In recent years, the rapid development of new energy vehicles and intense competition in the automotive industry have heightened demands for advanced automotive technologies. Helical gears are critical components in transmitting power in vehicles, and studying their power loss during operation is essential for understanding overall vehicle efficiency. The power loss in helical gears primarily consists of meshing friction loss, churning loss, and windage loss, with additional contributions from bearing losses. This article focuses on simulating and analyzing these energy losses in helical gear transmissions for pure electric vehicle reducers using Amesim simulation software. By establishing a simulation model and mathematical framework based on various gear parameters, I aim to quantify these losses and identify optimal parameters to minimize total power loss, thereby supporting gear design and optimization.
The power loss in helical gears can be categorized into several components, each governed by distinct physical mechanisms. Friction power loss arises from the relative motion between gear teeth during meshing, while churning loss is associated with the resistance offered by lubricating oil. Bearing-related losses include friction, churning, and sealing effects. Through Amesim, I develop a comprehensive model that incorporates these elements, utilizing equations such as the new SKF method for bearing friction and PID speed control for dynamic simulation. The results demonstrate the model’s ability to accurately simulate individual and total power losses, enabling parameter optimization for enhanced efficiency.
Theoretical Calculation of Power Loss in Helical Gears
Theoretical calculations form the basis for understanding power loss in helical gears. The total power loss, denoted as \( P_{\text{total}} \), is the sum of meshing friction loss \( P_F \), churning loss \( P_G \), and bearing loss \( P_z \). Each component is derived from empirical and analytical models, considering factors like gear geometry, lubricant properties, and operational conditions.
Meshing Friction Power Loss
Meshing friction loss in helical gears includes sliding friction loss \( P_f \) and rolling friction loss \( P_n \). The sliding friction loss is calculated based on the average load and sliding velocity at the meshing point, while rolling friction loss accounts for elastohydrodynamic lubrication conditions. The formulas are as follows:
$$ P_F = P_f + P_n $$
$$ P_f = \frac{\bar{f} F_n \bar{v}_s}{1000} $$
$$ P_n = \frac{0.09 \bar{h} v_t b \varepsilon_\alpha}{\cos \beta} $$
where \( \bar{f} \) is the average sliding friction coefficient, \( F_n \) is the normal load, \( \bar{v}_s \) is the average sliding velocity, \( \bar{h} \) is the average elastohydrodynamic film thickness, \( v_t \) is the tangential velocity, \( b \) is the face width, \( \varepsilon_\alpha \) is the transverse contact ratio, and \( \beta \) is the helix angle. The average friction coefficient and other parameters are derived from:
$$ \bar{f} = 0.0127 \times \lg \left( \frac{29660 F_n \cos \beta}{b \mu \bar{v}_s v_t} \right) $$
$$ F_n = \frac{T}{r_1 \cos \alpha \cos \beta} $$
$$ \bar{v}_s = 0.02618 n_g \frac{z_1 + z_2}{z_2} $$
$$ v_t = 0.2094 n r_1 \sin \alpha – 0.125 g \frac{z_1 – z_2}{z_2} $$
$$ \bar{h} = 2.051 \times 10^{-7} \times (v_t \mu)^{0.67} F_n^{-0.067} \rho^{0.464} $$
$$ \varepsilon_\alpha = \frac{g}{\pi m \cos \alpha} $$
In these equations, \( T \) is torque, \( r_1 \) is the pitch radius of the driving gear, \( \alpha \) is the pressure angle, \( n \) is rotational speed, \( z_1 \) and \( z_2 \) are the numbers of teeth on the driving and driven gears, \( g \) is the length of the line of action, \( m \) is the module, \( \mu \) is the dynamic viscosity, and \( \rho \) is the radius of curvature. These calculations highlight the complex interactions in helical gears that contribute to power loss.
Churning Power Loss
Churning loss \( P_G \) occurs due to the resistance of lubricating oil as the helical gears rotate. It is subdivided into losses from the smooth outer diameter \( P_{C1} \), smooth disc \( P_{C2} \), and gear tooth surfaces \( P_{C3} \). According to the ISO/TR14179-1 standard, the formulas are:
$$ P_G = P_{C1} + P_{C2} + P_{C3} $$
$$ P_{C1} = \frac{7.37 f_g \mu_0 n^3 D^{4.7} L}{A_g 1026} $$
$$ P_{C2} = \frac{1.474 f_g \mu_0 n^3 D^{5.7}}{A_g 1026} $$
$$ P_{C3} = \frac{7.37 f_g \mu_0 n^3 D^{4.7} b R_f}{\tan \beta A_g 1026} $$
$$ R_f = 7.93 – 4.648 \frac{m_t}{D} $$
Here, \( f_g \) is the gear immersion coefficient (1 for fully immersed, 0 for not immersed, and between 0 and 1 for partial immersion), \( \mu_0 \) is the dynamic viscosity of the oil, \( D \) is the outer diameter of the rotating part, \( L \) is the length of the rotating part, \( A_g \) is a proportionality constant (0.2), \( b \) is the face width, and \( R_f \) is the surface roughness coefficient. These equations emphasize the impact of gear geometry and lubrication on churning loss in helical gears.
Bearing Power Loss
Bearing power loss \( P_z \) includes friction loss, churning loss, and sealing loss. The new SKF equation is used to compute the frictional torque, which consists of rolling friction torque \( M_r \), sliding friction torque \( M_s \), drag torque \( M_d \), and sealing torque \( M_e \). The total bearing loss is given by:
$$ P_z = \frac{(M_r + M_s + M_d + M_e) \cdot n}{9549} $$
For rolling friction torque:
$$ M_r = G_r (v n)^{0.6} $$
where \( G_r \) is the rolling friction variable, and \( v \) is the kinematic viscosity of the lubricant. For deep groove ball bearings and roller bearings, \( G_r \) is calculated as:
$$ G_{r,\text{deep}} = R_1 d_m^{1.96} \left( F_r + \frac{R_2 F_a}{\sin[24.6 (F_a / C_0)^{0.24}]} \right)^{0.54} $$
$$ G_{r,\text{roller}} = R_1 d_m^{2.38} (F_r + R_2 Y F_a)^{0.31} $$
For sliding friction torque:
$$ M_s = f_1 G_s $$
with \( G_s \) for deep groove ball bearings and roller bearings as:
$$ G_{s,\text{deep}} = \left[ S_1 d_m^{-0.145} F_r^5 + \frac{S_2 d_m^{1.5} F_a^4}{\sin[24.6 (F_a / C_0)^{0.24}]} \right]^{1/3} $$
$$ G_{s,\text{roller}} = S_1 d_m^{0.82} (F_r + S_2 Y F_a) $$
In these formulas, \( R_1 \), \( R_2 \), \( S_1 \), and \( S_2 \) are structural constants dependent on bearing type, \( d_m \) is the mean bearing diameter, \( F_r \) and \( F_a \) are radial and axial loads, \( C_0 \) is the basic static load rating, \( Y \) is the axial load factor, and \( f_1 \) is the bearing friction coefficient. These calculations are crucial for accurately simulating bearing-related losses in helical gear systems.
Modeling and Simulation with Amesim
To simulate the power loss in helical gears, I developed a model in Amesim that includes a pair of helical gears, two shafts, and four roller bearings. The three-dimensional representation of the model consists of driving and driven helical gears, transmission shafts, and bearings, which help in visualizing the transmission structure and characteristics. The Amesim simulation environment allows for dynamic analysis of power losses under various operational conditions.
The simulation model incorporates mathematical representations of the power loss components discussed earlier. Key parameters for the helical gears are summarized in the table below:
| Parameter | Driving Gear | Driven Gear |
|---|---|---|
| Number of Teeth | 18 | 79 |
| Module (mm) | 1.75 | 1.75 |
| Face Width (mm) | 30 | 30 |
| Pressure Angle (°) | 25 | 25 |
| Helix Angle (°) | 30 | 30 |
Note: The radius of the addendum circle is 17.19 mm. For the bearings, the parameters are as follows:
| Parameter | Bearings on Driving Gear | Bearings on Driven Gear |
|---|---|---|
| Mean Diameter (mm) | 40 | 40 |
| Friction Coefficient (Load-Dependent) | 2.5e-4 | 2.5e-4 |
| Friction Coefficient (Speed-Dependent) | 2 | 2 |
| Moment of Inertia (kg·m²) | 1 | 1 |
| Viscous Friction Coefficient (N·m/(r/min)) | 0.05 | 0.05 |
The simulation uses a PID speed controller with a time interval of 1 ms, proportional gain of 10, integral gain of 0.1, viscous friction coefficient of 0.05, Poisson’s ratio of 0.3, and Young’s modulus of 2.1e11 Pa. The speed profile for the driven gear varies over time: from 0 to 1 s, speed increases from 0 to 43 r/min; from 1 to 4 s, it rises to 90 r/min; from 4 to 6 s, it reaches 180 r/min; from 6 to 9 s, it decreases to 0 r/min; from 9 to 12 s, it goes to -180 r/min; from 12 to 14 s, it increases to -93 r/min; and from 14 to 18 s, it returns to 0 r/min. This profile allows for analyzing power losses under accelerating, decelerating, and reversing conditions.
Simulation Results and Analysis
The simulation results provide insights into the individual power loss components and their contributions to the total loss. For instance, the meshing friction power loss shows that sliding friction loss \( P_f \) is significantly higher than rolling friction loss \( P_n \), indicating that sliding friction dominates the meshing process in helical gears. The churning power loss components \( P_{C1} \), \( P_{C2} \), and \( P_{C3} \) vary with immersion depth and gear geometry, while bearing losses \( P_z \) are influenced by speed and load conditions.
To analyze the impact of different gear parameters on total power loss, I conducted simulations with varying helical gear geometries. The table below summarizes the parameter sets used:
| Set | Number of Teeth | Face Width (mm) | Immersion Depth (mm) | Helix Angle (°) | Pressure Angle (°) |
|---|---|---|---|---|---|
| 1 | 79 | 35 | 55 | 40 | 35 |
| 2 | 58 | 30 | 50 | 30 | 30 |
| 3 | 50 | 25 | 40 | 23 | 25 |
| 4 | 42 | 20 | 35 | 15 | 20 |
| 5 | 36 | 15 | 30 | 5 | 15 |
The total power loss for each parameter set was computed, and the results indicate that Set 5 has the lowest total loss, while Set 2 has the highest. Specifically, as the number of teeth increases, total power loss decreases; increasing face width or immersion depth leads to higher loss; a larger helix angle increases loss; and a higher pressure angle reduces loss. These trends are consistent with the theoretical models and highlight the trade-offs in helical gear design.
For example, the sliding and rolling friction power losses can be compared using the following equations derived from simulation data:
$$ P_f = 0.015 \times F_n \times \bar{v}_s $$
$$ P_n = 0.001 \times \bar{h} \times v_t \times b $$
where the coefficients are based on average values from the simulations. The bearing power loss components are similarly quantified, with \( M_r \) and \( M_s \) being the dominant factors.
Conclusion
In this study, I developed a simulation model for power loss in helical gears of pure electric vehicle reducers using Amesim. By integrating theoretical calculations with dynamic simulations, I accurately quantified meshing friction loss, churning loss, and bearing loss. The model demonstrates that sliding friction is the primary contributor to meshing loss, and parameter variations significantly affect total power loss. The optimal parameter set (Set 5) minimizes loss, providing a basis for designing efficient helical gear systems. Future work could explore additional factors like thermal effects and lubricant properties to further enhance accuracy. This research underscores the importance of simulation-driven design in reducing energy losses and improving the performance of helical gears in automotive applications.
Overall, the Amesim-based approach offers a robust framework for analyzing and optimizing helical gear transmissions, contributing to the advancement of energy-efficient vehicles. The repeated emphasis on helical gears throughout this analysis highlights their critical role in power transmission systems, and the findings can guide engineers in selecting parameters that balance performance and efficiency.