In mechanical transmission systems, helical gears are widely used due to their smooth operation and high load capacity. However, friction between tooth surfaces significantly influences vibration, noise, and efficiency. This study focuses on analyzing the time-varying friction excitation in helical gears and its impact on meshing efficiency. We derive analytical methods for calculating friction forces and moments based on the time-varying contact line length and investigate how sliding friction affects meshing forces and efficiency under varying operating conditions.
The meshing process of helical gears involves continuous changes in the contact line length due to the helical angle and alternating tooth pairs. This time-varying characteristic leads to fluctuations in meshing stiffness, friction forces, and moments, which in turn affect the dynamic behavior and efficiency of the gear system. In this work, we develop a comprehensive model that incorporates the time-varying nature of contact lines and friction coefficients to evaluate these effects accurately.
To begin, we analyze the sliding friction on the meshing tooth surfaces of helical gears. During engagement, the relative sliding velocity between tooth surfaces changes direction as the contact point passes the pitch line. This results in varying friction forces and moments that influence the overall meshing performance. The friction force and moment calculations are based on dividing the contact line into segments on either side of the pitch line and summing the contributions from all simultaneously engaged tooth pairs.
The total friction force \( F_f \) and friction moments \( T_{fp} \) and \( T_{fg} \) for the driving and driven gears, respectively, are given by:
$$ F_f = \sum_{j=0}^{n-1} f_f(\mu + j p_{bt}) $$
$$ T_{fp} = \sum_{j=0}^{n-1} t_{fp}(\mu + j p_{bt}) $$
$$ T_{fg} = \sum_{j=0}^{n-1} t_{fg}(\mu + j p_{bt}) $$
where \( n \) is the number of simultaneously engaged tooth pairs, \( p_{bt} \) is the base pitch, and \( \mu \) is the meshing position coordinate. The single-tooth friction force \( f_f \) and moments \( t_{fp} \), \( t_{fg} \) are calculated as:
$$ f_f = \frac{F_n}{L} \left[ l_R(\mu) f_R(\mu) – l_L(\mu) f_L(\mu) \right] $$
$$ t_{fp} = \frac{F_n}{L} \left[ l_L(\mu) f_L(\mu) h_{Lp}(\mu) – l_R(\mu) f_R(\mu) h_{Rp}(\mu) \right] $$
$$ t_{fg} = \frac{F_n}{L} \left[ l_R(\mu) f_R(\mu) h_{Rg}(\mu) – l_L(\mu) f_L(\mu) h_{Lg}(\mu) \right] $$
Here, \( F_n \) is the normal force, \( L \) is the total contact line length, \( l_L \) and \( l_R \) are the lengths of the contact line segments on the left and right sides of the pitch line, \( f_L \) and \( f_R \) are the friction coefficients at the midpoints of these segments, and \( h_{Lp}, h_{Rp}, h_{Lg}, h_{Rg} \) are the friction arms for the driving and driven gears.
The time-varying contact line length is a critical factor in these calculations. For a single tooth pair, the contact line lengths on the left and right sides of the pitch line vary with the meshing position \( \mu \). The expressions for \( l_L(\mu) \) and \( l_R(\mu) \) are as follows:
$$ l_L(\mu) =
\begin{cases}
\frac{\mu}{\sin \beta_b} & \mu \leq \overline{B_2P} \\
\frac{\overline{B_2P}}{\sin \beta_b} & \overline{B_2P} < \mu \leq \varepsilon_\beta p_{bt} \\
\frac{\overline{B_2P} + \varepsilon_\beta p_{bt} – \mu}{\sin \beta_b} & \varepsilon_\beta p_{bt} < \mu \leq \overline{B_2P} + \varepsilon_\beta p_{bt} \\
0 & \overline{B_2P} + \varepsilon_\beta p_{bt} < \mu \leq \varepsilon_\gamma p_{bt}
\end{cases} $$
$$ l_R(\mu) =
\begin{cases}
0 & \mu \leq \overline{B_2P} \\
\frac{\mu – \overline{B_2P}}{\sin \beta_b} & \overline{B_2P} < \mu \leq \varepsilon_\alpha p_{bt} \\
\frac{\varepsilon_\alpha p_{bt} – \overline{B_2P}}{\sin \beta_b} & \varepsilon_\alpha p_{bt} < \mu \leq \overline{B_2P} + \varepsilon_\beta p_{bt} \\
\frac{\varepsilon_\gamma p_{bt} – \mu}{\sin \beta_b} & \overline{B_2P} + \varepsilon_\beta p_{bt} < \mu \leq \varepsilon_\gamma p_{bt}
\end{cases} $$
where \( \varepsilon_\alpha \), \( \varepsilon_\beta \), and \( \varepsilon_\gamma \) are the transverse, axial, and total contact ratios, respectively, and \( \beta_b \) is the base helix angle. The total contact line length \( L \) for all engaged tooth pairs is the sum of individual lengths over one base pitch period.
The friction arms for the driving gear are given by:
$$ h_{Lp}(\mu) =
\begin{cases}
\overline{N_1B_2} + \frac{\mu}{2} & \mu \leq \overline{B_2P} \\
\overline{N_1B_2} + \frac{\overline{B_2P}}{2} & \overline{B_2P} < \mu \leq \varepsilon_\beta p_{bt} \\
\overline{N_1B_2} + \frac{\overline{B_2P} – \varepsilon_\beta p_{bt} + \mu}{2} & \varepsilon_\beta p_{bt} < \mu \leq \overline{B_2P} + \varepsilon_\beta p_{bt} \\
0 & \overline{B_2P} + \varepsilon_\beta p_{bt} < \mu \leq \varepsilon_\gamma p_{bt}
\end{cases} $$
$$ h_{Rp}(\mu) =
\begin{cases}
0 & \mu \leq \overline{B_2P} \\
\overline{N_1B_2} + \frac{\overline{B_2P} + \mu}{2} & \overline{B_2P} < \mu \leq \varepsilon_\alpha p_{bt} \\
\overline{N_1B_2} + \frac{\overline{B_2P} + \varepsilon_\alpha p_{bt}}{2} & \varepsilon_\alpha p_{bt} < \mu \leq \overline{B_2P} + \varepsilon_\beta p_{bt} \\
\overline{N_1B_2} + \frac{\varepsilon_\alpha p_{bt} – \varepsilon_\beta p_{bt} + \mu}{2} & \overline{B_2P} + \varepsilon_\beta p_{bt} < \mu \leq \varepsilon_\gamma p_{bt}
\end{cases} $$
For the driven gear, the friction arms are derived as \( h_{Lg}(\mu) = \overline{N_1N_2} – h_{Lp}(\mu) \) and \( h_{Rg}(\mu) = \overline{N_1N_2} – h_{Rp}(\mu) \), where \( \overline{N_1N_2} \) is the total length of the path of contact.
The friction coefficient is modeled using an improved elastohydrodynamic lubrication (EHL) formula that accounts for surface roughness, lubricant viscosity, and other factors. The friction coefficient \( f \) at any point is expressed as:
$$ f = e^{f(R_S, P_h, \nu_0, S)} P_h^{b_2} |R_S|^{b_3} V_e^{b_6} \nu_0^{b_7} R^{b_8} $$
where
$$ f(R_S, P_h, \nu_0, S) = b_1 + b_4 |R_S| P_h \log_{10}(\nu_0) + b_5 e^{-|R_S| P_h \log_{10}(\nu_0)} + b_9 e^S $$
In this model, \( R_S \) is the slide-to-roll ratio, \( P_h \) is the maximum Hertzian contact pressure, \( \nu_0 \) is the dynamic viscosity of the lubricant, \( S \) is the root mean square surface roughness, \( R \) is the equivalent radius of curvature, and \( V_e \) is the entrainment velocity. The coefficients \( b_1 \) to \( b_9 \) are regression constants specific to the lubricant type.
The meshing efficiency of helical gears is defined as the ratio of output power to input power. Considering only sliding friction losses, the instantaneous efficiency \( \eta \) is given by:
$$ \eta = \frac{T_g \omega_g}{T_p \omega_p} = 1 – \frac{1}{T_p \omega_p} |F_f V_s| $$
where \( T_p \) and \( T_g \) are the torques on the driving and driven gears, \( \omega_p \) and \( \omega_g \) are their angular velocities, and \( V_s \) is the sliding velocity.
To analyze the effects of various parameters, we conduct simulations based on a case study of a metro gear transmission system. The geometric and operating parameters are summarized in the following table:
| Parameter | Driving Gear | Driven Gear |
|---|---|---|
| Number of Teeth | 16 | 107 |
| Module (mm) | 5.5 | 5.5 |
| Pressure Angle (°) | 20 | 20 |
| Helix Angle (°) | 17 | 17 |
| Face Width (mm) | 75 | 75 |
| Input Speed (rpm) | 1800 | – |
| Input Torque (N·m) | 1008 | – |
The material properties include an elastic modulus of 206 GPa and a Poisson’s ratio of 0.3. The surface roughness is 0.8 μm, resulting in an RMS roughness \( S \) of 1.13 μm. The lubricant is a 75W-90 transmission oil with a density of 0.86 kg/L and a kinematic viscosity of 15.7 mm²/s at 100°C, giving a dynamic viscosity \( \nu_0 \) of 13.5 × 10³ Pa·s.
The time-varying contact line length exhibits periodic fluctuations due to the helical geometry. The total contact line length \( L \) varies between a minimum of 113.4 mm and a maximum of 130 mm over one meshing cycle. This variation influences the friction forces and moments, as shown in the following analysis.
The friction force for a single tooth pair changes direction when the contact point crosses the pitch line. The total friction force peaks at transitions between single and double tooth engagement regions. Similarly, the friction moments for the driving and driven gears show opposing trends due to the constant sum of friction arms.
Considering friction effects, the total meshing force on the tooth surfaces is less than the normal force \( F_n \). This is derived from the moment equilibrium on the driving gear:
$$ T_p = F_n(\mu) \cos \beta_b r_{bp} – T_{fp}(\mu) \Rightarrow F_n(\mu) = \frac{T_p + T_{fp}(\mu)}{r_{bp} \cos \beta_b} $$
where \( r_{bp} \) is the base radius of the driving gear. Thus, the presence of friction reduces the effective meshing force.
The meshing efficiency varies throughout the engagement cycle. The maximum instantaneous efficiency reaches approximately 99.62%, coinciding with minimal friction force. The average meshing efficiency over one cycle is 99.53%, indicating that sliding friction accounts for a small but significant power loss.
We further investigate the influence of operating parameters on meshing force and efficiency. The following table summarizes the effects of surface roughness and lubricant viscosity on meshing force:
| Parameter | Variation | Effect on Meshing Force |
|---|---|---|
| Surface Roughness | Increase | Decrease |
| Lubricant Viscosity | Increase | Decrease |
These effects are more pronounced in regions with multiple tooth engagement. For instance, higher surface roughness or viscosity reduces the friction coefficient, leading to lower meshing forces. This is because increased roughness or viscosity alters the EHL conditions, reducing the effective friction.
The average meshing efficiency is affected by speed, torque, surface roughness, and lubricant viscosity. The following equations describe these relationships based on regression analysis:
$$ \eta_{\text{avg}} = c_0 + c_1 \omega_p + c_2 T_p + c_3 S + c_4 \nu_0 $$
where \( c_0 \) to \( c_4 \) are coefficients determined from simulations. Specifically, efficiency increases with speed and torque but decreases with surface roughness, especially at low temperatures where lubricant viscosity is high.
Geometric parameters such as face width and helix angle also play a crucial role. The table below shows how changes in these parameters affect the contact line length and meshing force fluctuations:
| Parameter | Variation | Effect on Contact Line Fluctuation | Effect on Meshing Force Fluctuation |
|---|---|---|---|
| Face Width | Increase | Increase | Increase |
| Helix Angle | Increase | Increase | Increase |
For example, when the face width is 60 mm and the helix angle is 13°, the axial contact ratio approaches unity, resulting in minimal fluctuations in contact line length and meshing force. This optimization reduces friction-induced vibrations and improves efficiency.
Additionally, module and pressure angle influence meshing efficiency. The following equations approximate these effects:
$$ \eta \propto m_n^{d_1} \alpha_n^{d_2} $$
where \( m_n \) is the module, \( \alpha_n \) is the pressure angle, and \( d_1 \), \( d_2 \) are positive exponents. Efficiency slightly increases with larger modules and pressure angles due to reduced specific sliding and improved load distribution.
In conclusion, the time-varying friction in helical gears significantly affects meshing forces and efficiency. The analytical models developed here provide a foundation for optimizing gear design and operating conditions. Future work could extend this approach to include thermal effects and more complex lubricant behaviors.
Overall, this study highlights the importance of considering time-varying friction in the design and analysis of helical gears. By accurately modeling these effects, engineers can enhance the performance, durability, and efficiency of gear transmission systems in various applications.