1. Introduction
Miter gear, also known as herringbone gear, possess distinct advantages such as large contact ratio, smooth torque transmission, high load-carrying capacity, and self-canceling axial forces, making them indispensable in critical industries like marine engineering, construction materials, metallurgy, and aerospace. Their unique structure, resembling two adjacent helical gears with opposite handings, introduces complex dynamic behaviors that are significantly influenced by manufacturing and installation errors. Understanding the dynamic response of herringbone gear transmission systems under various operational conditions is crucial for optimizing their design and minimizing vibrations and noise.
1.1 Research Objectives
The primary goal of this study is to develop a comprehensive dynamic model of herringbone gear transmission system that incorporates the interactions between the gearbox, bearings, shafts, and gear meshing. By integrating both numerical simulations and experimental validations, we aim to analyze the effects of geometric errors (such as axis tilt and eccentricity) and tooth surface modifications on the dynamic meshing forces and 箱体 (gearbox) vibration characteristics. The findings will provide theoretical and technical support for the vibration reduction and low-noise design of herringbone gear systems.
1.2 Literature Review
Extensive research has been conducted on the dynamics of herringbone gear systems. Liu et al. (2021) discussed the theoretical framework for gear transmission dynamics and investigated the influence of support layouts and gear parameters on the quasi-static and dynamic behaviors of herringbone gears using a multi-point meshing model. Lin et al. (2021) proposed a method to calculate the time-varying meshing stiffness of herringbone gears with profile modifications, highlighting the impact of relief groove width and modification parameters. Wang et al. (2014) analyzed the nonlinear dynamics of herringbone gear systems, focusing on the effects of contact ratio and backlash on dynamic responses.
However, most previous studies simplified the system by treating components as rigid bodies or using lumped parameter models, neglecting the flexibility of shafts and the dynamic interactions with the gearbox. There is a critical need for a more realistic model that includes flexible components and experimental validation to accurately capture the complex dynamics of herringbone gear transmissions.
2. Multi-Body Dynamics Modeling of Herringbone Gear Systems
2.1 System Configuration and Basic Parameters
The herringbone gearbox under study consists of two gear shafts with herringbone gear and four spherical roller bearings, as shown in Figure 1 (schematic). The input shaft is supported by one fixed and one floating bearing, while the output shaft uses two floating bearings to accommodate axial movements. The basic parameters of the herringbone gears are summarized in Table 1.
| Parameter | Value |
|---|---|
| Number of Teeth (Z₁/Z₂) | 25/32 |
| Module (mm) | 25 |
| Helix Angle (°) | 26 |
| Pressure Angle (°) | 20 |
| Face Width (mm) | 240×2 + 250 |
Table 1: Basic Parameters of the herringbone Gear
2.2 Rigid-Flexible Coupling Model
2.2.1 Shaft and Gear Modeling
The shafts are discretized into flexible beam elements using finite element method (FEM), with the input shaft divided into 100 elements and the output shaft into 106 elements. Each beam element has six degrees of freedom (three translations and three rotations). The gears are treated as rigid bodies and connected to the flexible shafts using RBE2 rigid elements, which rigidly couple the gear center and the shaft nodes.
2.2.2 Bearing and Gearbox Modeling
Bearings are modeled as linear spring-damper systems with stiffness and damping coefficients derived from manufacturer data. The gearbox housing, due to its complex geometry, is simplified by removing non-critical features (e.g., chamfers, oil holes) and meshed using tetrahedral elements. The finite element model of the gearbox contains 9,351 nodes and 29,871 elements, with material properties: Young’s modulus \(E = 200 \, \text{GPa}\), Poisson’s ratio \(\nu = 0.285\).
2.2.3 Meshing Contact Model
The dynamic meshing between herringbone gear is simulated using a contact model that considers time-varying meshing stiffness and friction. The contact stiffness is calculated based on the Hertzian contact theory, and the friction coefficient is set to 0.15 to account for surface interactions. The meshing force \(F_m\) at each contact point is given by:\(F_m = k(t) \delta(t) + c \dot{\delta}(t)\) where \(k(t)\) is the time-varying meshing stiffness, \(\delta(t)\) is the meshing displacement, and c is the damping coefficient.
2.3 Governing Equations of Motion
Using the Lagrangian approach, the equations of motion for the herringbone gear transmission system are derived as:\([M]\{\ddot{x}\} + [C]\{\dot{x}\} + [K]\{x\} = \{P(t)\}\) where \([M]\), \([C]\), and \([K]\) are the mass, damping, and stiffness matrices, respectively; \(\{x\}\) is the displacement vector; and \(\{P(t)\}\) is the vector of external forces, including meshing forces and bearing reactions.
2.3.1 Kinetic Energy of the System
The kinetic energy T of the system includes the translational and rotational energies of the gears and shafts:\(T = \frac{1}{2} \sum_{i=1}^{n} m_i (\dot{x}_i^2 + \dot{y}_i^2 + \dot{z}_i^2) + \frac{1}{2} \sum_{i=1}^{n} \boldsymbol{\omega}_i^T [J_i] \boldsymbol{\omega}_i\) where \(m_i\) is the mass of component i, \(\boldsymbol{\omega}_i\) is the angular velocity vector, and \([J_i]\) is the moment of inertia tensor.
2.3.2 Potential Energy and Damping
The potential energy accounts for the elastic deformations of the shafts and gearbox, while the damping matrix includes both structural damping and contact damping. The Rayleigh damping model is used, assuming \([C] = \alpha [M] + \beta [K]\), where \(\alpha\) and \(\beta\) are damping coefficients.
3. Simulation and Experimental Validation
3.1 Simulation Setup
The simulation is performed under a constant input speed of 300 rpm and a gradually applied output torque of 560 kNm over 0.2 seconds. The time-domain response is solved using the Newmark-β method with a simulation time of 1.5 seconds and 3,000 time steps. The vibration acceleration at the bearing seat (node 65181) is recorded for both frequency and time-domain analyses.
3.2 Experimental Setup
The test herringbone gearbox is installed in a steel mill hot-rolling line, connected to an electric motor and rolling mill via couplings (Figure 2). Accelerometers are mounted on the gearbox housing to measure vibration signals during steady-state operation. The acquired signals are processed using Fast Fourier Transform (FFT) to obtain the frequency spectrum, with two typical operating conditions (226.2 Hz and 177.13 Hz meshing frequencies) considered.
3.3 Comparison of Simulation and Experiment
Figure 3 shows the vibration acceleration spectra from both simulation and experiment. The simulation accurately captures the dominant meshing frequency (250 Hz) and its harmonics, which align well with the experimental results. While the experimental spectrum contains more low-amplitude background frequencies due to real-world noise, the peak values at meshing frequencies match closely, validating the accuracy of the dynamic model.
4. Effects of Geometric Errors on Dynamic Response
4.1 Axis Tilt Error
An axis tilt error of 0.03° is introduced to one half of herringbone gear, simulating misalignment during installation. The meshing forces in the X and Y directions exhibit significant low-frequency fluctuations with a 180° phase difference between the two halves (Figure 4), indicating balanced loading due to the floating bearing design. The frequency spectrum of the meshing force (Figure 5) shows a prominent rotational frequency peak (7.81 Hz) and sidebands around the meshing frequency, reflecting the periodic variation in contact position caused by the tilt.
The corresponding vibration acceleration at the bearing seat (Figure 6) maintains the dominant meshing frequency peak (1433 mm/s²) with no significant increase in rotational frequency response. This is attributed to the system’s low-pass filtering effect and the self-canceling nature of axial forces in herringbone gear.
4.2 Eccentricity Error
A radial eccentricity of 0.2 mm is applied to one half of the gear, altering the instantaneous center distance during meshing. The meshing force time history (Figure 7) shows similar phase opposition between the two gear halves but with a distinct low-frequency component at 26.91 Hz, which is correlated with the contact ratio of the gears (Figure 8). The frequency spectrum (Figure 9) highlights a new peak at 26.91 Hz, indicating a non-rotational frequency related to the meshing stiffness variation due to eccentricity.
The vibration response under eccentricity (Figure 10) shows a slight increase in the meshing frequency peak (1444 mm/s²) but no dominant low-frequency vibrations, again due to the balanced loading and flexible support effects.
5. Influence of Tooth Surface Modification
5.1 Drum Shaping Modification
Drum shaping, a type of tooth end modification, is applied to reduce load concentration and improve meshing smoothness. A drum amount of 0.02 mm is applied to both tooth surfaces, and the dynamic response is compared with the unmodified case.
5.1.1 Meshing Force Reduction
As shown in Table 2, the meshing force peak at the meshing frequency decreases significantly after modification: from 7019 N to 5411 N under axis tilt, and from 6371 N to 5273 N under eccentricity. This reduction is due to the improved contact pattern and reduced edge loading caused by drum shaping.
| Condition | Meshing Force Peak (N) | |
|---|---|---|
| Unmodified | Modified | |
| Axis Tilt | 7019 | 5411 |
| Eccentricity | 6371 | 5273 |
Table 2: Meshing Force Peaks Before and After Drum Shaping
5.1.2 Vibration Reduction
The vibration acceleration at the bearing seat also shows notable improvements (Figure 11). Under axis tilt, the peak acceleration drops from 1433 mm/s² to 1207 mm/s², and under eccentricity, it decreases from 1444 mm/s² to 1218 mm/s². The modification effectively mitigates high-frequency vibrations by smoothing the meshing impact and reducing dynamic loads.
6. Conclusions
In this study, a comprehensive rigid-flexible coupling dynamic model of herringbone gear transmission system was developed, integrating gear meshing, shaft flexibility, bearing dynamics, and gearbox flexibility. Key findings include:
- Model Validation: The simulated vibration spectra closely match experimental results, confirming the accuracy of the multi-body dynamics model in capturing real-world dynamic behaviors.
- Error Effects: Both axis tilt and eccentricity introduce low-frequency fluctuations in meshing forces, with phase differences of 180° between the two gear halves, leading to self-canceling effects on 箱体 vibrations. Axis tilt primarily excites rotational frequencies, while eccentricity generates non-rotational frequencies related to the contact ratio.
- Modification Benefits: Drum shaping modification significantly reduces meshing force peaks and vibration accelerations at the meshing frequency, demonstrating its effectiveness in improving dynamic performance and reducing noise.
This research provides a robust framework for analyzing herringbone gear dynamics and offers practical guidelines for designing low-vibration, high-reliability transmission systems. Future work will focus on investigating the effects of bolt preload and lubricant viscosity on system dynamics, further enhancing the model’s applicability to real-world engineering scenarios.