In this study, we investigate the influence of contact ratio on the nonlinear vibration characteristics of herringbone gear transmission systems. Herringbone gears, also known as double-helical gears, are widely used in high-power and high-speed applications due to their ability to cancel axial thrust forces and provide smooth load transmission. However, their dynamic behavior is complicated by time-varying mesh stiffness, tooth backlash, mesh impact, and coupling between bending, torsion, and axial vibrations. Increasing the contact ratio is often considered a way to enhance load sharing and reduce noise, but its effect on nonlinear system dynamics, especially when considering mesh impact and backlash, has not been fully explored. Our work aims to fill this gap by developing a comprehensive nonlinear dynamic model for herringbone gears, incorporating a new corner meshing impact model that accounts for contact ratio, and analyzing the vibration responses under different contact ratios.
We first perform tooth contact analysis (TCA) and loaded tooth contact analysis (LTCA) to compute the integrated mesh stiffness and single-tooth mesh stiffness, taking into account assembly misalignments. Based on these results, we propose a meshing impact calculation model that explicitly includes the contact ratio. A 12-degree-of-freedom (12-DOF) bending-torsion-shaft coupled nonlinear vibration model is then established for the herringbone gear pair, considering time-varying mesh stiffness, corner meshing impact, and gear backlash. A marine single-stage herringbone gear set is taken as an example, and the contact ratio is adjusted by varying the tooth height modification coefficients. The computed mesh impact forces from our proposed model are compared with those from a literature model to verify its effectiveness. Finally, we analyze the influence of contact ratio on the system’s circumferential mesh vibration and pinion axial vibration under constant load conditions.
1. Modeling of Corner Meshing Impact Considering Contact Ratio
During gear meshing, due to transmission errors and tooth deflections, the actual points of engagement and disengagement deviate from the theoretical ones, causing corner meshing impact. This impact is more pronounced at the entry side (mesh-in impact). We focus on mesh-in impact, as it has been shown to be more significant than mesh-out impact. The impact arises from both the base pitch error and the change in the number of contacting tooth pairs; the latter is treated as a stiffness excitation in our dynamic model.
Rather than using the traditional “tooth deflection–load history” approach, we integrate TCA and LTCA to obtain the loaded transmission error at the mesh-in point, which represents the normal tooth deflection at impact. The impact position and impact velocity are derived following the geometry presented in the literature. The key contribution of our work is the development of an impact model that accounts for the fact that when a new tooth pair enters meshing, there may already be one or more pairs in contact (common in helical/herringbone gears due to high contact ratio). The impact energy is distributed between the impacting tooth pair and the existing contacting pairs, as shown in Figure 1.
According to impact mechanics, the kinetic energy at the mesh-in point is given by:
$$
E_k = \frac{1}{2} \frac{I_1 I_2 v_s^2}{(I_1 r’_{b2}^2 + I_2 r_{b1}^2)} = \frac{1}{2} q_s \delta_s^2 + \frac{1}{2} q_r \delta_r^2
$$
where:
- $E_k$ is the impact kinetic energy,
- $I_1$, $I_2$ are the moments of inertia of the pinion and gear,
- $v_s$ is the impact velocity at point D (the entry corner contact point),
- $r_{b1}$ is the base radius of the pinion under normal meshing,
- $r’_{b2}$ is the instantaneous base radius of the gear during corner contact,
- $q_s$ is the compliance of the single impacting tooth pair at point D, obtained from the loaded tooth contact analysis by dividing the integrated mesh stiffness by the load sharing ratio,
- $q_r$ is the combined compliance of the other tooth pairs that are already in contact at the instant of impact,
- $\delta_s$ is the deflection of the impacting tooth pair,
- $\delta_r$ is the deflection of the remaining tooth pairs,
- $\theta$ is the angle between the instantaneous line of action at the corner contact point and the theoretical line of action, determined from the geometry.
The angle $\theta$ is computed as:
$$
\theta = \arccos\left( \frac{r_{b2}’}{r_{O2D}} \right) – \angle PO_2D – \alpha
$$
where $r_{O2D}$ is the tip circle radius of the driven gear, $\alpha$ is the pressure angle, and $\angle PO_2D$ is the geometric angle obtained from the mesh geometry.
The maximum impact force $F_s$ is then obtained by equating the work done to the kinetic energy:
$$
F_s = \frac{v_s \sqrt{I_1 I_2}}{\sqrt{(I_1 r_{b2}’^2 + I_2 r_{b1}^2)(q_s + q_r \cos^2 \theta)}}
$$
This formula accounts for the cushioning effect of the already meshing tooth pairs, which is particularly important when the contact ratio is high. Our model thus provides a more accurate prediction of the mesh impact force compared to models that neglect the presence of other tooth pairs.
2. Bending-Torsion-Axial Coupled Nonlinear Dynamic Model of Herringbone Gears
In a herringbone gear transmission, the left-hand and right-hand helical gear pairs cannot be perfectly symmetrical due to manufacturing and assembly errors. This results in coupled bending, torsion, and axial vibrations. The pinion is often mounted in a floating arrangement. We develop a 12-DOF lumped-parameter model as shown schematically in Figure 2.
The generalized displacement vector is:
$$
\{\delta\} = \{ y_{p1}, z_{p1}, \theta_{p1}, y_{g1}, z_{g1}, \theta_{g1}, y_{p2}, z_{p2}, \theta_{p2}, y_{g2}, z_{g2}, \theta_{g2} \}^T
$$
where subscripts $p$ and $g$ denote pinion and gear, respectively, and 1 and 2 denote the left and right helical gear pairs. $y$, $z$, and $\theta$ represent translational vibrations in the $y$ (tangential) and $z$ (axial) directions, and torsional vibration, respectively.
The governing differential equations for each lumped mass are derived using Newton’s second law. For the left pinion pair (index 1):
$$
\begin{aligned}
m_{p1} \ddot{y}_{p1} + c_{p1y} \dot{y}_{p1} + k_{p1y} y_{p1} &= -F_{y1} \\
m_{p1} \ddot{z}_{p1} + c_{p12z} (\dot{z}_{p1} – \dot{z}_{p2}) + k_{p12z} (z_{p1} – z_{p2}) &= -F_{z1} \\
I_{p1} \ddot{\theta}_{p1} + c_{p1t} \dot{\theta}_{p1} + k_{p1t} \theta_{p1} &= -F_{y1} R_p + T_{p1} – F_{s1} R_p
\end{aligned}
$$
For the left gear pair (index 1):
$$
\begin{aligned}
m_{g1} \ddot{y}_{g1} + c_{g1y} \dot{y}_{g1} + k_{g1y} y_{g1} &= F_{y1} \\
m_{g1} \ddot{z}_{g1} + c_{g1z} \dot{z}_{g1} + k_{g1z} z_{g1} + c_{g12z} (\dot{z}_{g1} – \dot{z}_{g2}) + k_{g12z} (z_{g1} – z_{g2}) &= F_{z1} \\
I_{g1} \ddot{\theta}_{g1} + c_{g1t} \dot{\theta}_{g1} + k_{g1t} \theta_{g1} &= F_{y1} R_g – T_{g1} + F_{s1} R_g
\end{aligned}
$$
Similar equations hold for the right-hand pair (index 2), with appropriate sign conventions for the axial coupling forces.
The dynamic mesh forces in the tangential and axial directions are expressed as:
$$
\begin{aligned}
F_{y1} &= \cos\beta_1 \, c_{m1} \left[ \cos\beta_1 (\dot{y}_{p1} – \dot{y}_{g1} + R_p\dot{\theta}_{p1} – R_g\dot{\theta}_{g1}) + \sin\beta_1 (\dot{z}_{p1} – \dot{z}_{g1}) \right] \\
&\quad + \cos\beta_1 \, k_{m1} \, f(\Delta_1) \\
F_{z1} &= \sin\beta_1 \, c_{m1} \left[ \cos\beta_1 (\dot{y}_{p1} – \dot{y}_{g1} + R_p\dot{\theta}_{p1} – R_g\dot{\theta}_{g1}) + \sin\beta_1 (\dot{z}_{p1} – \dot{z}_{g1}) \right] \\
&\quad + \sin\beta_1 \, k_{m1} \, f(\Delta_1)
\end{aligned}
$$
where $\beta$ is the helix angle, $k_m(t)$ is the time-varying mesh stiffness, $c_m$ is the mesh damping coefficient (assumed constant), $R_p$ and $R_g$ are the pitch radii, $T$ is the applied torque, $F_s$ is the mesh impact force from the corner contact model, and $f(\Delta)$ is a piecewise linear function representing gear backlash. The relative displacement along the line of action $\Delta_1$ is:
$$
\Delta_1 = \cos\beta_1 (y_{p1} – y_{g1} + R_p\theta_{p1} – R_g\theta_{g1}) + \sin\beta_1 (z_{p1} – z_{g1})
$$
The backlash function $f(\Delta)$ is defined as:
$$
f(\Delta) = \begin{cases}
\Delta – b, & \Delta > b \\
0, & |\Delta| \leq b \\
\Delta + b, & \Delta < -b
\end{cases}
$$
where $b$ is half the total backlash. The mesh stiffness $k_m(t)$ is obtained from the LTCA results, which inherently include the effects of manufacturing errors and misalignment through the loaded tooth contact analysis; thus, the error excitation is integrated into the stiffness excitation.
3. Verification and Parametric Study
The parameters of the example marine single-stage herringbone gear pair are listed in Table 1.
| Parameter | Pinion (Driving) | Gear (Driven) |
|---|---|---|
| Normal module (mm) | 6 | |
| Transverse pressure angle (°) | 20 | |
| Helix angle (°) | 24.43 | |
| Backlash (μm) | 2 | |
| Load torque (N·m) | 800 | |
| Damping ratio | 0.1 | |
| Density (g/cm³) | 7.85 | |
| Number of teeth | 17 | 44 |
| Hand of teeth (left/right) | Left/Right | Right/Left |
| Face width (mm) | 75 | 75 |
| Moment of inertia (kg·m²) | 0.065 | 3.70 |
| Nominal speed (r/min) | 2000 | 772 |
To study the effect of contact ratio, we vary the tooth height modification coefficient of the pinion, resulting in five different contact ratios. The mesh impact forces computed by our proposed model and by a literature model are compared in Table 2.
| Pinion modification coefficient | Contact ratio $\varepsilon$ | Impact force from literature model [8] (kN) | Impact force from our model (kN) |
|---|---|---|---|
| +0.25 | 2.72 | 9.255 | 8.896 |
| +0.12 | 2.91 | 8.661 | 8.378 |
| 0 | 3.30 | 6.593 | 6.469 |
| -0.13 | 3.68 | 5.180 | 5.121 |
| -0.22 | 4.07 | 2.107 | 2.096 |
It is observed that as the contact ratio increases, the mesh impact force decreases significantly. This is because a higher contact ratio leads to a larger combined mesh stiffness, which reduces tooth deflection and thus the effective base pitch error that causes impact. The comparison between the two models shows that our model predicts slightly lower impact forces, with deviations ranging from 1% to 5%. The deviation is more pronounced at lower contact ratios, because the existing tooth pairs have higher compliance and provide a more significant cushioning effect, which our model accounts for but the literature model neglects.
Next, we investigate the system’s vibration response. The nonlinear differential equations are nondimensionalized and solved using the variable-step fourth-order Runge-Kutta method. The vibration acceleration of the left mesh pair in the circumferential direction and the axial direction of the pinion are analyzed for different contact ratios. The root mean square (RMS) values of these accelerations are summarized in Table 3.
| Contact ratio $\varepsilon$ | Circumferential mesh acceleration RMS (m/s²) | Pinion axial acceleration RMS (m/s²) |
|---|---|---|
| 2.72 | 32.7 | 2.07 |
| 3.30 | 15.8 | 1.02 |
| 4.07 | 34.4 | 2.56 |
When the contact ratio increases from 2.72 to 3.30, both the circumferential and axial vibration accelerations decrease substantially. This is attributed to the increased mesh stiffness and reduced mesh impact force, leading to smoother meshing. However, when the contact ratio further increases to 4.07, the vibrations increase again. The RMS circumferential acceleration rises to 34.4 m/s², and the axial acceleration rises to 2.56 m/s², even exceeding the values at the lowest contact ratio. The phase plane plots for the case $\varepsilon=4.07$ reveal a strongly nonlinear behavior with multiple loops, indicating that the system has entered a regime where the dynamic backlash is fully engaged. Under constant load, the higher mesh stiffness at $\varepsilon=4.07$ results in smaller tooth deflection, which is insufficient to overcome the backlash nonlinearity. Consequently, the gear pairs experience repeated impacts due to backlash, amplifying the vibration level.
From a design perspective, this implies that simply increasing the contact ratio is not always beneficial for vibration reduction. There exists an optimal contact ratio that balances the benefits of reduced mesh impact and the adverse effects of backlash-induced impacts. For the specific gear set studied here, the contact ratio around 3.30 yields the lowest vibration levels.
4. Discussion and Implications
Our results highlight the importance of considering the combined effects of mesh stiffness, mesh impact, and backlash in herringbone gear dynamics. The proposed corner meshing impact model, which accounts for the cushioning effect of simultaneously meshing tooth pairs, provides more realistic impact forces, especially for high-contact-ratio designs. The 12-DOF nonlinear model captures the essential coupling between bending, torsion, and axial motions, which is characteristic of herringbone gears due to the opposing helix directions.
In practical applications, selecting an appropriate contact ratio (e.g., through tooth profile modifications, addendum modifications, or pressure angle adjustments) can significantly influence noise and vibration levels. For instance, in ship propulsion systems where quiet operation is critical, our analysis suggests that a contact ratio near 3.3 may be preferable to either lower or higher values. However, the exact optimal value will depend on the specific load, speed, and damping conditions.
It should be noted that our model assumes perfect manufacturing quality except for the imposed tooth modifications. In reality, additional sources of excitation such as pitch errors, runout, and bearing clearances will interact with contact ratio effects. Future work could incorporate these stochastic excitations and also investigate the influence of varying load conditions on the optimal contact ratio.
5. Conclusions
We have presented a comprehensive study on the effect of contact ratio on the nonlinear dynamic behavior of herringbone gear systems. The main conclusions are:
- A new mesh impact model for corner contact is developed, which accounts for the buffer effect of other meshing tooth pairs. This model shows better accuracy at lower contact ratios compared to existing models.
- Under constant load, when the contact ratio is below a certain threshold (around 3.3 in this example), the elastic tooth deflection is sufficient to suppress backlash nonlinearity. Increasing the contact ratio within this range reduces mesh impact forces and lowers both circumferential and axial vibrations.
- When the contact ratio exceeds that threshold (e.g., 4.07), the higher mesh stiffness leads to smaller tooth deflections that cannot overcome the backlash, causing the system to enter a strongly nonlinear regime with elevated vibration levels.
- Thus, an optimal contact ratio exists that minimizes vibration for a given load condition. Designers of herringbone gears should carefully select the contact ratio to achieve both high load capacity and low noise.
Our findings provide insights for the vibration and noise reduction of herringbone gear transmissions and can be extended to other high-contact-ratio gear types such as helical gears.