Gear transmission systems are among the most critical and widely used mechanical transmission devices in various industries, including machinery, transportation, and aerospace. Designing high-efficiency, low-noise, and high-performance helical gears is a key objective in gear research. The dynamic meshing excitation caused by factors such as varying contact tooth pairs, load-induced deformation, manufacturing inaccuracies, and assembly errors leads to vibration and noise, which are significant indicators of dynamic performance. Helical gears, with their complex tooth profiles, exhibit dynamic meshing excitations due to these factors. Tooth modification techniques, such as profile and lead modifications, can mitigate the effects of these factors on dynamic激励 and vibration noise to some extent. Additionally, research indicates that friction excitation is another major source of vibration and noise in gear systems, closely linked to dynamic response and gear life. After modification, both stiffness excitation and friction excitation in helical gears change, but the impact of these changes on dynamic response and vibration noise has not been thoroughly investigated. Therefore, establishing a coupled dynamic model that integrates helical gear modification and friction excitation for vibration noise analysis holds significant theoretical and practical value for developing low-vibration, low-noise gear transmission products.
In this study, I focus on helical gear systems, analyzing tooth profile modification, lead modification, friction excitation, and meshing stiffness excitation. I develop a coupled dynamic model that incorporates helical gear modification and friction excitation to investigate internal dynamic激励, dynamic response, and vibration noise. The main research work includes determining optimal modification parameters for helical gears, calculating time-varying contact lines, friction forces, and friction torques, and analyzing the effects of different spiral angles and contact tooth widths. Based on the meshing stiffness calculation method for helical gears, I compute the meshing stiffness before and after modification. Using a lumped mass method, I establish a coupled dynamic model and solve it with the Runge-Kutta method to analyze dynamic responses. Furthermore, I create an acoustic boundary element model to simulate vibration noise based on dynamic meshing forces from the coupled dynamic response analysis, examining the effects of modification and different friction coefficients on vibration noise.
Helical gears are essential components in transmission systems, and their dynamic behavior is influenced by various internal and external excitations. Internal excitations include stiffness excitation, error excitation, and impact excitation, which arise from the time-varying nature of meshing stiffness, manufacturing errors, and engagement impacts. Static transmission error (STE) is a critical parameter defined as the deviation between the actual and theoretical positions of the driven gear under load, serving as a primary source of vibration and noise. In helical gears, the contact line varies with time, leading to fluctuations in meshing stiffness and friction forces. Tooth modification, such as profile and lead modifications, aims to optimize load distribution and reduce transmission error fluctuations. For instance, profile modification involves linear or parabolic adjustments to the tooth tip or root, while lead modification includes crowning or helix angle corrections. These modifications help minimize entry and exit impacts, thereby lowering vibration and noise levels.
To calculate the optimal modification parameters, I use Romax software to determine the best values for linear profile modification, parabolic profile modification, and lead crowning. The helical gear system parameters are summarized in Table 1.
| Parameter | Pinion | Gear |
|---|---|---|
| Module (mm) | 3 | 3 |
| Number of Teeth | 23 | 51 |
| Helix Angle (°) | 18.42 | 18.42 |
| Tooth Width (mm) | 40 | 35 |
| Pressure Angle (°) | 20 | 20 |
| Input Torque (N·m) | 300 | – |
| Input Speed (r/min) | 600 | – |
After optimization, the static transmission error for linear profile modification shows a peak-to-peak value of 0.26 μm, compared to 1.01 μm before modification. Similarly, parabolic profile modification reduces the fluctuation to 0.31 μm, and lead crowning to 0.32 μm. This demonstrates that modification effectively minimizes transmission error variations, enhancing gear stability. The meshing stiffness of helical gears is calculated based on the unit contact line length in the transverse plane. The single-tooth meshing stiffness $$k_d$$ is derived using the formula:
$$k_d = \frac{F_m}{\delta}$$
where $$F_m$$ is the meshing force per unit length and $$\delta$$ is the deformation. The comprehensive meshing stiffness $$k_z$$ for multiple tooth pairs is obtained by summing the individual stiffnesses over the contact lines. For helical gears, the time-varying contact line length $$L(t)$$ is computed considering the gear geometry and engagement process. Assuming uniform load distribution along the contact line, the friction force $$F_f(t)$$ and friction torque $$T_f(t)$$ are calculated as:
$$F_f(t) = \mu \frac{F_n(t)}{L_z(t)} \left( L_{\text{right}}(t) – L_{\text{left}}(t) \right)$$
$$T_f(t) = \mu \frac{F_n(t)}{L_z(t)} \left( L_{\text{right}}(t) H_{\text{right}}(t) – L_{\text{left}}(t) H_{\text{left}}(t) \right)$$
where $$\mu$$ is the friction coefficient, $$F_n(t)$$ is the normal force, $$L_z(t)$$ is the total contact line length, and $$H$$ represents the friction arm. The effects of different helix angles and contact tooth widths on these parameters are analyzed. For example, increasing the helix angle from 18° to 20° enlarges the variation range of the contact line length, while increasing the contact tooth width from 35 mm to 37 mm raises the overall amplitude of the contact line length. The friction force and torque decrease with larger helix angles and tooth widths for single-tooth pairs but increase for the total meshing tooth pairs.
To model the dynamic behavior, I establish an 8-degree-of-freedom coupled dynamic model that includes translational and rotational vibrations. The equations of motion are derived using Newton’s second law and the d’Alembert principle. The dynamic meshing force $$F_m(t)$$ is expressed as:
$$F_m(t) = k_m(t) \left( y_p(t) – y_g(t) + R_{bp} \theta_p(t) – R_{bg} \theta_g(t) – e(t) \right) + c_m(t) \left( \dot{y}_p(t) – \dot{y}_g(t) + R_{bp} \dot{\theta}_p(t) – R_{bg} \dot{\theta}_g(t) – \dot{e}(t) \right)$$
where $$k_m(t)$$ and $$c_m(t)$$ are the time-varying meshing stiffness and damping, $$y$$ and $$\theta$$ are displacements and rotations, $$R_b$$ is the base circle radius, and $$e(t)$$ is the static transmission error. The friction forces $$F_{fp}$$ and $$F_{fg}$$ are incorporated into the model, affecting the vibrations in the direction perpendicular to the meshing line. The dynamic transmission error (DTE) is calculated as:
$$\text{DTE}(t) = y_p(t) – y_g(t) + R_{bp} \theta_p(t) + R_{bg} \theta_g(t)$$
Using the Runge-Kutta method, I solve the dynamic equations to analyze the system’s response. The results show that friction excitation induces periodic vibrations perpendicular to the meshing line and increases vibration amplitudes along the meshing line and axial direction. After modification, the oscillation amplitudes of friction forces and dynamic meshing forces decrease, reducing vibrations in all directions. For instance, with a friction coefficient of 0.1, the dynamic meshing force fluctuates between 8900 N and 9800 N before modification, but after modification, the range narrows to 9000–9700 N. Similarly, the dynamic transmission error shows reduced oscillations post-modification. Variations in friction coefficient also impact the dynamic response; increasing the coefficient from 0.05 to 0.1 raises vibration amplitudes and dynamic meshing force fluctuations.
For noise analysis, I develop an acoustic boundary element model in Virtual.Lab Acoustics. The structural finite element model of the helical gear system is simplified into equivalent cylinders with tetrahedral meshing. The acoustic boundary element model uses a 2D surface mesh with a maximum element size of 5 mm to ensure accuracy up to 1150 Hz. Field points are placed 20 cm from the gear surface to capture sound pressure levels. The dynamic meshing forces from the coupled dynamic analysis are applied as boundary conditions to compute vibration responses, which are then mapped to the acoustic model. The sound pressure levels at key frequencies (230 Hz, 460 Hz, 690 Hz, 920 Hz, and 1150 Hz) are evaluated. The results indicate that modification reduces sound pressure levels by 1–3 dB across all frequencies. For example, at 230 Hz, the sound pressure decreases from 70.1 dB to 68.1 dB after modification. Moreover, higher friction coefficients lead to increased noise levels; at 1150 Hz, the sound pressure rises from 83.6 dB (μ=0) to 92.1 dB (μ=0.1). This underscores the importance of considering friction excitation in noise reduction strategies for helical gears.
In conclusion, my research demonstrates that tooth modification effectively reduces static transmission error fluctuations and vibration noise in helical gears. The coupled dynamic model incorporating modification and friction excitation provides insights into the system’s dynamic behavior, highlighting the significant role of friction in exacerbating vibrations and noise. Future work should focus on experimental validation and incorporating additional factors like time-varying friction coefficients and lubrication effects to enhance model accuracy. This study contributes to the development of quieter and more efficient helical gear systems, aligning with the goals of advanced mechanical transmission design.