This paper presents a comprehensive study on the optimization of dynamic performance of herringbone gear transmissions with high contact ratio. The herringbone gear is widely used in aerospace, marine, and other high-power transmission systems due to its high load capacity, smooth operation, and self-balanced axial forces. In this work, based on loaded tooth contact analysis (LTCA) technology, a bending-torsional-axial coupled nonlinear dynamic model of the herringbone gear pair is established using the lumped mass method, considering the axial floating characteristics of the pinion, time-varying mesh stiffness, meshing impact excitation, and error excitation. A novel genetic algorithm with a fitness approximation mechanism (FAGA) is proposed to efficiently optimize the design parameters and modification coefficients of the herringbone gear. The root mean square (RMS) of vibration acceleration along the line of action is adopted as the objective function. The optimization results show a significant reduction in vibration acceleration and noise level, validated by experimental tests on a noise and vibration reduction test rig.
1. High Contact Ratio Design Analysis of Herringbone Gears
The contact ratio of a herringbone gear transmission is a critical factor influencing its load-sharing capability and transmission smoothness. The total contact ratio consists of the transverse contact ratio εα and the overlap (axial) contact ratio εβ. Increasing the addendum coefficient and helix angle can effectively enhance the contact ratio. For aviation herringbone gears, the pinion often adopts a floating structure to balance axial forces, allowing a larger helix angle (typically 25° to 40°). In this design, an equal-shift gear transmission is adopted: the pinion uses a positive addendum modification (xn1 > 0) while the gear uses a negative one (xn2 < 0). This approach maintains a compact structure while improving load capacity. The design variables selected for optimization include the helix angle β, the normal addendum modification coefficient xn1, the normal addendum coefficient han*, and the parabola modification coefficients A, B, C for tooth profile and lead modification.
2. Dynamic Modeling of Herringbone Gear Transmission
The dynamic behavior of herringbone gears is influenced by three main excitations: stiffness excitation, error excitation, and meshing impact excitation. The time-varying mesh stiffness is obtained through a loaded tooth contact analysis (LTCA) based on the finite element flexibility matrix and nonlinear programming. The stiffness data over one mesh cycle are fitted using a Fourier series. The error excitation includes tooth profile errors and base pitch errors, determined from the gear manufacturing accuracy class. The meshing impact excitation is calculated considering the initial meshing position, impact velocity, and impact force, with emphasis on the meshing-in impact due to its dominant effect.
The lumped mass method is used to establish a 12-degree-of-freedom (DOF) bending-torsional-axial coupled dynamic model for the herringbone gear pair (see Figure 1 in the original text). The model accounts for the axial floating of the pinion and the interaction between the left and right helical gear halves. The generalized displacement vector is expressed as:
$$ \{ \delta \} = \{ y_{p1}, z_{p1}, \theta_{p1}, y_{g1}, z_{g1}, \theta_{g1}, y_{p2}, z_{p2}, \theta_{p2}, y_{g2}, z_{g2}, \theta_{g2} \}^T $$
The dimensionless equations of motion are derived by introducing a dimensionless time τ = tωn and a nominal displacement scale bc. The resulting dimensionless differential equations are solved using the fourth-order Runge-Kutta method (ODE45 in MATLAB). The steady-state vibration acceleration responses along the line of action and the axial direction are obtained.
3. Optimization Mathematical Model
3.1 Design Variables
Based on the high contact ratio design analysis, six design variables are selected:
$$ X = [x_1, x_2, x_3, x_4, x_5, x_6]^T = [\beta, x_{n1}, h_{an}^{*}, A, B, C]^T $$
where β is the helix angle, xn1 is the normal addendum modification coefficient of the pinion, han* is the normal addendum coefficient, A and B are the quadratic and constant terms of the parabola tooth profile modification, and C is the quadratic coefficient of the lead modification. The parabola modification is applied only to the pinion for economic reasons, with modification equations:
Tooth profile: $$ y = A x^2 + B $$
Lead direction: $$ y = C x^2 $$ (where x is the axial coordinate, with the modification vertex at the midpoint of the gear width).
3.2 Objective Function
The vibration noise of the gear transmission is proportional to the RMS value of the vibration acceleration. The objective function is the RMS of the mesh-line vibration acceleration of the left-side gear over one mesh cycle:
$$ \min f_1(x) = \sqrt{ \frac{1}{n} \sum_{i=1}^{n} \ddot{x}_i^2 } $$
3.3 Constraints
Constraints include strength conditions (contact and bending stress), scuffing resistance, sliding ratio limits, minimum contact ratio (ensuring high contact ratio), maximum dynamic load, and gear geometry constraints (e.g., minimum tooth tip thickness, no undercut, etc.).
4. Fitness Approximation Genetic Algorithm (FAGA)
Traditional genetic algorithms require numerous fitness evaluations, each involving a time-consuming LTCA and dynamic simulation. To overcome this, a fitness approximation genetic algorithm (FAGA) is proposed. The algorithm maintains a historical population database. For each individual i, a shared region Ωi is defined using an adaptive sharing radius rshare. If the predicted credibility R(i) of the fitness value exceeds a threshold R*, the fitness is estimated using a weighted average of the fitness values of individuals in the shared region. Otherwise, the true fitness is computed. The credibility of predicted fitness decays over generations to ensure accuracy. The algorithm reduces the number of true evaluations by about 60–65% as validated by benchmark tests (Table 1).
| Benchmark Function | Max. Num. | Eval. Num. (avg) | Percentage |
|---|---|---|---|
| Goldstein-Price | 20,000 | 7,167.5 | 35.84% |
| Six-Hump Camel-Back | 20,000 | 7,560.5 | 37.80% |
| Shekel’s Foxholes | 20,000 | 7,212.05 | 36.06% |
The FAGA parameters are set as: population size 100, maximum generations 50, crossover probability 0.8, mutation probability 0.3, credibility threshold R*=0.6, redundancy threshold Ir*=1e-7, and credibility decay factor β=0.9.
5. Optimization Example and Results
An aviation single-stage herringbone gear pair with the initial parameters listed in Table 2 is optimized. Both pinion and gear are made of carburized and quenched steel, Grade 5 accuracy.
| Parameter | Before Optimization | After Optimization |
|---|---|---|
| z1/z2 | 31/103 | 31/103 |
| mn | 4.5 | 4.5 |
| β (°) | 31 | 34 |
| han* | 1.0 | 1.297 |
| xn1/xn2 | 0/0 | 0.4203/−0.4203 |
| A (tooth profile) | 0 | 0.005 |
| B (tooth profile) | 0 | 0.03 |
| C (lead) | 0 | 4.0E−6 |
| εγ / εα | 7.3275 / 1.3985 | 8.4004 / 1.7254 |
| RMS acceleration (m/s2) | 28.3746 | 18.9331 |
The optimization increases the total contact ratio εγ by 14.6% and the transverse contact ratio εα by 23.4%. The RMS vibration acceleration along the line of action decreases by 33%. The dynamic responses of the herringbone gear pair before and after optimization are shown in Figure 6 and Figure 7 (not reproduced here). The vibration amplitudes in both the circumferential and axial directions are significantly reduced due to the combined profile and lead modifications.
6. Experimental Validation
To verify the noise reduction effect, the optimized herringbone gears are manufactured and tested on a power-circulating gear test rig. The test rig consists of a DC motor (200 kW, 300–1200 rpm), a gearbox with a ratio of 1:3.322, and a load device. Six microphones are placed 1 m away from the gearbox surface at different locations. Noise levels are measured under two torque levels (2000 N·m and 1000 N·m) at motor speeds of 500, 750, and 1000 rpm. The average noise values are summarized in Tables 3 and 4.
| Measurement Point | Before Optimization | After Optimization |
|---|---|---|
| 1 | 125.32 | 119.82 |
| 2 | 122.13 | 116.06 |
| 3 | 127.54 | 120.14 |
| 4 | 125.51 | 119.50 |
| 5 | 123.47 | 117.93 |
| 6 | 125.11 | 118.35 |
| Measurement Point | Before Optimization | After Optimization |
|---|---|---|
| 1 | 115.46 | 110.09 |
| 2 | 117.87 | 111.16 |
| 3 | 113.40 | 107.05 |
| 4 | 114.17 | 108.67 |
| 5 | 115.13 | 109.93 |
| 6 | 112.07 | 106.96 |
The optimized herringbone gear pair achieves a noise reduction of approximately 5–7 dB under both torque conditions. This experimental result confirms that the proposed dynamic optimization method effectively reduces vibration and noise of the herringbone gear transmission.
7. Conclusion
A systematic optimization approach for the dynamic behavior of herringbone gears with high contact ratio is presented. The key contributions include: (1) a comprehensive bending-torsional-axial coupled dynamic model that considers the axial floating of the pinion and individual modifications; (2) a fitness approximation genetic algorithm that significantly reduces the computational cost while maintaining accuracy; (3) a successful application to an aviation herringbone gear pair, resulting in a 33% reduction in vibration acceleration RMS and a 5–7 dB noise reduction in experiments. The work demonstrates that proper modification combined with high contact ratio design can substantially improve the dynamic performance of herringbone gear transmissions, which is crucial for noise-sensitive applications such as aerospace power trains.