We investigate the relationship between internal nonlinear dynamic excitations, such as tooth surface friction and pitting spalling, and the external vibration response of a helical gear system. Based on the variation of the contact line of helical gear, we derive the calculation methods for friction excitation and pitting spalling stiffness excitation. An eight-degree-of-freedom translation-torsion coupling dynamic model of the helical gear pair is established using the lumped mass method, and the influences of tooth surface friction and pitting spalling on the dynamic response of the helical gear system are analyzed.
1. Internal Nonlinear Excitation of Helical Gear
1.1 Tooth Surface Friction Excitation of Helical Gear
The friction direction on the meshing plane of helical gear is opposite on two sides of the pitch line. We define friction on the right side as positive and on the left as negative. The total friction force and friction torque for a single tooth pair are expressed as:
$$
\begin{aligned}
f_f(\mu) &=\mu\frac{F_n}{L(\mu)}\left[l_R(\mu)-l_L(\mu)\right] \\
T_{fp}(\mu) &=\mu\frac{F_n}{L(\mu)}\left[l_L(\mu)h_{Lp}(\mu)-l_R(\mu)h_{Rp}(\mu)\right] \\
T_{fg}(\mu) &=\mu\frac{F_n}{L(\mu)}\left[l_R(\mu)h_{Rg}(\mu)-l_L(\mu)h_{Lg}(\mu)\right]
\end{aligned}
$$
Here, \(\mu\) is the meshing position coordinate, \(F_n\) the normal load, \(\mu\) the friction coefficient, \(L\) the total contact line length, \(l_L\) and \(l_R\) the contact line lengths on the left and right sides of the pitch line, and \(h_{Lp},h_{Rp},h_{Lg},h_{Rg}\) the friction arms of the driving and driven gears. Based on the meshing periodicity, the total friction force and torque within one base pitch are:
$$
\begin{aligned}
F_f(\mu) &=\sum_{i=0}^{N-1} f_f(\mu+i\cdot p_{bt}),\quad 0\leq\mu<p_{bt}\\ $$="" &="\sum_{i=0}^{N-1}"
where \(N\) is the number of simultaneously meshing tooth pairs, and \(p_{bt}\) the transverse base pitch.
1.2 Pitting Spalling Stiffness Excitation of Helical Gear
The unit length meshing stiffness of a single tooth pair of helical gear is approximated by a parabolic distribution:
$$
k(\mu) = k_0\left[1-4\alpha_k\left(\frac{\mu}{p_{bt}}-\frac{1}{4}\right)^2\right]
$$
where \(k_0\) is the maximum stiffness per unit length, \(\alpha_k = 0.55\). The single tooth pair stiffness is then \(k_{\text{single}}(\mu)=k(\mu)l(\mu)k_{\max}\). The total meshing stiffness of helical gear is:
$$
K(\mu)=\sum_{i=0}^{N-1}k_{\text{single}}\left(\mu+i\cdot p_{bt}\right)
$$
When pitting spalling occurs, the contact line reduces. The reduced length due to a rectangular pitting pit can be expressed as:
$$
\Delta l_p(\mu)=
\begin{cases}
0, &\mu\leq x_1\\
l_s, &x_1<\mu\leq x_2\\
0, &x_2<\mu\leq\varepsilon_\gamma p_{bt}
\end{cases}
$$
where \(l_s\) is the spalling length along the contact line, and \(x_1,x_2\) define the spalling region on the meshing line. The actual contact line length becomes \(l_p(\mu)=l(\mu)-\Delta l_p(\mu)\), and the corresponding single tooth stiffness is recalculated.
2. Eight-DOF Translation-Torsion Coupling Dynamic Model of Helical Gear Pair
Figure 1 shows the dynamic model of a helical gear pair with 8 degrees of freedom (DOF): translational displacements along x (perpendicular to line of action, OLOA), y (line of action, LOA), and z (axial), plus torsional displacement about the z-axis for both driving and driven gears.
The displacement vector is:
$$
\mathbf{X} = \left\{x_p, y_p, z_p, \theta_p, x_g, y_g, z_g, \theta_g\right\}^T
$$
Normal relative displacement along the meshing direction is:
$$
\begin{aligned}
\delta_n &= \delta_y\cos\beta_b + \delta_z\sin\beta_b \\
\delta_y &= y_p – y_g + r_{bp}\theta_p – r_{bg}\theta_g \\
\delta_z &= z_p – z_g
\end{aligned}
$$
The dynamic meshing force \(F_m = k_m\delta_n + c_m\dot{\delta}_n\), where \(k_m\) is the time-varying mesh stiffness and \(c_m\) the mesh damping. The force components are:
$$
F_y = F_m\cos\beta_b,\quad F_z = F_m\sin\beta_b
$$
The equations of motion for the helical gear system considering friction are:
$$
\begin{aligned}
m_p\ddot{x}_p + c_{px}\dot{x}_p + k_{px}x_p &= F_f \\
m_p\ddot{y}_p + c_{py}\dot{y}_p + k_{py}y_p &= -F_y \\
m_p\ddot{z}_p + c_{pz}\dot{z}_p + k_{pz}z_p &= -F_z \\
J_p\ddot{\theta}_p &= T_p – F_y r_{bp} + T_{fp} \\
m_g\ddot{x}_g + c_{gx}\dot{x}_g + k_{gx}x_g &= -F_f \\
m_g\ddot{y}_g + c_{gy}\dot{y}_g + k_{gy}y_g &= F_y \\
m_g\ddot{z}_g + c_{gz}\dot{z}_g + k_{gz}z_g &= F_z \\
J_g\ddot{\theta}_g &= -T_g + F_y r_{bg} – T_{fg}
\end{aligned}
$$
Table 1 presents the system parameters used in this study.
| Parameter | Driving gear | Driven gear |
|---|---|---|
| Number of teeth | 16 | 107 |
| Module (mm) | 5.5 | |
| Face width (mm) | 80 | 75 |
| Pressure angle (°) | 20 | |
| Helix angle (°) | 17 | |
| Mass (kg) | 5.1 | 171 |
| Moment of inertia (kg·m²) | 0.0067 | 8.1 |
| Support stiffness (N/m) | 1.2×10⁹ | |
| Support damping (N·s/m) | 4×10³ | |
| Input speed (r/min) | 1800 | |
| Input torque (N·m) | 1008 | |
The mesh damping is calculated as:
$$
c_m = 2\xi_m\sqrt{\frac{k_m}{\frac{1}{m_p} + \frac{1}{m_g} + \frac{r_{bp}^2}{J_p} + \frac{r_{bg}^2}{J_g}}}
$$
where \(\xi_m\) is the mesh damping ratio (0.03–0.17). The mesh frequency \(f_m = (n\cdot z)/60\) (Hz), with \(n\) the speed in r/min and \(z\) the number of teeth.
3. Results and Discussion
3.1 Effect of Support Stiffness on Helical Gear Vibration
We first evaluate the sensitivity of the LOA vibration displacement of the driving gear to the support stiffness. As shown in Figure 4 (not reproduced here), when the support stiffness is lower than 0.5×10⁹ N/m, the displacement fluctuates strongly. For stiffness values above 1.2×10⁹ N/m, the response stabilizes. Therefore, we adopt 1.2×10⁹ N/m for subsequent analyses to reduce the sensitivity to support stiffness.
3.2 Effect of Tooth Friction on Helical Gear Dynamic Response
Figure 5 shows the influence of friction on the dynamic meshing force. Including friction reduces the mean dynamic meshing force by about 0.6% but significantly increases the amplitudes at the mesh frequency \(f_m\) (480 Hz) and its second harmonic \(2f_m\) in the frequency domain. Thus, tooth surface friction intensifies the vibration of the helical gear.
Figure 6 indicates that friction increases the peak-to-peak value of the dynamic transmission error (DTE) and the amplitude at \(f_m\). Consequently, friction degrades the transmission accuracy of the helical gear.
Figure 7 presents the effect of friction on the displacement response of the driving helical gear. Without friction, the OLOA displacement is nearly zero; with friction, periodical vibration appears with components at mesh frequency and its harmonics. For the LOA displacement, friction reduces the mean value but amplifies the oscillatory amplitude. This demonstrates that due to the bending-torsion-axis coupling of helical gear, friction not only excites OLOA vibration but also influences the LOA response.
3.3 Effect of Pitting Spalling on Helical Gear Vibration
We compare three cases: healthy gear without friction, spalled gear without friction, and spalled gear with friction. The spalling length \(l_s\) is set to 10 mm and width \(w_s\) to 5 mm.
- Dynamic transmission error (Fig. 8): The DTE of the healthy helical gear shows periodic fluctuations. When spalling occurs, clear impact signals appear at the entry and exit regions of the spall. Including friction reduces the impact amplitude slightly but increases the overall fluctuation. In the frequency domain, the spalled gear produces sidebands around the mesh frequency and its harmonics; friction further elevates the peak amplitudes at \(f_m\) and \(2f_m\).
- Dynamic meshing force (Fig. 9): Similar impact patterns are observed. Under the coupling of spalling and friction, the impact amplitude increases. Frequency-domain sidebands appear, and the mesh frequency amplitudes are amplified with friction.
- LOA displacement of driving helical gear (Fig. 10): The trend matches the dynamic meshing force because the driving gear has higher speed and the transmission ratio is large. The spalling-induced impacts are evident.
3.4 Effect of Spalling Length on Helical Gear Vibration
We simulate four spalling lengths: 0 mm (healthy), 6 mm, 10 mm, and 14 mm, with a fixed width of 5 mm and including friction. The results are summarized in Table 2.
| Spalling length (mm) | DTE impact amplitude (relative) | Meshing force impact amplitude (relative) | LOA displacement impact amplitude (relative) |
|---|---|---|---|
| 0 | 1.0 | 1.0 | 1.0 |
| 6 | 1.35 | 1.28 | 1.31 |
| 10 | 1.72 | 1.63 | 1.68 |
| 14 | 2.15 | 2.04 | 2.10 |
As shown in Fig. 11 and Table 2, the impact amplitude in DTE, dynamic meshing force, and LOA displacement increases monotonically with spalling length. In the frequency domain, the sideband amplitudes around the mesh frequency also grow. This indicates that the spalling severity can be identified by monitoring the impact strength in the vibration signals of the helical gear.
4. Conclusion
We have presented a comprehensive study on the vibration characteristics of a helical gear system subjected to coupling excitation of tooth friction and pitting spalling. The following conclusions are drawn:
- Tooth surface friction amplifies the dynamic meshing force and increases the dynamic transmission error. Due to the coupling effect of helical gear, friction not only triggers OLOA vibration but also significantly affects LOA displacement.
- Pitting spalling introduces clear impact signals at the fault region. The coupling of friction intensifies these impact vibrations. The impact amplitude grows with increasing spalling length.
- The frequency-domain signatures, such as sidebands around mesh frequency, provide diagnostic indicators for spalling faults in helical gear transmissions.
These findings offer valuable insights for the detection and monitoring of pitting spalling faults in helical gear systems under realistic operating conditions with friction.