In this study, I investigate the large-period three-dimensional (3D) vibration characteristics of herringbone gear transmissions caused by asymmetric tooth pitch deviation. Herringbone gears are widely used in heavy machinery, aerospace, and marine applications due to their high load capacity, large contact ratio, and low axial force. However, manufacturing deviations inevitably lead to asymmetric tooth pitch errors between the left and right helical gear pairs. This asymmetry induces distinct meshing states, axial displacements, and meshing impacts that significantly affect the dynamic performance of the herringbone gear system. To reveal these effects, I propose a comprehensive large-period load-bearing contact analysis (LCA) model for asymmetric tooth pitch deviation, and then establish a dynamic model of the herringbone gear system. Through numerical simulations, I analyze the 3D vibration responses under various loads and rotational speeds, highlighting the influence of asymmetric meshing impacts on the left and right helical gear pairs. The results provide deep insights into the vibration mechanisms of herringbone gears with long-period errors and offer guidance for vibration reduction design.
1. Mathematical Description and Periodicity of Asymmetric Tooth Pitch Deviation
The tooth pitch cumulative deviation is measured along the tangent direction of the reference circle in the transverse plane. For a herringbone gear pair, the relative tooth pitch deviation along the normal direction of the meshing line is expressed as:
$$ \lambda = (F_{pt1} – F_{pt2}) \cos \beta_b \cos \alpha_t $$
where \(F_{pt1}\) and \(F_{pt2}\) are the cumulative pitch deviations of the pinion and gear, respectively; \(\beta_b\) is the base helix angle; and \(\alpha_t\) is the transverse pressure angle. The short-period fluctuation cycle corresponds to one meshing period, while the long-period fluctuation cycle is determined by the tooth numbers of both gears and the rotational periods of the shafts. The meshing period \(T_m\), rotational periods of pinion \(T_1\) and gear \(T_2\), and the large period \(T_l\) are given by:
$$ T_m = \frac{60}{n_1 Z_1}, \quad T_1 = \frac{60}{n_1}, \quad T_2 = \frac{60 Z_2}{n_1 Z_1}, \quad T_l = T_m \cdot \text{lcm}(Z_1, Z_2) $$
where \(Z_1\) and \(Z_2\) are the tooth numbers of the pinion and gear; \(n_1\) is the pinion speed; and \(\text{lcm}\) denotes the least common multiple. The basic parameters of the herringbone gear pair are listed in Table 1.
| Parameter | Pinion | Gear |
|---|---|---|
| Number of teeth | 17 | 44 |
| Normal modulus (mm) | 6 | 6 |
| Normal pressure angle (°) | 20 | 20 |
| Helix angle (°) | 24.43 | -24.43 |
| Face width (mm) | 55 | 55 |
| Relief groove width (mm) | 58 | 58 |
| Relief groove diameter (mm) | 92 | 260 |
Since \(Z_1\) and \(Z_2\) are coprime, a large period contains \(\text{lcm}(Z_1, Z_2) = 748\) meshing cycles. The relative pitch deviations along the normal direction for the left and right helical gear pairs differ significantly, forming the asymmetric tooth pitch deviation of the herringbone gear pair.
2. Large-Period Load-Bearing Contact Analysis Model
Considering the asymmetric tooth pitch deviation, the tooth surface gaps for simultaneously meshing tooth pairs in one meshing cycle consist of three parts: (i) the initial tooth surface gaps calculated from geometric contact analysis; (ii) the meshing gaps caused by the relative pitch deviation for different tooth pairs; and (iii) the normal gap increment due to the axial floating of the pinion. For a given meshing instant, assume the left and right helical gear pairs have \(m\) and \(q\) discrete contact points, respectively, with total \(N = m + q\) points. The new displacement compatibility condition is:
$$ -F_k p_k + eZ + I d_k + A \delta_p = w_k + \lambda_k $$
where \(k\) denotes the tooth pair; \(w_k\) is the initial gap; \(\lambda_k\) is the matrix of meshing gaps from pitch deviation; \(F_k\) is the \(N \times N\) normal compliance matrix; \(p_k\) is the normal load vector; \(e\) is the \(N \times 1\) unit vector; \(Z\) is the normal line displacement; \(I\) is the identity matrix; \(d_k\) is the deformed gap; and \(A\) is a matrix with entries \(\pm \tan\beta \cos\alpha_t\). The large-period LCA problem is formulated as an optimization:
$$ X_0 = \min \sum_{j=1}^{N+2} X_j $$
subject to the constraints:
$$
\begin{cases}
– F p + e Z + I d + A \delta_p + I X_N = w + \lambda \\
e^T p + X_{N+1} = P \\
\sum_{j=1}^m p_{jl} \cos \gamma_{jl} – \sum_{j=1}^q p_{jr} \cos \gamma_{jr} + X_{N+2} = 0 \\
p_j > 0 \; (d_j = 0),\quad p_j = 0 \; (d_j > 0), \\
p_j \ge 0,\quad d_j \ge 0,\quad \delta_p \ge 0,\quad X_j \ge 0
\end{cases}
$$
Solving this model repeatedly for \(\text{lcm}(Z_1, Z_2)\) meshing cycles yields the large-period comprehensive meshing stiffness, axial displacement \(\delta_p\), and composite meshing error for the herringbone gear pair. The comprehensive meshing stiffness of the left and right helical gear pairs can be expressed as:
$$ k_{m1} = \frac{T}{r_{p2} \cos \alpha_n \cos \beta (Z – \delta_{TE1})}, \quad k_{m2} = \frac{T}{r_{p2} \cos \alpha_n \cos \beta (Z – \delta_{TE2})} $$
where \(T\) is the load torque on the gear; \(\delta_{TE1}\) and \(\delta_{TE2}\) are the composite meshing errors of the left and right pairs; \(\alpha_n\) is the normal pressure angle; \(\beta\) is the reference helix angle; and \(r_{p2}\) is the reference radius of the gear.
3. Calculation of Large-Period Meshing Impact Force
The meshing impact force occurs at the initial point of engagement. For a herringbone gear with asymmetric tooth pitch deviation, the equivalent gap of the engaging tooth pair and the elastic deformation of other tooth pairs vary within each meshing cycle of the large period. The maximum impact force \(f_s\) for each helical gear pair is calculated using the formula:
$$ f_s = \left( \frac{c + 1}{2} \cdot \frac{I_1 I_2}{I_1 r_{b2}^2 + I_2 r_{b1}^2} v_s^2 k_s \cdot \frac{1}{c} \right)^{c/(c+1)} $$
where \(I_1\) and \(I_2\) are the moments of inertia of the pinion and gear; \(k_s\) is the meshing stiffness at the impact point; \(c\) is the deformation coefficient under static load; \(v_s\) is the relative velocity at the impact point; and \(r_{b1}, r_{b2}\) are the base radii. The asymmetric meshing impact forces for the left and right helical gear pairs over the large period are obtained by cycling through all \(\text{lcm}(Z_1, Z_2)\) meshing cycles.
4. Dynamic Model of the Herringbone Gear System
Figure 1 illustrates the dynamic model of the herringbone gear transmission system. The pinion and gear are each split into left (L) and right (R) parts. The equivalent support stiffness and damping in the x, y, z directions are denoted by \(k_{xij}, k_{yij}, k_{zij}\) and \(c_{xij}, c_{yij}, c_{zij}\) (\(i=1,2; j=L,R\)). The bending, tension-compression, and torsional stiffness of the central relief groove are \(k_{bi}, k_{pi}, k_{ti}\) with corresponding damping \(c_{bi}, c_{pi}, c_{ti}\). The comprehensive meshing stiffness and damping for each helical gear pair are \(k_{12j}, c_{12j}\); the meshing impact forces are \(f_{s1}, f_{s2}\); the composite meshing error excitations are \(e_{TE1}, e_{TE2}\); and the base helix angles are \(\beta_{bL} = \beta_b, \beta_{bR} = -\beta_b\).
The relative vibration displacements along the normal direction for the left and right helical gear pairs are:
$$
\begin{aligned}
\lambda_{12L} &= [(x_{1L} – x_{2L}) \sin \psi_{12L} + (y_{1L} – y_{2L}) \cos \psi_{12L} + (r_{b1L} \theta_{1L} + r_{b2L} \theta_{2L})] \cos \beta_{bL} \\
&\quad + (-z_{1L} + z_{2L}) \sin \beta_{bL} – \delta_p \sin \beta_{bL} – e_{TE1} \\
\lambda_{12R} &= [(x_{1R} – x_{2R}) \sin \psi_{12R} + (y_{1R} – y_{2R}) \cos \psi_{12R} + (r_{b1R} \theta_{1R} + r_{b2R} \theta_{2R})] \cos \beta_{bR} \\
&\quad + (-z_{1R} + z_{2R}) \sin \beta_{bR} – \delta_p \sin \beta_{bR} – e_{TE2}
\end{aligned}
$$
where \(x_{ij}, y_{ij}, z_{ij}\) are the translational vibration displacements of the mass centers; \(\theta_{ij}\) are the torsional displacements; \(\psi_{12L}\) and \(\psi_{12R}\) are the angles between the transverse meshing line and the y-axis; \(\delta_p\) is the large-period axial displacement excitation.
Applying Newton’s second law, the dynamic equations for the pinion left part (1L) are:
$$
\begin{aligned}
m_{1L} \ddot{x}_{1L} + c_{x1L} \dot{x}_{1L} + k_{x1L} x_{1L} + c_{b1}(\dot{x}_{1L} – \dot{x}_{1R}) + k_{b1}(x_{1L} – x_{1R}) + F_{nLx} + f_{s1} \cos \beta_{bL} \sin \psi_{12L} &= 0 \\
m_{1L} \ddot{y}_{1L} + c_{y1L} \dot{y}_{1L} + k_{y1L} y_{1L} + c_{b1}(\dot{y}_{1L} – \dot{y}_{1R}) + k_{b1}(y_{1L} – y_{1R}) + F_{nLy} + f_{s1} \cos \beta_{bL} \cos \psi_{12L} &= 0 \\
m_{1L} \ddot{z}_{1L} + c_{z1L} \dot{z}_{1L} + k_{z1L} z_{1L} + c_{p1}(\dot{z}_{1L} – \dot{z}_{1R}) + k_{p1}(z_{1L} – z_{1R}) – F_{nLz} – f_{s1} \sin \beta_{bL} &= 0 \\
I_{1L} \ddot{\theta}_{1L} + c_{t1}(\dot{\theta}_{1L} – \dot{\theta}_{1R}) + k_{t1}(\theta_{1L} – \theta_{1R}) + (F_{nL} + f_{s1}) r_{b1L} \cos \beta_{bL} &= T_{1L}
\end{aligned}
$$
Similar equations hold for the pinion right part (1R), gear left part (2L), and gear right part (2R). The normal mesh forces are \(F_{nj} = c_{12j} \dot{\lambda}_{12j} + k_{mj} \lambda_{12j}\) with \(j = L,R\). The transverse vibration acceleration (end-face acceleration) is defined as:
$$
\begin{aligned}
a_{12L} &= (\ddot{x}_{1L} – \ddot{x}_{2L}) \sin \psi_{12L} + (\ddot{y}_{1L} – \ddot{y}_{2L}) \cos \psi_{12L} + (r_{b1L} \ddot{\theta}_{1L} + r_{b2L} \ddot{\theta}_{2L}) \\
a_{12R} &= (\ddot{x}_{1R} – \ddot{x}_{2R}) \sin \psi_{12R} + (\ddot{y}_{1R} – \ddot{y}_{2R}) \cos \psi_{12R} + (r_{b1R} \ddot{\theta}_{1R} + r_{b2R} \ddot{\theta}_{2R})
\end{aligned}
$$
The dynamic equations are solved using the Runge-Kutta numerical integration method to obtain the large-period dynamic response. The main system parameters are listed in Table 2.
| Parameter | Pinion | Gear |
|---|---|---|
| Mass (kg) | 7.11 | 64.75 |
| Moment of inertia (kg·m²) | 0.012 | 0.7279 |
| Support stiffness \(k_x\) (N/m) | 2.42×10⁸ | 3.83×10⁸ |
| Support stiffness \(k_y\) (N/m) | 2.42×10⁸ | 3.83×10⁸ |
| Support stiffness \(k_z\) (N/m) | — | 2.96×10⁸ |
| Bending stiffness \(k_b\) (N/m) | 1.11×10¹⁰ | 7.11×10¹¹ |
| Tension-compression stiffness \(k_p\) (N/m) | 2.36×10¹⁰ | 1.89×10¹¹ |
| Torsional stiffness \(k_t\) (N·m/rad) | 9.61×10⁶ | 6.13×10⁸ |
5. Three-Dimensional Vibration Characteristics under Different Loads
I first analyze the 3D vibration responses at a fixed pinion speed of 2000 r/min under gear torques of 828 N·m, 2000 N·m, and 5000 N·m. The meshing frequency \(f_m = 566.67\) Hz, pinion shaft frequency \(f_1 = 33.33\) Hz, and gear shaft frequency \(f_2 = 12.88\) Hz. The vibration displacement and acceleration spectra for the left helical gear pair are examined. The results show that in the displacement spectra, the shaft frequency components (especially \(f_2\)) are most prominent, while the meshing frequency and its harmonics are relatively small. In the axial vibration displacement spectra, sidebands around the meshing frequency appear, such as \(2f_m – 5f_1\) and \(2f_m – 2f_1\). With increasing load, the amplitudes of meshing frequency and its harmonics gradually increase, while the sideband amplitudes decrease. This occurs because the elastic deformation of teeth becomes dominant over the pitch deviation effect at higher loads, increasing the effective contact ratio and meshing stiffness.
The acceleration spectra show that the shaft frequency components are almost invisible. In the end-face acceleration spectra, the meshing frequency and its harmonics are clear, with a series of sidebands on both sides. The \(2f_m\) sidebands are particularly complex, indicating amplitude modulation by both the pinion and gear shaft frequencies. The axial acceleration spectra exhibit weaker meshing components, and at lower loads, the sidebands dominate. As load increases, the meshing frequency amplitudes become larger and sidebands diminish.
I further compute the root-mean-square (RMS) values of end-face and axial accelerations for six load levels (650, 828, 984, 1113, 2000, and 5000 N·m). The RMS axial acceleration is larger than the end-face acceleration under all loads. Both increase monotonically with load, but the growth rate of axial acceleration is higher at low loads and slows down at high loads.
6. Three-Dimensional Vibration Characteristics under Different Speeds
Next, I fix the gear torque at 5000 N·m and vary the pinion speed from 800 to 20,000 r/min. The RMS values of 3D acceleration are plotted as functions of speed. The end-face acceleration RMS increases rapidly with speed, showing two distinct peaks at approximately half the resonance speed (\(N_r/2\)) and the main resonance speed (\(N_r\)). In contrast, the axial acceleration RMS increases slowly with speed and exhibits multiple irregular peaks that do not align with \(N_r\) or \(N_r/2\). This indicates that the axial vibration is more sensitive to the asymmetric meshing impact excitation, which becomes dominant at high speeds.
At speeds below the main resonance, the left and right helical gear pairs exhibit nearly identical 3D acceleration RMS curves. However, when the speed exceeds the resonance, the curves diverge noticeably. This divergence is attributed to the increasingly asymmetric meshing impact forces between the left and right sides.
7. Comparison of Left and Right Helical Gear Pairs under Asymmetric Meshing Impacts
To illustrate the effect of asymmetric meshing impacts, I compare the dynamic responses of the left and right helical gear pairs at two speed conditions: 2000 r/min and 17,200 r/min, both under 5000 N·m load. At 2000 r/min, the left and right end-face acceleration RMS values are 67.84 m/s² and 68.75 m/s², respectively, while axial acceleration RMS values are 111.53 m/s² and 111.50 m/s². The differences are negligible. The dynamic mesh force and dynamic load factor are also similar: left maximum dynamic load factor 1.1267, right 1.1445, a difference of only 0.0178.
At 17,200 r/min (above resonance), the left end-face acceleration RMS reaches 2379.46 m/s² and the right reaches 2428.73 m/s²; left axial RMS 176.33 m/s², right 183.21 m/s². The dynamic load factor difference increases: left 1.4161, right 1.4597, a difference of 0.0436. This confirms that at high speeds, the asymmetric meshing impact forces dominate the 3D vibration, causing distinct behaviors between the two helical gear pairs.
8. Conclusion
In this work, I have systematically investigated the large-period three-dimensional vibration characteristics of herringbone gears with asymmetric tooth pitch deviation. The key findings are summarized as follows:
- The spectra of 3D vibration displacement are dominated by shaft frequency components, while meshing frequency and its harmonics are relatively small. In contrast, the end-face acceleration spectra exhibit clear meshing frequency components with rich sidebands due to frequency modulation by both shaft frequencies.
- With increasing load, the amplitudes of meshing frequency and its harmonics in both displacement and acceleration spectra increase, while sideband amplitudes decrease. The RMS of both end-face and axial acceleration increases monotonically with load.
- As the pinion speed exceeds the main resonance speed, the asymmetric meshing impact forces become the dominant excitation, leading to significant differences in the 3D vibration characteristics between the left and right helical gear pairs.
- The axial vibration is more sensitive to asymmetric impacts, showing irregular peaks in the RMS vs. speed curves that do not correspond to the traditional resonance speeds.
These insights are crucial for the design and condition monitoring of herringbone gear transmissions, especially when high-speed operation and manufacturing deviations are present. The proposed large-period LCA model and dynamic analysis method provide a powerful tool for predicting and mitigating vibration in such systems.