑dimensional tooth surface modification for herringbone gears, combining analytical modeling, multi‑objective optimization, and experimental testing. The following conclusions are drawn:
- A fourth‑order parabolic three‑dimensional modification, fitted by cubic B‑spline surfaces, can accurately represent the modified tooth geometry including realistic errors.
- The 12‑DOF bending–torsion–axial coupled dynamic model of a herringbone gear pair successfully incorporates transmission error and off‑line mesh impact excitations.
- Proper tooth surface modification reduces the transmission error fluctuation amplitude and the off‑line impact force, leading to a 20.42% reduction in the RMS vibration acceleration along the line of action.
- The closed‑power‑flow test rig with Heidenhain circular gratings provides reliable measurement of the relative vibration. The experimental results confirm the theoretical trends, with a maximum error of 14.5%.
- Modification optimization must consider the entire operating torque range; a single modification set may not perform optimally under all loads.
The proposed methodology offers a systematic approach to design low‑vibration herringbone gears for high‑precision, high‑speed applications.