Abstract
This article comprehensively investigates the effect of modified designs on the dynamic characteristics of helical gear systems. By considering different modification coefficients, a dynamics model for helical gear is established based on Lagrangian dynamics. Analytical methods are applied to solve the time-varying meshing stiffness of modified helical gear, and the influence of varying modification coefficients on meshing stiffness is analyzed. The integration of the time-varying meshing stiffness with the dynamics model reveals the impact of various modified designs on the dynamic characteristics of the helical gear system. The results demonstrate that positive drive modifications reduce time-varying meshing stiffness and meshing forces, while negative drive modifications increase both. This study offers valuable insights into the design and optimization of helical gear transmissions.
Keywords: helical gear, modification coefficient, potential energy method, time-varying meshing stiffness, dynamic characteristic, contact ratio
1. Introduction
Helical gear, known for their compact structure, high load-bearing capacity, and efficient power transmission, are crucial components in various mechanical systems, including aerospace, railway transportation, and marine equipment. To enhance performance and durability, gear modification techniques, such as profile shifting, are often employed. Modification not only prevents undercutting but also improves the load-bearing capacity and allows for adjustment of the center distance. Understanding the dynamic behavior of modified helical gear is essential for optimal design and maintenance.
Time-varying meshing stiffness (TVMS), a critical internal excitation source, significantly impacts the vibration, noise, and fatigue life of helical gear systems. Various methods, including experimental, finite element, and analytical approaches, have been used to evaluate TVMS. However, the analytical method stands out for its balance between computational efficiency and accuracy.
This study aims to:
- Establish an analytical model for TVMS calculation of modified helical gear.
- Investigate the influence of different modification coefficients on TVMS.
- Analyze the dynamic response of helical gear systems under various modified designs.
2. Literature Review
2.1 Helical Gear Dynamics
Helical gear have attracted significant research attention due to their superior performance in power transmission. Many scholars have analyzed the dynamics of helical gear systems using various methods [1-3]. For example, Li and Wang [1] provided a comprehensive review of gear system dynamics, emphasizing vibration, impact, and noise. Zhang et al. studied the dynamic behavior of helical gear systems, highlighting the influence of system parameters.
2.2 Time-Varying Meshing Stiffness (TVMS)
TVMS is a critical factor influencing gear dynamics. Several methods exist for its calculation, including experimental, finite element, and analytical approaches [4-7]. Experimental methods, though accurate, are costly and complex. Finite element methods, while versatile, can be computationally intensive. Analytical methods, on the other hand, offer a balance between accuracy and efficiency, making them a popular choice [8-10].
2.3 Gear Modification
Gear modification techniques, such as profile shifting, can significantly improve gear performance. However, the effect of modification on TVMS and dynamic behavior remains understudied. Existing research has primarily focused on the influence of standard gear parameters [11-13], leaving a gap in understanding modified helical gear.
3. Time-Varying Meshing Stiffness Calculation for Modified Helical Gear
3.1 Basic Geometry of Modified Helical Gear
Modified helical gear maintain the involute profile of standard gears but with altered tooth positions. the geometric relationship between standard and modified gears. The modified gear has adjusted tooth heights, root fillets, and center distances.
Key parameters for modified gears include:
- Tooth addendum height (h_ai)
- Dedendum height (h_fi)
- Center distance (a_i)
- Transverse pressure angle (α_t)
These parameters are governed by equations (1) to (4) in the reference document.
3.2 Analytical Calculation of TVMS
The TVMS of modified helical gear is computed using the potential energy method combined with the slice method. Hertz contact stiffness (k_h), fillet-basis stiffness (k_f), bending stiffness (k_b), shear stiffness (k_s), and axial compression stiffness (k_a) are considered. The combined mesh stiffness (k_m) is calculated as:
km=(kh,i1+kb1,i1+ks1,i1+ka1,i1+kf1,i1+kb2,i1+ks2,i1+ka2,i1+kf2,i1)−1
where the subscripts i represent different slices along the tooth width.
3.3 Consideration of Contact Ratio
The contact ratio (ε), a crucial parameter for smooth gear operation, affects TVMS. For a helical gear pair with ε ranging from 2 to 3, the meshing process involves both double- and triple-tooth contact regions. TVMS is calculated separately for these regions.
4. Dynamics Model of Modified Helical Gear Systems
4.1 Generalized Coordinates and Lagrange Function
The dynamics model considers eight degrees of freedom, including three translational and one rotational degree of freedom for each gear. The generalized coordinates are:
mathbfq={xp,yp,zp,θp,xg,yg,zg,θg}
The Lagrangian function (L) is defined as the difference between kinetic energy (E_k) and potential energy (E_p):
L=Ek−Ep=21q˙TMq˙−21qTKq
where M and K are the mass and stiffness matrices, respectively.
4.2 Dissipation Function and Equations of Motion
Dissipation is considered through the Rayleigh dissipation function (D):
D=21q˙TCq˙
where C is the damping matrix.
The equations of motion are derived from the Lagrange-D’Alembert principle:
fracddt(∂q˙∂L)−∂q∂L+∂q˙∂D=Q
where Q represents the generalized forces, including meshing force (F_m), friction force (F_f), driving torque (T_p), and load torque (T_g).
5. Influence of Modified Designs on Dynamic Characteristics
5.1 TVMS and Meshing Forces
The effect of modification coefficients on TVMS and meshing forces. When the modification leads to a decrease in TVMS (positive drive), the meshing force fluctuation reduces, indicating smoother operation. Conversely, negative drive increases TVMS and meshing forces.
5.2 Frequency Domain Analysis
Frequency domain analysis provides insights into the dynamic response. The frequency response of TVMS and meshing forces, respectively. Positive drive modifications lead to reduced frequency components and faster attenuation, while negative drive modifications increase the frequency content.
5.3 Statistical Indicators
To quantify the dynamic response, statistical indicators such as root mean square (RMS) and kurtosis value (KV) are evaluated. The variation of these indicators with modification coefficients.
6. Experimental Validation
Experimental validation ensures the accuracy of the analytical model. A test rig, is used to measure the dynamic response of modified helical gears under varying conditions.
The experimental results align well with the analytical predictions, confirming the model’s validity.
7. Conclusion
This study investigates the influence of modified designs on the dynamic characteristics of helical gear systems. Analytical methods are employed to calculate the time-varying meshing stiffness of modified gears, and a dynamics model is established based on Lagrangian dynamics. The results show that:
- TVMS Variation: Positive drive modifications reduce TVMS and meshing forces, while negative drive modifications increase them.
- Frequency Response: In the frequency domain, positive drive modifications lead to reduced frequency components and faster attenuation, while negative drive modifications increase frequency content.
- Statistical Indicators: The RMS and KV indicators vary systematically with modification coefficients, offering a quantitative assessment of dynamic performance.
The proposed model and analytical framework provide valuable insights into the design and optimization of helical gear systems, contributing to improved performance and reliability.