1. Introduction and Research Background
Herringbone gears, characterized by their two mirrored helical gear sections with opposite helix angles, possess significant advantages in load-carrying capacity and the ability to eliminate axial forces inherent in helical gear transmissions. These unique attributes have led to their extensive application across heavy machinery transmission industries, including aviation engines, vehicle engineering, and ship propulsion systems. The continuous advancement of modern industrial systems has imposed increasingly stringent requirements on mechanical transmission systems, particularly in terms of transmission smoothness and noise suppression. The construction of a scientific vibration model for herringbone gear transmission systems enables the prediction of dynamic performance at the design stage, effectively reducing transmission noise. This study conducts an in-depth investigation into the axial vibration characteristics of herringbone gear transmission systems through dynamic simulation, vibration response analysis, and experimental verification.
In this research, I focus on a pair of herringbone gears and their associated gearbox as the primary research objects. The Abaqus software is utilized for finite element analysis to examine the meshing stiffness of herringbone gears and the dynamic response of the gearbox. Experimental verification is conducted on a closed-power-flow herringbone gear transmission test bench. The research contributes to understanding the vibration behavior of herringbone gear systems under various operating conditions, particularly considering the impact of symmetry deviation, tooth profile modification, load torque variations, and input speed changes.
2. Literature Review and International Research Status
2.1 International Research on Gear Meshing Stiffness Computation
The meshing stiffness of gears is a critical parameter in analyzing gear system vibration. Sirichai formally defined gear meshing stiffness as the resistance of gear teeth to elastic deformation under torque loading. This resistance directly influences transmission error, thereby causing vibration and noise. Weber, Banaschck, and Ishikawa developed formulas that treated gear tooth cross-sections as cantilever beams of trapezoidal or rectangular shape. Umezawa established an equivalent method considering gear bodies as variable cross-section cantilever beams, deriving the distribution law of tooth root bending moment along the tooth width direction. Choi utilized isoparametric plate elements to construct reliable finite element analysis models for both spur and helical gears. Kuang proposed a universal formula for calculating the meshing stiffness of cylindrical gears based on finite element results for various gear configurations. Pimsarn presented a method approximating the contact region as geometric overlap of two hypothetical rigid bodies, estimating meshing forces and stiffness. Hedlund developed a parametric numerical model based on Hertzian contact theory and structural analysis. Rincon decomposed gear contact deformation into local and global components for comprehensive stiffness calculations.
2.2 International Research on Gear System Dynamics
International research on gear dynamics has evolved from linear mechanical models to complex nonlinear dynamic models. Nakamura constructed the first gear dynamic model considering nonlinear gear clearance effects. Dubowsky analyzed gear clearance dynamics, while Azar studied the time-varying nature of meshing stiffness and simulated spur gear motion characteristics. Comparin developed a clearance dynamic model addressing single and equivalent stiffness configurations. Kahraman extensively studied nonlinear frequency responses of gears with variable stiffness, identifying jump phenomena. Choy employed modal synthesis methods for multi-stage transmission system analysis. Blankenship constructed multi-degree-of-freedom models considering multiple clearance directions. Li combined elastohydrodynamic lubrication models with dynamics to investigate friction effects on gear dynamic responses.
2.3 Domestic Research Progress
Domestic research on gear dynamics has achieved remarkable progress. Tang Zengbao derived calculation formulas for helical gear meshing stiffness considering both bending and contact deformations. Lin Tengjiao conducted secondary development of gear meshing pair modeling and finite element contact analysis. He Zhaohua systematically analyzed factors influencing herringbone gear meshing stiffness. Chang Lehao proposed an improved gear meshing stiffness algorithm based on finite element and elastic contact theory. Yuan Bing constructed dynamic models of herringbone gear systems considering cumulative pitch errors. Liu Xuan investigated the influence of small gear axial movement on tooth surface load distribution in herringbone gears.
2.4 Experimental Research
Experimental studies on gear transmission systems have progressed significantly. Zhu Xiaolu established closed-power-flow test benches for load testing. Wang Rui developed high-precision transmission error measurement test benches. Dalpiaz utilized acceleration sensors for gearbox surface vibration measurement and fault analysis. Kang developed novel three-piece herringbone gear test devices for phase adjustment. Liu established test benches for measuring vibrations of multiple helical gear pairs under different working conditions.
3. Meshing Stiffness Analysis of Herringbone Gears Based on Abaqus
3.1 Definition and Significance of Meshing Stiffness
Gear meshing stiffness represents the total stiffness of engaging tooth pairs during a complete meshing cycle, primarily dependent on elastic deformation of single-tooth engagement, comprehensive elastic deformation, and the contact ratio of gears during meshing. For herringbone gears, the contact ratio typically exceeds 2, meaning multiple tooth pairs engage simultaneously. The number of engaging tooth pairs alternates during gear rotation, resulting in periodic variations of meshing stiffness.
The comprehensive elastic deformation of a meshing tooth pair can be expressed as:
$$\delta_n = \frac{\Delta\varphi \cdot m_n z_2 \cos\alpha_t}{2\cos^2\beta}$$
where Δφ represents the quasi-static transmission error:
$$\Delta\varphi = \varphi_2 – \frac{z_1}{z_2}\varphi_1$$
The normal meshing stiffness is calculated as:
$$K_n = \frac{F_n}{\delta_n}$$
where Fn represents the total normal meshing force on the tooth surface.
3.2 Herringbone Gear Parameters and Model Establishment
Herringbone gears are structurally composed of two helical gears with opposite helix angles of equal magnitude, effectively canceling axial forces. The basic parameters for the herringbone gear pair in this study are presented in the table below:
| Parameter | Pinion | Gear |
|—|—|—|
| Number of teeth | 27 | 81 |
| Normal module (mm) | 3.5 | 3.5 |
| Normal pressure angle (°) | 20 | 20 |
| Center distance (mm) | 218.2 | 218.2 |
| Helix angle (°) | 30 | 30 |
The involute curve for gear tooth profile is expressed in polar coordinates:
$$\begin{cases} R = \dfrac{R_b}{\cos\alpha} \\ \theta = \tan\alpha – \alpha \end{cases}$$
Converting to rectangular coordinates:
$$\begin{cases} x = R_b(\cos\omega_k + \omega_k \sin\omega_k) \\ y = R_b(\sin\omega_k – \omega_k \cos\omega_k) \end{cases}$$
The three-dimensional models of the herringbone gear pair were established using Pro/E software. During normal operation, the input angular velocity ω is applied to the driving gear, while the driven gear experiences load torque TL. For practical applications, the herringbone gear can be considered as two sets of helical gears each experiencing torque TL/2, meaning the total meshing stiffness equals the sum of both helical gear pairs’ stiffness.
3.3 Finite Element Mesh Generation
Due to the complex structure of herringbone gears with helical angles, obtaining high-quality hexahedral meshes requires careful geometric decomposition. The HyperMesh software was utilized for mesh generation. Individual gear teeth were first partitioned, generating end-face surface meshes. Solid mapping functions generated volume meshes for helical gears, followed by symmetric replication to create the herringbone gear body mesh. The mesh model consists of refined tooth regions and coarser wheel body regions, optimizing computational efficiency while maintaining accuracy.
3.4 Analysis of Meshing Stiffness under Eccentric Loading
Due to manufacturing errors, symmetry deviation fA exists in actual herringbone gears. This deviation is defined as the distance between the actual intersection point of the helix line with the tooth surface and the center plane. The presence of symmetry deviation leads to uneven load distribution between the two sides of the herringbone gear.
The analysis results indicate that with symmetry deviation values of 0, 10 μm, and 20 μm, the comprehensive meshing stiffness of both gear sides decreases with increasing deviation. The load-sharing imbalance caused by symmetry deviation results in uneven contact forces between gear sides, consequently reducing the tooth meshing stiffness curve.
3.5 Effect of Tooth Profile Modification on Meshing Stiffness
Tooth profile modification effectively improves tooth surface meshing conditions, reduces meshing stiffness variation amplitude, and decreases gear dynamic load power. The modification curve selected is the Walker curve:
$$\Delta = \Delta_{max}\left(\frac{x}{h}\right)^n$$
where Δmax is the maximum modification amount, h is the modification height, and n is the modification curve exponent. When n=2, the modification curve becomes parabolic, smoothly connecting with the initial involute profile and effectively reducing vibration and impact.
The analysis revealed that with maximum modification amounts of 0, 5 μm, and 10 μm (with h=3mm), the comprehensive meshing stiffness varies differently. When Δmax=5μm, the comprehensive meshing stiffness slightly increases compared to unmodified conditions. However, when Δmax=10μm, the stiffness decreases, with the maximum stiffness value reducing by approximately 0.95%. This demonstrates that tooth profile modification can effectively mitigate meshing stiffness variations and reduce gear dynamic loads.
4. Dynamic Model Establishment of Herringbone Gear System
4.1 Fundamentals of Gear System Dynamics
Gear system dynamics examines the kinematic behavior of gear systems during operation. The research framework encompasses dynamic excitation, system modeling, and dynamic response analysis. Dynamic excitation includes external excitation from input torque and load resistance torque, as well as internal excitation including meshing stiffness excitation, transmission error excitation, and meshing impact excitation.
For gear system modeling, commonly used methods include: single-degree-of-freedom models for dynamic load coefficient determination, gear torsional vibration models for evaluating gear pair meshing issues, transmission system models considering gear pairs, shafts, and bearings, and complete gear system models for analyzing overall system dynamics.
4.2 Establishment of Twelve-Degree-of-Freedom Dynamic Model
Using the concentrated parameter method, I established a bending-torsion-axial coupled twelve-degree-of-freedom dynamic model for herringbone gear transmission. The model considers the herringbone gear as two sets of helical gears with opposite helix directions. Due to manufacturing constraints causing incomplete symmetry, the meshing stiffness of the two sides differs.
The generalized displacement array of the system is expressed as:
$$\{\delta\} = \{y_1, z_1, \theta_1, y_2, z_2, \theta_2, y_3, z_3, \theta_3, y_4, z_4, \theta_4\}$$
According to Newton’s laws of motion, the system dynamics equations are established. For the left driving gear (gear 1): For the transmission system, the dynamics equations are formulated for each mass component. The key dynamic meshing forces are:
$$\begin{cases} F_{n1} = c_{m1}\left[\cos\beta_1(\dot{y}_1 – \dot{y}_2 + R_{b1}\dot{\theta}_1 – R_{b2}\dot{\theta}_2) + \sin\beta_2(\dot{z}_1 – \dot{z}_2)\right] + k_{m1}f[\cos\beta_2(y_1 – y_2 + R_{b1}\theta_1 – R_{b2}\theta_2) + \sin\beta_2(z_1 – z_2 + \varepsilon_z)] \\ F_{y1} = \cos\beta_2 F_{n1} \\ F_{z1} = \sin\beta_2 F_{n1} \end{cases}$$
For the right meshing pair:
$$\begin{cases} F_{n2} = c_{m2}\left[\cos\beta_1(\dot{y}_3 – \dot{y}_4 + R_{b1}\dot{\theta}_3 – R_{b2}\dot{\theta}_4) + \sin\beta_2(\dot{z}_3 – \dot{z}_4)\right] + k_{m1}f[\cos\beta_2(y_3 – y_4 + R_{b1}\theta_3 – R_{b2}\theta_4) + \sin\beta_2(z_3 – z_4)] \\ F_{y2} = \cos\beta_2 F_{n2} \\ F_{z2} = \sin\beta_2 F_{n2} \end{cases}$$
4.3 Key Parameters in the Dynamic Model
The meshing damping varies with time and is calculated as:
$$C_m(t) = 2\zeta\sqrt{\frac{k_m(t) I_p I_g}{r_{pb}^2 I_p + r_{gb}^2 I_g}}$$
The backlash function representing clearance displacement in the meshing line direction:
$$f[x] = \begin{cases} X – \delta, & X > \delta \\ 0, & |X| \leq \delta \\ X – \delta, & X < \delta \end{cases}$$
The meshing impact force at the initial meshing point is calculated as:
$$F_s = v_s\sqrt{\frac{J_1 J_2}{(J_1 r_{b2}’^2 + J_2 r_{b1}^2)(q_s + \cos^2\theta \cdot q_P)}}$$
The dynamics parameters of the herringbone gear transmission system are presented below:
| Parameter | Pinion | Gear |
|—|—|—|
| Meshing damping coefficient ζ | 0.1 | 0.1 |
| Normal backlash 2δ (μm) | 180 | 180 |
| Center distance installation error (μm) | 8 | 8 |
| Radial support stiffness (×10⁸ N/m) | 6.34 | 6.34 |
| Axial support stiffness (×10⁸ N/m) | / | 2.03 |
4.4 Vibration Analysis in Different Directions
Solving the dynamic equations reveals the vibration response of the herringbone gear transmission system under various conditions. The analysis focuses on vibration patterns in different directions affected by meshing stiffness, meshing impact, load torque, and input speed.
4.4.1 Effect of Meshing Stiffness on System Vibration
The axial and radial vibration displacement and velocity curves were obtained for the driving and driven gears under different eccentric load conditions. The results indicate that with increasing load imbalance between left and right gear sides, both meshing displacement and velocity increase. The vibration displacement and velocity amplitudes of the driving shaft exceed those of the driven shaft.
4.4.2 Effect of Meshing Impact on System Vibration
The axial vibration velocity changes under different load torques (200 N·m, 250 N·m, and 300 N·m) were analyzed with only meshing impact considered. The results demonstrate that with increasing load torque, the velocity fluctuation amplitude increases significantly.
4.4.3 Effect of Load Torque on System Vibration
The axial vibration curves were extracted under different load torques while keeping other parameters constant. The analysis reveals that vibration velocity increases with increasing torque, and larger load torques produce greater velocity fluctuations.
4.4.4 Effect of Input Speed on System Vibration
The axial vibration curves were analyzed at input speeds of 800 r/min, 1200 r/min, and 1500 r/min. The results show that vibration velocity increases with increasing input speed, with larger speeds producing greater velocity fluctuations.
5. Gearbox Modal and Vibration Response Analysis
5.1 Modal Analysis Theory and Modal Superposition Method
For continuous structures with complex shapes, finite element analysis is typically employed. After discretization into n degrees of freedom, the dynamic equation of the system is:
$$[M]\{\ddot{\delta}\} + [C]\{\dot{\delta}\} + [K]\{\delta\} = \{P\}$$
For free vibration with damping neglected:
$$[M]\{\ddot{\delta}\} + [K]\{\delta\} = 0$$
Assuming harmonic motion solution:
$$\{\delta\} = \{X\}\cos(\omega t + \varphi)$$
The characteristic equation:
$$([K] – \omega^2[M])\{X\} = 0$$
Using modal superposition with Rayleigh damping:
$$[C] = \alpha[M] + \beta[K]$$
The modal coordinates transform the coupled equations into n independent single-degree-of-freedom equations:
$$\ddot{q}_i(t) + 2\zeta_i\omega_i\dot{q}_i(t) + \omega_i^2 q_i(t) = Q_i(t), \quad i = 1, 2, \ldots, n$$
5.2 Gearbox Model Establishment and Modal Analysis
The gearbox body was constructed using Q235A steel with the following material parameters:
| Material | Density (kg/m³) | Elastic Modulus (GPa) | Poisson’s Ratio |
|—|—|—|—|
| Q235A | 7.85×10³ | 210 | 0.3 |
The gearbox three-dimensional model was established, and finite element mesh generation was performed after geometric simplification and cleanup. The lower gearbox bolt holes were constrained to simulate ground connection. The first four modes of the gearbox were extracted:
| Mode Order | Natural Frequency (Hz) | Mode Shape Description |
|—|—|—|
| 1 | 248.24 | Axial oscillation |
| 2 | 447.29 | Gearbox twisting |
| 3 | 608.32 | Base deformation |
| 4 | 620.89 | Twisting combined with axial oscillation |
The analysis reveals that the gearbox base deformation and axial oscillation dominate the mode shapes, highlighting the importance of axial vibration analysis in herringbone gear systems.
5.3 Bearing Reaction Force Extraction
From the dynamic model established in Chapter 4, the bearing reaction forces at the driving and driven shaft positions were extracted. The analysis condition was input speed of 1000 r/min and load torque of 200 N·m. To reduce computational complexity, data from the 0.02s to 0.04s interval (200 data points) was used after the system had stabilized.
The bearing reaction force curves exhibit periodic variations due to the periodic nature of gear meshing transmission. Due to the consideration of symmetry deviation, the reaction force trends of driving and driven shafts are not identical. The driving shaft bearing reaction forces are relatively larger than those of the driven shaft because the pinion has smaller radial dimensions.
5.4 Gearbox Radial and Axial Vibration Response
Reference nodes were selected at the bearing seat positions of both driving and driven shafts. The vibration displacement and velocity curves along radial and axial directions were obtained using modal superposition method.
The analysis shows that:
1. The vibration displacement and velocity values at the driving shaft bearing position exceed those at the driven shaft position
2. The presence of reinforcing ribs near reference points enhances the radial stiffness of the gearbox in those regions
3. The driving shaft exhibits larger axial vibration displacement and velocity amplitudes compared to the driven shaft
6. Experimental Verification of Herringbone Gear Vibration
6.1 Test Bench Design and Layout
A closed-power-flow test bench was constructed for herringbone gear dynamic loading experiments. The closed-power-flow configuration enables energy recirculation within the system, providing continuous loading capability while achieving energy savings. This design aligns with sustainable development principles.
The test bench system consists of:
1. TCT1000 measurement and control system
2. Driving and loading motors with frequency conversion inverter systems
3. Torque and speed sensors
4. Mechanical components including test platform, couplings, and protective covers
The system utilizes a DC common bus configuration where electrical energy is recycled. The torque sensors collect input and output signals from the tested gearbox in real-time, enabling dynamic loading while analyzing the gearbox efficiency.
6.2 Test Content and Arrangement
The vibration measurement employed CYT9200 vibration velocity sensors with a measurement range of 0-50 mm/s and a sampling frequency of 2000 Hz. Sensors were positioned at three measurement points on the gearbox at 90°, 180°, and 270° positions around the input shaft axis.
The experimental procedure included:
1. Testing at constant torque of 200 N·m with varying speeds (800, 1200, 1500 r/min) at each measurement point
2. Testing at constant speed of 1000 r/min with varying load torques (100, 200, 270, 300, 350, 400 N·m)
3. Testing with herringbone gears having known symmetry deviation errors
6.3 Measurement Results and Analysis
The experimental data was analyzed to investigate vibration characteristics under different operating conditions.
For eccentric loading conditions at speeds of 800 r/min, 1200 r/min, and 1500 r/min, the axial vibration velocity varies across measurement points. At lower speeds (800 r/min and 1200 r/min), the vibration velocity changes between measurement points remain relatively small. However, at 1500 r/min, the vibration velocity increases progressively with measurement point changes, demonstrating that gear axial vibration velocity increases with speed under eccentric loading conditions.
For different input speeds at constant torque, the experimental results confirm that:
– Increasing speed leads to increased vibration velocity when measurement point and torque remain constant
– At 1500 r/min, the axial vibration velocity fluctuations are notably more pronounced compared to lower speeds
For different load torques at constant speed (1000 r/min), the results show:
– Increasing torque leads to increased vibration velocity when measurement point and speed remain constant
– Larger torque values produce greater axial vibration velocity fluctuations
A direct comparison between measured parameters and theoretical predictions at input speed of 1000 r/min and load torque of 200 N·m shows that the overall shape of the response curves matches well. Although small discrepancies exist, the reasonable agreement between the model and experimental results remains evident, thereby validating the theoretical approach.
7. Conclusions and Future Prospects
This research conducted a comprehensive analysis of herringbone gear transmission systems, focusing on meshing stiffness calculation, dynamic modeling, vibration response analysis, and experimental verification. The primary conclusions drawn from this work are:
1. Through finite element analysis of herringbone gear meshing stiffness, it was found that symmetry deviation causes eccentric loading, resulting in uneven contact forces between gear sides and reduction of the tooth meshing stiffness curve. Under eccentric loading conditions, the meshing stiffness curve significantly decreases compared to normal conditions, with notable amplitude differences between left and right sides. Tooth profile modification with appropriate modification amounts can reduce meshing stiffness variations and decrease gear dynamic loads.
2. A twelve-degree-of-freedom bending-torsion-axial coupled dynamic model of herringbone gear transmission was established, considering multiple physical factors including backlash, meshing impact, comprehensive meshing stiffness, and meshing damping. The vibration response analysis revealed that meshing stiffness significantly affects both axial and radial vibrations, with meshing impact influencing axial vibration characteristics. Load torque and input speed both contribute to increased vibration velocity with increasing magnitudes.
3. Gearbox constraint modal analysis showed that the primary vibration modes involve base deformation and axial oscillation. The vibration response analysis revealed that the reinforcing rib-connected bearing seat positions are relatively weak points in terms of radial vibration amplitude. The driving shaft exhibits larger axial vibration displacement and velocity amplitudes compared to the driven shaft.
4. Experimental verification on the closed-power-flow test bench confirmed the theoretical predictions. The experiments demonstrated that under eccentric meshing conditions, vibration velocity increases with input speed. At constant measurement point and torque, increasing speed leads to increased vibration velocity, with more pronounced fluctuations at 1500 r/min. At constant measurement point and speed, increasing torque leads to increased vibration velocity with larger fluctuations at higher torques. The direct comparison between measured and predicted results confirmed the feasibility and rationality of the theoretical approach.
Future research directions could include:
1. Parameterized modeling of herringbone gears and secondary development of existing stiffness calculation modules for more comprehensive analysis
2. Finite element or boundary element acoustic analysis to study the radiated noise distribution of the gearbox
3. Extension of the current model to high-speed, heavy-load conditions applicable to aviation applications, including more detailed experimental investigation of herringbone gear systems with symmetry deviation under extreme operating conditions