Herringbone gears are widely used in heavy-load and high-reliability equipment, such as marine propulsion systems, aircraft engines, and rotorcraft, due to their high contact ratio, large load-carrying capacity, and smooth transmission. In this study, I focus on a multi-parallel shaft herringbone gear power split transmission system, which is a common configuration for splitting input power into multiple paths. The system consists of five herringbone gears arranged in parallel, as described in the system schematic. A time-varying nonlinear dynamic model is established using a combined approach of lumped parameter method, multi-body dynamics, and finite element method. This model accounts for the time-varying mesh stiffness and backlash nonlinearities. I then analyze the dynamic response characteristics under internal excitation, including dynamic transmission error fluctuations, dynamic meshing forces, gear center trajectories, and bearing forces. The results reveal significant load imbalance between the left and right halves of the herringbone gears, which is the primary source of vibration and noise.
Introduction
Herringbone gears are essentially two helical gears with opposite helix angles joined together, allowing them to cancel axial thrust forces. This makes herringbone gears particularly attractive for high-power transmission systems where bearing loads must be minimized. The power split transmission system studied here employs multiple parallel shafts to distribute power from a single input to two outputs, thereby reducing the load on individual gear pairs and improving overall reliability. However, such systems are prone to dynamic issues such as uneven load sharing, vibration, and noise due to the inherent flexibility and time-varying nature of the mesh.
Dynamic modeling of herringbone gear systems is more complex than that of spur or helical gears because of the need to consider the coupling between the left and right helical halves. Many previous studies have focused on spur or helical gear dynamics, but relatively few address the specific behavior of herringbone gears. For instance, Ajmi and Velex developed a quasi-static and dynamic model for double helical gears by treating them as two opposite-handed helical gears connected by a flexible shaft. Sondkar and Kahraman proposed a dynamic model for a double-helical planetary gear set. Liu et al. analyzed the effects of friction and tooth profile errors on herringbone gear dynamics. However, most of these works employ linear dynamic models and ignore the time-varying nature of mesh stiffness and nonlinear effects such as backlash. In high-speed applications, these nonlinearities become significant and must be included for accurate predictions.
In this paper, I present a comprehensive time-varying nonlinear dynamic model for a multi-parallel shaft herringbone gear power split transmission system. The model integrates lumped parameter elements for gear tooth meshing, multi-body dynamics for gear bodies, and finite element flexibility for shafts and bearings. I calculate time-varying mesh stiffness based on the ISO 6336-6 standard, accounting for bending, shear, compression, and contact deformations. The system dynamics are solved in the time domain to obtain vibration responses. Through detailed analysis, I quantify the dynamic transmission errors, mesh forces, gear center orbits, and bearing forces, with particular emphasis on the left-right asymmetry within each herringbone gear. The findings provide crucial insights for optimizing the design of such systems to reduce vibration and noise.
System Description and Modeling
System Configuration
The multi-parallel shaft power split system comprises five herringbone gears labeled G1, G2, G3, G4, and G5. The input power is delivered to gear G3, which then splits into two paths: one through G2 to G1, and the other through G4 to G5. Gears G1 and G5 are connected to separate loads. All gears have identical module, pressure angle, and helix angle. The key design parameters are summarized in Table 1.
| Gear | Number of teeth | Module / mm | Helix angle / (°) | Pressure angle / (°) | Backlash / mm | Inner diameter / mm | Face width / mm | Groove width / mm |
|---|---|---|---|---|---|---|---|---|
| 1, 5 | 105 | 7 | 29.958 | 20 | 0.47 | 190 | 120 | 80 |
| 1, 4 | 197 | 7 | 29.958 | 20 | 0.47 | 150 | 120 | 80 |
| 3 | 32 | 7 | 29.958 | 20 | 0.47 | 100 | 120 | 80 |
Mesh Frequency and Static Mesh Force
The mesh frequency of a gear pair is given by:
$$ f_m = \frac{Z n}{2\pi} $$
where \( Z \) is the number of teeth on the driving gear and \( n \) is its rotational speed in rad/s. For the input gear (G3) rotating at 3000 rpm (314.16 rad/s), the mesh frequency is calculated as 1,600 Hz.
The static normal mesh force for a single gear pair under torque \( T \) is:
$$ F_n = \frac{2T}{d \cos(\alpha_t)} $$
where \( \alpha_t = \arctan(\tan \alpha_n / \cos \beta) \), with \( \alpha_n = 20^\circ \) and \( \beta = 29.958^\circ \). The tangential force is:
$$ F_t = F_n \cos \alpha_n $$
With an input torque of 3,183 N·m and a pitch diameter of 224 mm (for gear G3), the static normal mesh force is 6,677 N for each mesh pair (G3–G2 and G3–G4).
Time-Varying Nonlinear Dynamic Model
I model each herringbone gear as two separate helical gears (left and right halves) connected by a flexible shaft representing the groove region. The tooth contact is represented by nonlinear springs with time-varying stiffness calculated from ISO 6336-6. The stiffness variation during one mesh cycle for each gear pair is shown in Figure 2 (conceptualized in the text). The gear bodies are considered rigid, while shafts and bearings are modeled as spring-damper elements. The overall system model includes the following degrees of freedom: translational displacements in X, Y, Z directions and rotational displacements for each half of each gear. The equation of motion can be written in matrix form as:
$$ \mathbf{M}\ddot{\mathbf{q}} + \mathbf{C}\dot{\mathbf{q}} + \mathbf{K}(t, \mathbf{q})\mathbf{q} = \mathbf{F}(t) $$
where \(\mathbf{M}\) is the mass matrix, \(\mathbf{C}\) the damping matrix, \(\mathbf{K}(t, \mathbf{q})\) the nonlinear time-varying stiffness matrix, and \(\mathbf{F}(t)\) the external excitation including input torque and load torques. The backlash nonlinearity is incorporated through a piecewise function relating mesh force to relative displacement across the gear pair.
Table 2 lists the time-varying mesh stiffness values at different mesh positions for representative gear pairs. The stiffness peaks when two pairs of teeth are in contact and drops when only one pair is engaged.
| Mesh position | Stiffness (G1–G2) / (N/m) | Stiffness (G2–G3) / (N/m) | Stiffness (G3–G4) / (N/m) | Stiffness (G4–G5) / (N/m) |
|---|---|---|---|---|
| Single tooth contact | 2.1×10⁹ | 1.8×10⁹ | 1.8×10⁹ | 2.1×10⁹ |
| Double tooth contact | 3.5×10⁹ | 3.0×10⁹ | 3.0×10⁹ | 3.5×10⁹ |
Dynamic Response Under Internal Excitation
Dynamic Transmission Error
Dynamic transmission error (DTE) is defined as the difference between the actual relative rotation of two gears and the ideal kinematic relationship:
$$ \text{DTE} = r_{b1}\theta_1 – r_{b2}\theta_2 $$
where \( r_{b1} \) and \( r_{b2} \) are the base circle radii, and \( \theta_1, \theta_2 \) are the dynamic rotational displacements. DTE is a key indicator of transmission smoothness. Table 3 summarizes the computed DTE fluctuations for each gear pair in the system. The largest fluctuation (1.6 μm) occurs on the right side of the G3–G4 pair. All DTE spectra exhibit sidebands at approximately 16 Hz, which corresponds to the mesh frequency divided by the number of teeth of G1 (105), indicating modulation effects from the rotating gear.
| Gear pair | Side | Fluctuation / μm | Total fluctuation / μm |
|---|---|---|---|
| G1–G2 | Left | 0.32 | 6.49 |
| Right | 0.33 | 6.49 | |
| G2–G3 | Left | 0.94 | 6.49 |
| Right | 0.92 | 6.49 | |
| G3–G4 | Left | 0.47 | 6.49 |
| Right | 1.60 | 6.49 | |
| G4–G5 | Left | 0.95 | 6.49 |
| Right | 0.96 | 6.49 |
Dynamic Meshing Force
The dynamic meshing force between each gear pair exhibits periodic fluctuations due to the time-varying mesh stiffness and the inherent nonlinearities. Figure 5 (referenced in the text) shows the time histories for all herringbone gear pairs. A clear left-right imbalance is observed, with the left side (closer to the input) generally experiencing higher forces. The most severe imbalance occurs in the gear pairs involving G3 (G2–G3 and G3–G4). Table 4 lists the fluctuation amplitudes of the dynamic meshing forces. The maximum amplitude of 1,403 N occurs on the left side of the G3–G4 pair.
| Gear pair | Side | Fluctuation / N | Total fluctuation / N |
|---|---|---|---|
| G1–G2 | Left | 478 | 7,298 |
| Right | 458 | 7,298 | |
| G2–G3 | Left | 1,371 | 7,298 |
| Right | 1,377 | 7,298 | |
| G3–G4 | Left | 1,403 | 7,298 |
| Right | 1,296 | 7,298 | |
| G4–G5 | Left | 455 | 7,298 |
| Right | 460 | 7,298 |
Gear Center Trajectories
The motion of the gear centers in the radial plane (Y-Z) provides insight into the vibration modes. Figure 6 (conceptual) shows the orbit plots for each half of the five herringbone gears. The largest deviation between left and right centers occurs for gear G1, where the left-side center deviates by up to 0.8 μm. Gear G3 exhibits the most chaotic behavior, with its center trajectory scattered inside an approximate circle, indicating significant vibration excitation at multiple frequencies. This chaotic behavior is consistent with the high mesh force fluctuations observed for G3.
Bearing Forces
Bearings are modeled as linear springs and dampers. Table 5 lists the mean radial forces on the bearings supporting each gear. For gears G1 and G5, the right-side bearings carry larger radial loads, while gears G2, G3, and G4 show nearly equal loads on both sides. The dynamic components of the radial forces exhibit oscillations at the mesh frequency and its harmonics, as seen in Figure 7 for gear G2, where the maximum amplitude occurs at 16 Hz.
| Gear | Bearing side | Mean radial force (Y-direction) / N | Mean radial force (Z-direction) / N |
|---|---|---|---|
| G1 | Left | −2,020 | −10,625 |
| Right | −3,150 | −16,525 | |
| G2 | Left | 450 | −10,900 |
| Right | −450 | −11,100 | |
| G3 | Left | 0 | −700 |
| Right | 0 | −700 | |
| G4 | Left | −450 | −35,750 |
| Right | 450 | −35,550 | |
| G5 | Left | 2,020 | −750 |
| Right | 3,150 | −1,775 |
Axial forces on bearings are shown in Figure 8. The largest axial force occurs on the input gear G3, with a difference of nearly 30 N between its left and right bearings, consistent with the severe load imbalance observed in the G3 meshing pairs. For other gears, axial forces are relatively small but still exhibit oscillatory behavior.
Discussion on Load Imbalance in Herringbone Gears
The most striking finding from the dynamic analysis is the consistent left-right asymmetry within each herringbone gear. This imbalance is not due to manufacturing errors but arises from the dynamic interaction between the two helical halves and the flexible connection (the groove region). The left half, being closer to the power input, experiences higher dynamic loads because it receives the torque earlier in the transmission path. The flexible shaft between the halves allows relative torsional and lateral motion, leading to unequal load distribution. This effect is amplified in gear G3, which is the input gear and therefore experiences the highest dynamic excitation.
To quantify the imbalance, we define a load sharing ratio \( \lambda \) as the ratio of the dynamic mesh force on the left side to that on the right side. For the G3–G4 pair, \( \lambda \) reaches 1.08 (1,403 N left vs 1,296 N right), and for the G3–G2 pair it is approximately 0.99, indicating a near-equal share but still with a phase shift. The axial force imbalance on G3 bearings further confirms the presence of a net axial force that is not canceled, contrary to the ideal behavior of a perfectly balanced herringbone gear under static conditions.
Conclusions
A time-varying nonlinear dynamic model of a multi-parallel shaft herringbone gear power split transmission system has been established by treating each herringbone gear as two helical halves connected by a flexible shaft. The model combines lumped parameter, multi-body, and finite element techniques, providing a general framework applicable to other herringbone gear systems.
Quantitative analysis of dynamic responses under internal excitation reveals that:
- Dynamic transmission errors exhibit fluctuations up to 1.6 μm on the right side of the G3–G4 pair, with sidebands at 16 Hz, indicating potential vibration and noise sources.
- Dynamic meshing forces show clear left-right imbalance, with the left half generally carrying higher loads. The maximum fluctuation amplitude of 1,403 N occurs on the left side of the G3–G4 pair.
- Gear center trajectories indicate the most severe vibration for G3, whose center motion is scattered in a circular pattern.
- Bearing radial forces are asymmetric for gears G1 and G5, while axial forces exhibit a net imbalance of about 30 N on G3, confirming the load-sharing problem within herringbone gears.
These findings highlight that the current design leads to significant left-right load imbalance in the herringbone gears, particularly at the input stage. This imbalance is a primary source of vibration and noise. The risk points and evaluation indicators provided here serve as a basis for future optimization, such as tooth profile modifications, shaft stiffness adjustments, or bearing preload tuning, to improve the dynamic performance of multi-parallel shaft herringbone gear systems.