In the evolving landscape of automotive engineering, the electric vehicle car has emerged as a pivotal innovation, driving advancements in powertrain efficiency and reliability. As an integral component, the gear transmission system in an electric vehicle car must exhibit high performance under dynamic loads. Among various configurations, the two-stage helical gear transmission is widely adopted due to its superior load capacity, smooth operation, and high transmission efficiency. However, dynamic issues such as meshing impact—a transient force caused by minute clearances, surface irregularities, or manufacturing errors during gear engagement—can significantly influence the system’s vibrational behavior and overall stability. In this paper, I investigate the dynamic characteristics of a two-stage helical gear transmission system for an electric vehicle car, with a focus on meshing impact effects. By deriving meshing impact forces and establishing a comprehensive dynamic model, I aim to analyze how these impacts propagate through the system under varying operational speeds, thereby providing insights for optimizing electric vehicle car drivetrain design.
The importance of this study stems from the critical role that gear dynamics play in the longevity and noise-vibration-harshness (NVH) performance of an electric vehicle car. Unlike internal combustion engines, electric motors in an electric vehicle car deliver instantaneous torque, which can exacerbate dynamic excitations like meshing impact. Prior research has extensively explored meshing impact in single gear pairs, but there is a gap in understanding its coupled effects in multi-stage systems, such as those common in electric vehicle car transmissions. This work addresses that gap by considering a two-stage helical gear arrangement, typical in electric vehicle car applications, where impacts at one stage may influence another due to mechanical coupling. I employ a lumped-mass approach to model the system, incorporating bending, torsion, and axial degrees of freedom, and solve the resulting equations using numerical methods. The findings highlight how meshing impact intensities with speed and varies between gear pairs, offering practical guidance for mitigating vibrations in electric vehicle car powertrains.
Meshing impact in gears primarily occurs during engagement and disengagement, with engagement impact being more dominant due to the initial collision of tooth surfaces. For helical gears, this impact arises from factors like load-induced deformations, machining inaccuracies, and changes in meshing states. In the context of an electric vehicle car, where lightweight and compact designs are prioritized, such impacts can lead to premature wear, increased noise, and reduced efficiency. Therefore, accurately calculating meshing impact forces is essential. Based on geometric and dynamic principles, the meshing impact force for a helical gear pair can be derived. For engagement impact, the maximum force depends on the instantaneous inertia, velocities, and compliance of the gear teeth. Let \( J_n \) represent the instantaneous moment of inertia for the driving (\( n=1 \)) and driven (\( n=2 \)) gears, given by:
$$ J_n = \pi \rho b^2 (r_{bn}^4 – r_{hn}^4), \quad n=1,2 $$
where \( \rho \) is the material density, \( b \) is the face width, \( r_{bn} \) is the base circle radius, and \( r_{hn} \) is the hub inner radius. The induced mass \( m_n \) along the meshing line is:
$$ m_n = \frac{J_n}{r_{bn}^2}, \quad n=1,2 $$
The kinetic energy \( E_k \) at the impact point is:
$$ E_k = \frac{1}{2} \cdot \frac{J_1 J_2}{(J_1 r’_{b2}^2 + J_2 r_{b1}^2)} v_s^2 $$
Here, \( v_s \) is the velocity at the off-line engagement point, and \( r’_{b2} \) is the instantaneous base circle radius of the driven gear. The angle \( \theta \) between the impact direction and the meshing line is:
$$ \theta = \arccos \frac{r’_{b2}}{r_{O_2D}} – \angle PO_2D – \alpha $$
where \( \alpha \) is the pressure angle. Finally, the maximum meshing impact force \( F_s \) is:
$$ F_s = v_s \sqrt{\frac{J_1 J_2}{(J_1 r’_{b2}^2 + J_2 r_{b1}^2)} (q_s + \cos^2\theta \cdot q_q)} $$
In this equation, \( q_s \) denotes the single-tooth-pair compliance at the initial engagement point, and \( q_q \) represents the comprehensive compliance of other tooth pairs during engagement. These parameters are derived from the overall meshing stiffness and load distribution factors. This formulation allows for computing impact forces at any point between the start and end of engagement, providing a basis for dynamic analysis in an electric vehicle car transmission system.
To model the dynamic behavior of the two-stage helical gear transmission in an electric vehicle car, I develop a 16-degree-of-freedom (DOF) lumped-mass model. This approach treats gears as rigid disks with mass and inertia, while the teeth and supports are represented by springs and dampers. The system includes two helical gear pairs: the first-stage pair (Gear Pair 12) connected to the input, and the second-stage pair (Gear Pair 34) connected to the output, with an intermediate elastic shaft linking the two stages. The coordinate system \( O-xyz \) is defined with \( x \) along the gear axis, \( y \) along the tooth thickness, and \( z \) along the tooth height. The model incorporates support stiffness \( K_{xi}, K_{yi}, K_{zi} \) and damping \( C_{xi}, C_{yi}, C_{zi} \) for each gear \( i \) (where \( i=1,2,3,4 \)), time-varying meshing stiffness \( K_{12}, K_{34} \) and damping \( C_{12}, C_{34} \) for the gear pairs, and torsional stiffness \( K_{23} \) and damping \( C_{23} \) for the intermediate shaft. These parameters are determined based on standard mechanical principles relevant to electric vehicle car components.
The generalized displacement vector \( \mathbf{U} \) encompasses 12 translational and 4 rotational DOFs:
$$ \mathbf{U} = [x_1, y_1, z_1, \phi_1, x_2, y_2, z_2, \phi_2, x_3, y_3, z_3, \phi_3, x_4, y_4, z_4, \phi_4] $$
where \( x_i, y_i, z_i \) are the translational displacements, and \( \phi_i \) are the rotational angles around each gear’s axis. The relative displacement along the meshing line for Gear Pairs 12 and 34, denoted \( \delta_{12} \) and \( \delta_{34} \), accounts for vibrations and installation errors \( \Delta x_i, \Delta y_i, \Delta z_i \):
$$ \delta_{12} = (\phi_1 r_{b1} – \phi_2 r_{b2}) \cos \beta_{12} – (y_1 – y_2 + \Delta y_1 – \Delta y_2) \sin \zeta_{12} \cos \beta_{12} – (z_1 – z_2 + \Delta z_1 – \Delta z_2) \cos \zeta_{12} \cos \beta_{12} – (x_1 – x_2 + \Delta x_1 – \Delta x_2) \sin \beta_{12} $$
$$ \delta_{34} = (\phi_3 r_{b3} – \phi_4 r_{b4}) \cos \beta_{34} + (y_3 – y_4 + \Delta y_3 – \Delta y_4) \sin \zeta_{34} \cos \beta_{34} – (z_3 – z_4 + \Delta z_3 – \Delta z_4) \cos \zeta_{34} \cos \beta_{34} + (x_3 – x_4 + \Delta x_3 – \Delta x_4) \sin \beta_{34} $$
Here, \( \beta_{12} \) and \( \beta_{34} \) are the helix angles, and \( \zeta_{12} \) and \( \zeta_{34} \) are the angles between the meshing line and the z-axis. The elastic meshing force \( P_{jk} \) and damping force \( D_{jk} \) for gear pair \( jk \) are:
$$ P_{jk} = K_{jk} \delta_{jk} $$
$$ D_{jk} = C_{jk} \dot{\delta}_{jk} $$
where \( \dot{\delta}_{jk} \) is the relative velocity. The total dynamic meshing force \( F_{jk} \) is:
$$ F_{jk} = P_{jk} + D_{jk} $$
Considering meshing impact forces \( F_{s12} \) and \( F_{s34} \) derived earlier, the coupled bending-torsional-axial dynamic differential equations for the electric vehicle car transmission system are established. For each gear, the equations of motion incorporate inertia, support forces, meshing forces, and impact forces. For instance, for Gear 1 (input gear):
$$ M_1 \ddot{x}_1 + C_{x1} \dot{x}_1 + K_{x1} x_1 = F_{12} \sin \beta_{12} $$
$$ M_1 \ddot{y}_1 + C_{y1} \dot{y}_1 + K_{y1} y_1 = F_{12} \cos \beta_{12} \sin \zeta_{12} $$
$$ M_1 \ddot{z}_1 + C_{z1} \dot{z}_1 + K_{z1} z_1 = F_{12} \cos \beta_{12} \cos \zeta_{12} $$
$$ I_1 \ddot{\phi}_1 = T_1 – F_{12} \cos \beta_{12} r_{b1} – F_{s12} r_{b1} $$
Similar equations are formulated for Gears 2, 3, and 4, with appropriate signs and terms for the intermediate shaft and output. The complete system comprises 16 equations that describe the dynamic interactions. To solve these, I apply the Runge-Kutta method after eliminating rigid-body displacements and non-dimensionalizing the equations to avoid numerical issues. The relative accelerations along the meshing lines, \( a_{12} \) and \( a_{34} \), are computed as:
$$ a_{12} = [(\ddot{\phi}_1 r_{b1} – \ddot{\phi}_2 r_{b2}) – (\ddot{y}_1 – \ddot{y}_2) \sin \zeta_{12} – (\ddot{z}_1 – \ddot{z}_2) \cos \zeta_{12}] \cos \beta_{12} – (\ddot{x}_1 – \ddot{x}_2) \sin \beta_{12} $$
$$ a_{34} = [(\ddot{\phi}_3 r_{b3} – \ddot{\phi}_4 r_{b4}) – (\ddot{y}_3 – \ddot{y}_4) \sin \zeta_{34} – (\ddot{z}_3 – \ddot{z}_4) \cos \zeta_{34}] \cos \beta_{34} + (\ddot{x}_3 – \ddot{x}_4) \sin \beta_{34} $$
These accelerations serve as key indicators of dynamic response under meshing impact in the electric vehicle car gear system.
For the numerical analysis, I consider a typical two-stage helical gear transmission used in an electric vehicle car. The system parameters, including gear geometry, material properties, and stiffness values, are summarized in the table below. These parameters are representative of compact, high-efficiency designs common in modern electric vehicle car applications.
| Parameter | Gear 1 (Input) | Gear 2 | Gear 3 | Gear 4 (Output) |
|---|---|---|---|---|
| Handedness | Left | Right | Left | Right |
| Module \( m \) (mm) | 4 | 4 | 4 | 4 |
| Number of Teeth \( z \) | 23 | 80 | 22 | 87 |
| Pressure Angle \( \alpha \) (°) | 20 | 20 | 20 | 20 |
| Helix Angle \( \beta \) (°) | 20 | 20 | 20 | 20 |
| Addendum \( h_a \) (mm) | 4 | 4 | 4 | 4 |
| Dedendum \( h_f \) (mm) | 5 | 5 | 5 | 5 |
| Face Width \( b \) (mm) | 33 | 31.5 | 40 | 38 |
| Radial Support Stiffness (y,z) (N/m) | 2.0850×10⁹ | 1.8269×10⁹ | 1.8569×10⁹ | 2.3698×10⁹ |
| Axial Support Stiffness (x) (N/m) | 2.2159×10⁹ | 1.9949×10⁹ | 1.9949×10⁹ | 1.4097×10⁹ |
The input torque is set to 100 N·m, simulating a common load condition in an electric vehicle car. Meshing impact forces for Gear Pairs 12 and 34 are calculated at different input speeds using the derived formulas, as shown in the following table. This illustrates how impact forces vary with rotational speed, a critical factor for electric vehicle car performance under acceleration.
| Input Speed (r/min) | Meshing Impact Force \( F_{s12} \) (N) for Gear Pair 12 | Meshing Impact Force \( F_{s34} \) (N) for Gear Pair 34 |
|---|---|---|
| 3,000 | 1,245.94 | 89.21 |
| 6,000 | 2,419.89 | 178.41 |
| 12,000 | 4,983.77 | 357.21 |
I analyze the dynamic response by computing the meshing-line direction accelerations for both gear pairs under input speeds of 3,000, 6,000, and 12,000 r/min. These speeds represent low, medium, and high operational ranges in an electric vehicle car. The results reveal significant insights into meshing impact effects. At 3,000 r/min, the acceleration peaks for Gear Pair 12 and Gear Pair 34 are 686.21 m/s² and 223.35 m/s², respectively. As speed increases to 6,000 r/min, these peaks rise to 1,135.75 m/s² and 500.58 m/s². At 12,000 r/min, typical of high-speed cruising in an electric vehicle car, the peaks escalate further to 2,291.61 m/s² for Gear Pair 12 and 1,131.77 m/s² for Gear Pair 34. This trend demonstrates that higher rotational speeds amplify meshing impact, leading to greater accelerations and potential instability in the electric vehicle car transmission system.
Notably, Gear Pair 12, connected to the input, exhibits higher acceleration peaks than Gear Pair 34 at all speeds, indicating that input-stage impacts are more severe. This is crucial for electric vehicle car design, as the input stage often experiences sharper torque fluctuations from the electric motor. Moreover, at the same speed, the influence of Gear Pair 12’s meshing impact on Gear Pair 34 is stronger than the reverse. For example, at 3,000 r/min, Gear Pair 12’s impact noticeably affects Gear Pair 34’s acceleration profile, whereas Gear Pair 34’s impact on Gear Pair 12 is minimal. This asymmetry stems from the higher kinetic energy and force transmission at the input side of an electric vehicle car drivetrain. The frequency analysis shows that the mutual influence between gear pairs occurs at their respective meshing frequencies, aligning with theoretical expectations for coupled helical gear systems in an electric vehicle car.
The implications for electric vehicle car engineering are profound. As speeds increase, meshing impact can exacerbate noise, vibration, and wear, potentially compromising the reliability and comfort of the electric vehicle car. Therefore, design strategies such as profile modifications, optimized stiffness matching, or damping enhancements should focus on the input-stage gears to mitigate impact effects. Additionally, the coupling between stages suggests that dynamic evaluations must consider the entire transmission system, rather than isolated pairs, to ensure robust performance in an electric vehicle car.
In conclusion, this study provides a detailed dynamic characterization of a two-stage helical gear transmission for an electric vehicle car, with emphasis on meshing impact. I derive meshing impact forces and develop a 16-DOF model to simulate the system’s behavior. The analysis reveals that meshing impact intensifies with rotational speed, particularly at the input stage (Gear Pair 12), and significantly influences the dynamic response of both gear pairs. These findings underscore the importance of accounting for meshing impact in the design and optimization of electric vehicle car powertrains, aiming to enhance durability and reduce NVH. Future work could explore the effects of different load conditions or gear geometries specific to electric vehicle car applications, further advancing the performance of these critical systems.
Throughout this paper, the term “electric vehicle car” has been emphasized to highlight the application context. The methodologies and results presented here offer a foundation for improving gear transmission dynamics in the rapidly evolving field of electric vehicle car technology. By addressing meshing impact, engineers can contribute to quieter, more efficient, and longer-lasting electric vehicle car drivetrains, supporting the global transition to sustainable transportation.