In the field of heavy-duty marine transmission systems, the herringbone gear is widely adopted due to its high load capacity, smooth operation, and ability to cancel axial thrust forces. However, under varying load and speed conditions, the dynamic behavior of the herringbone gear system significantly influences tooth root stress fluctuations, which directly affect fatigue life and reliability. In this work, we establish a comprehensive nonlinear dynamic model for a herringbone gear transmission system, incorporating multiple excitation sources including transmission error, time-varying mesh stiffness, corner mesh impact, and backlash. A novel calculation method for tooth root dynamic stress is proposed, which accounts for the influence of real-time dynamic load on the load distribution coefficient between meshing tooth pairs. The accuracy of the proposed method is validated by comparing numerical simulations with experimental measurements on a marine herringbone gear pair.
Our dynamic model considers the coupled bending-torsion-axial vibrations of the herringbone gear system. The system is discretized into a twelve-degree-of-freedom (12-DOF) lumped parameter model, where each of the two helical gear pairs (left and right sides) is represented by translational and rotational coordinates. The generalized displacement vector is defined as:
$$ \{\delta\} = \{ y_{p1}, z_{p1}, \theta_{p1}, y_{g1}, z_{g1}, \theta_{g1}, y_{p2}, z_{p2}, \theta_{p2}, y_{g2}, z_{g2}, \theta_{g2} \}^T $$
where \(y_{ij}\) and \(z_{ij}\) represent the transverse and axial translational displacements of the pinion (p) or gear (g) for the left (j=1) and right (j=2) sides, and \(\theta_{ij}\) are the torsional displacements. The equations of motion are derived using Newton’s second law. For the left-side pinion, we have:
$$ m_p \ddot{y}_{p1} + c_{p1y} \dot{y}_{p1} + k_{p1y} y_{p1} = -F_{y1} $$
$$ m_p \ddot{z}_{p1} + c_{p12z} (\dot{z}_{p1} – \dot{z}_{p2}) + k_{p12z} (z_{p1} – z_{p2}) = -F_{z1} $$
$$ I_p \ddot{\theta}_{p1} = -F_{y1} R_p + T_{p1} $$
Similarly, for the left-side gear:
$$ m_g \ddot{y}_{g1} + c_{g1y} \dot{y}_{g1} + k_{g1y} y_{g1} = F_{y1} $$
$$ m_g \ddot{z}_{g1} + c_{g1z} \dot{z}_{g1} + k_{g1z} z_{g1} + c_{g12z} (\dot{z}_{g1} – \dot{z}_{g2}) + k_{g12z} (z_{g1} – z_{g2}) = F_{z1} $$
$$ I_g \ddot{\theta}_{g1} = F_{y1} R_g – T_{g1} $$
The equations for the right-side pinion and gear are analogous. In these equations, \(m_p\), \(m_g\), \(I_p\), and \(I_g\) are the masses and moments of inertia of the pinion and gear; \(R_p\) and \(R_g\) are the base circle radii; \(c\) and \(k\) represent damping and stiffness coefficients of shafts and bearings; and \(F_{y1}\), \(F_{y2}\), \(F_{z1}\), \(F_{z2}\) are the dynamic mesh forces in the tangential and axial directions, which include the corner mesh impact excitation.
The tangential and axial dynamic mesh forces for the left side are expressed as:
$$ F_{y1} = \cos\beta \, c_m (\dot{y}_{p1} + R_p \dot{\theta}_{p1} – \dot{y}_{g1} – R_g \dot{\theta}_{g1}) + \cos\beta \, k_m(t) f(y_{p1} + R_p \theta_{p1} – y_{g1} – R_g \theta_{g1}) – F_s(t) $$
$$ F_{z1} = \sin\beta \, c_m [\tan\beta (\dot{y}_{p1} + R_p \dot{\theta}_{p1} – \dot{y}_{g1} – R_g \dot{\theta}_{g1}) + \dot{z}_{p1} – \dot{z}_{g1}] + \sin\beta \, k_m(t) f[\tan\beta (y_{p1} + R_p \theta_{p1} – y_{g1} – R_g \theta_{g1}) + z_{p1} – z_{g1}] – \tan\beta \, F_s(t) $$
where \(\beta\) is the helix angle, \(c_m\) is the mesh damping coefficient, \(k_m(t)\) is the time-varying mesh stiffness obtained from loaded tooth contact analysis (LTCA), \(F_s(t)\) is the corner mesh impact force, and \(f(x)\) is a piecewise nonlinear function representing backlash. The dynamic mesh force for the right side is defined similarly. The total dynamic load on a meshing tooth pair is given by:
$$ F_d = k_m(t) \cdot (y_{p1} – y_{g1} + R_p \theta_{p1} – R_g \theta_{g1}) + F_s(t) $$
The gear parameters used in our simulation are listed in Table 1.
| Parameter | Pinion (driving) | Gear (driven) |
|---|---|---|
| Normal module (mm) | 8 | 8 |
| Transverse pressure angle (°) | 20 | 20 |
| Helix angle (°) | 16.26 | 16.26 |
| Backlash (μm) | 2 | 2 |
| Load torque (N·m) | 1088 | 1088 |
| Damping ratio | 0.1 | 0.1 |
| Density (g/cm³) | 7.85 | 7.85 |
| Number of teeth | 16 | 32 |
| Tooth hand | Left/Right | Right/Left |
| Face width (mm) | 35 | 35 |
| Moment of inertia (kg·m²) | 0.049 | 2.600 |
| Rated speed (r/min) | 1000 | 500 |
We solved the 12-DOF nonlinear differential equations using a variable-step fourth-order Runge-Kutta numerical integration method. The dynamic load response of the left-side meshing pair is shown in the figure below. The time zero corresponds to the beginning of mesh for the tooth pair under consideration.
To compute the tooth root dynamic stress, we propose an efficient method that accounts for the effect of dynamic load on the load distribution among simultaneously contacting tooth pairs. The static load distribution coefficient is insufficient for high-speed, heavy-duty herringbone gear systems because the dynamic load variation alters the share of load on each contact line. Instead of performing LTCA for every instantaneous dynamic load (which is computationally expensive), we partition the dynamic load range (from minimum to maximum) into five equal intervals. For each interval, we run LTCA once to obtain the load distribution coefficients for all contact lines along the tooth profile. By fitting these coefficients as a function of the applied load, we can then determine the actual load carried by any single tooth pair at any mesh position. This method balances accuracy and efficiency.
The dynamic load on a single meshing tooth pair of the left side is illustrated in Figure 3 (not shown here, but described). The tooth root stress at the midpoint of the tensile side of the driven gear tooth is calculated by:
$$ \sigma_{vi} = F_{vi} \cdot \sigma_{ei} \quad (i=1,\ldots,n) $$
where \(\sigma_{ei}\) is the unit-load stress at the root point (obtained from LTCA), and \(F_{vi}\) is the dynamic load carried by the i-th contact line. The total number of contact lines from mesh entry to exit is n.
Experimental verification was performed on a gear dynamic test rig. The test gearbox housed the same herringbone gear pair as in Table 1. Strain gauges were mounted at the root midpoint of the driven gear tooth on the tensile side. The test rig consisted of a driving motor, a gearbox, and a magnetic powder brake for loading. Signals were collected by a data acquisition system and processed.
We conducted experiments under various torque and speed conditions. Table 2 compares the maximum tooth root dynamic stress obtained from simulation and experiment at three different load torques, with a constant driving speed of 4500 r/min.
| Load Torque (N·m) | Simulation (MPa) | Experiment (MPa) | Relative Error (%) |
|---|---|---|---|
| 240 | 23.64 | 25.7 | 8.0 |
| 640 | 54.86 | 60.0 | 8.6 |
| 840 | 69.34 | 72.6 | 4.5 |
It is observed that the maximum tooth root dynamic stress increases with load torque. At the lowest torque (240 N·m), the relative fluctuation of stress is larger than that at 640 N·m because the tooth deflection is insufficient to overcome the backlash, leading to stronger dynamic excitation. At higher torque (840 N·m), the backlash effect is largely suppressed, and the stress fluctuation diminishes. The simulation results agree well with experimental data, with errors under 9%.
Similar comparisons were performed at different driving speeds while keeping the load torque constant at 1088 N·m. The results are listed in Table 3.
| Driving Speed (r/min) | Simulation (MPa) | Experiment (MPa) | Relative Error (%) |
|---|---|---|---|
| 1500 | 90.20 | 93.7 | 3.7 |
| 2000 | 94.60 | 103.5 | 8.6 |
| 2500 | 98.68 | 107.1 | 7.9 |
The tooth root dynamic stress increases with speed, and the stress fluctuations become more pronounced. Our numerical results capture the trend well, though some deviations exist at higher speeds due to additional dynamic effects like gyroscopic coupling and thermal deformation not included in the model.
We note that at mesh entry and exit (corresponding to angular positions around -11.5° and 11° in the figures), the discrepancy between simulation and experiment becomes larger. This is because manufacturing errors and thermal expansion in actual gears amplify the mesh impact, which our model approximates with a simplified impact force function.
Through this study, we conclude that the proposed dynamic model and tooth root stress calculation method provide reliable predictions for herringbone gear systems. The key findings are:
- The maximum tooth root dynamic stress increases with both load torque and driving speed.
- Due to backlash, the relative fluctuation of stress initially decreases and then increases as the load torque rises from low to high values.
- The efficient load distribution method based on dynamic load partitioning yields stress predictions that are within 9% of experimental measurements, making it suitable for engineering design.
- Further refinement of the mesh impact model and inclusion of thermal effects could improve accuracy at high speeds.
In summary, our work provides a practical framework for evaluating the tooth root dynamic stress in herringbone gear transmissions under realistic operating conditions. The combination of a comprehensive dynamic model and an efficient stress calculation method enables designers to assess fatigue risk and optimize gear geometry for improved reliability.