Abstract: This paper focuses on the differences in the number of engaged teeth between high contact ratio (HCR) spur gear and conventional spur gear. By utilizing the finite element method to solve the time-varying mesh stiffness, a coupled dynamic model for HCR spur gear is established. The dynamic responses of spur gear system are obtained using the Newmark-β time domain integration method, and a time-frequency domain analysis is conducted. Comparisons with conventional spur gear reveal the variation patterns of dynamic load factors (DLFs) for HCR spur gear under different rotational speeds and error influences. The results indicate significant differences in mesh stiffness and improved system stability for HCR spur gear.
1. Introduction
Spur gear is crucial components in mechanical transmissions, known for their smooth transmission, high power capacity, and long service life. They are widely used in aerospace, automotive, and other fields. To meet the increasing demands for high-speed, high-efficiency, and lightweight mechanical systems, traditional spur gear is often inadequate. By adjusting parameters such as gear modification coefficients and addendum coefficients to achieve a contact ratio above 2, such gears are referred to as high contact ratio (HCR) spur gear. Compared to conventional low contact ratio (LCR) spur gear with a contact ratio between 1 and 2, HCR spur gear have an increased number of simultaneously engaged tooth pairs, which enhances load capacity and reduces vibration and noise. The vibration characteristics during meshing directly affect the dynamic load performance of the entire system, thus studying the dynamic characteristics of HCR spur gear is of practical significance.
Table 1 summarizes previous research efforts on HCR spur gear.
| Research Area | Key References |
|---|---|
| Design of HCR Spur Gear | RAJESH et al. [1], Zhang Lin [2], Li Fajia et al. [3] |
| Load Capacity Analysis | FRANULOVIC et al. [4], HUSSEIN et al. [5], MARIMUTHU et al. [6] |
| Mesh Stiffness and DLF | SANCHEZ et al. [8], PEDRERO et al. [9], DOGAN [10] |
2. Gear System Dynamic Model Construction
2.1 Gear Analysis Model Construction
The research object in this paper is a pair of HCR spur gear, and for comparison, a pair of conventional LCR spur gear with the same basic parameters is also established. The spur gear parameters are shown in Tables 2 and 3.
Table 2: Parameters of HCR Spur Gear
| Parameter | Value |
|---|---|
| Module (mm) | 2.3 |
| Center Distance (mm) | 91.5 |
| Number of Teeth (z1/z2) | 33/46 |
| Contact Ratio | 2.14 |
| Pressure Angle (°) | 18 |
| Modification Coefficient | 0.3/-0.008 |
| Addendum (mm) | 3.544/2.996 |
| Full Tooth Height (mm) | 6.465 |
| Face Width (mm) | 29/25 |
Table 3: Parameters of LCR Spur Gear (with similar basic parameters for comparison purposes)
| Parameter | Value (Assuming Similar to HCR for Comparison) |
|---|---|
| (Same as HCR, except…) | |
| Contact Ratio | ~1.5 (typically for LCR) |
| (Other parameters may vary slightly for practical LCR design) |
The HCR spur gear is processed using a “modification” method, altering tooth profiles by increasing the addendum coefficient and reducing the pressure angle to achieve a contact ratio above 2.
2.2 Time-Varying Mesh Stiffness Calculation
Mesh stiffness is an internal dynamic excitation during gear meshing and is crucial for accurate dynamic characteristic analysis. Prior to dynamic analysis, accurately solving the time-varying mesh stiffness is essential.
The finite element method is employed to solve gear mesh stiffness. The spur gear material is 20CrMnTiH, and a hexahedral mesh is used. The driving gear is the target surface, and the driven gear is the contact surface. The tooth friction coefficient is selected as 0.15, and the mesh in the tooth contact area is subdivided. The inner circle of the driven gear is fixed, and a torque is applied to the inner circle of the driving gear.
(a) Finite Element Mesh Model and Mesh Schematic
(b) Time-Varying Mesh Stiffness
The spur gear mesh stiffness is calculated using the formula:
kn=μFn=σ1⋅rbFn=θ⋅rb2T
where Fn and μ are the equivalent force and deformation in the mesh line direction, T, θ, and rb are the torque on the driving gear, the deformation angle, and the base circle radius, respectively.
The time-varying mesh stiffness results for LCR and HCR spur gear. Both LCR and HCR gear stiffness curves exhibit a square wave pattern with alternating values.
For LCR spur gear, the mesh cycle alternates between double-tooth and single-tooth meshing. Double-tooth meshing stiffness is approximately 3.62×108 N/m, while single-tooth meshing stiffness is 2.51×108 N/m. The stiffness change is drastic during transitions, causing stiffness excitation and periodic elastic deformation variations, which may lead to significant vibrations and reduced smoothness.
In contrast, HCR spur gear transition between triple-tooth and double-tooth meshing, with stiffness values of 4.62×108 N/m and 3.82×108 N/m, respectively. HCR spur gear maintain multi-tooth contact throughout the mesh cycle, reducing stiffness changes during tooth pair transitions, which positively affects system vibration reduction and transmission smoothness.
To investigate the load’s influence on mesh stiffness, the finite element method applies different load torques to the driving gear. Tthat as the load increases, the average stiffness of HCR spur gear also increases, with a more pronounced stiffness variation between triple-tooth and double-tooth contacts. The stiffness grows rapidly at lower loads and then stabilizes as the load increases, indicating that heavy loads can alter gear meshing states but excessive stiffness changes may adversely affect system stability.
2.3 Gear System Dynamic Model
Based on actual gear transmission relationships, excluding friction, smaller and more flexible components like shafts and bearings are equivalent to springs, while larger and less flexible components like spur gear is equivalent to mass blocks. This results in a coupled analysis model for spur gear pairs.
In this model, the input end is defined as p, the output end as g, and the gear mesh line direction as the Y-direction. The support stiffness and damping of shafts, bearings, and the housing are represented by combined equivalent values kpy, kgy, cpy, and cgy.
This model provides a foundation for analyzing the dynamic responses and dynamic load characteristics of HCR spur gear compared to LCR spur gear.
Conclusion
This paper presents a comprehensive analysis of the dynamic load characteristics of HCR spur gear. By constructing a coupled dynamic model and utilizing the finite element method, significant differences in mesh stiffness and dynamic responses between HCR and LCR spur gear is revealed. The results indicate that HCR spur gear exhibit reduced dynamic response fluctuations and improved system stability, with dynamic load factors decreasing compared to LCR gears. As rotational speed increases and errors accumulate, dynamic load factors rise, particularly within resonance speed ranges. This study contributes to advancing the design theory and analysis methods for HCR spur gear.