## 1 Introduction  
Stress distribution plays a pivotal role in the design of gear end surfaces [1; 2; 3], profoundly affecting the gear's performance and lifespan. Two primary determinants shape this distribution: the applied load and the material properties of the gear. Typically, the loads imposed on gears originate from power sources such as engines or motors on the input shaft. These loads give rise to compressive and shear stresses on the gear, culminating in varied stress distributions. Furthermore, the intrinsic attributes of the gear material, including its strength, hardness, and toughness, further delineate this distribution. Stress distribution in gears plays a pivotal role in determining their longevity, reliability, and susceptibility to fatigue. When gears are subjected to loads, the distribution of stress can significantly influence their fatigue lives and the associated risk of fracture [4; 5; 6]. An optimized stress distribution enhances the gear's lifespan and reduces failure risks. Many studies have emphasized the importance of understanding the underlying failure mechanisms in gear systems to enhance their reliability, efficiency, and longevity, with fatigue failure, particularly contact and bending fatigue, being a recurring concern. [PERSON] and [PERSON] utilized FEA to delve into the stress distribution in high-speed helical gear shafts, pinpointing fatigue fracture as the primary failure mode [7]. Similarly, the research by [PERSON] et al. indicated that the maximum stress location correlates with the failure site [8]. FEA also surfaced as a crucial tool in other studies, such as those by [PERSON] et al., which inspected aluminum alloy brake reducers for reusable launch vehicles, and by [PERSON] et al., who explored stress sensitivity and fatigue life in harmonic gear drives. These analyses underlined the necessity to grasp material attributes and grain orientation to foresee ductile overload or fatigue failures [9]. In a contrasting perspective, [PERSON] et al. highlighted the repercussions of misalignment, such as tooth breakage and pitting due to surface contact fatigue [10]. Meanwhile, [PERSON] et al. concentrated on structural stress analysis techniques and the prediction of fatigue life in high reduction ratio reducers, juxtaposing FEA-derived outcomes with formula-calculated stress and fatigue life estimates [11]. Furthermore, specific fatigue forms, like rolling contact fatigue in RV reducer crankshafts, have been investigated. [PERSON] et al. introduced a 3D elasto-plastic contact model, factoring in hardness gradients and initial residual stress for better fatigue life prediction [12]. Additionally, [PERSON] et al. examined the breakdown of a secondary driving helical gear in electric vehicles, linking the failure to elevated contact stress and impact load caused by subpar material properties [13]. Collectively, these studies underscore the multifaceted aspects of gear failures. Addressing fatigue requires a holistic comprehension of stress distribution, material characteristics, and operational conditions. Consequently, the use of advanced analytical tools like FEA becomes indispensable in designing, diagnosing, and refining gear systems across various applications. Stress distribution in gears profoundly impacts their transmission efficiencies, closely relating to gear friction, wear, and energy losses [14; 15]. Optimizing this distribution can lead to improved transmission and energy conversion efficiencies. A surge of research has focused on the interplay between stress and transmission performance in various reducers, aiming to boost efficiency, reliability, and load capacity through innovative mechanical designs and mathematical modeling. [PERSON] and [PERSON]'s examination of the China Bearing Reducer (CBR) underscored the significance of the clearance between the cycloid teeth and pin on the contact force and stress [16]. Using finite element methods, [PERSON] et al. found that in RV reducers, the eccentric shaft's stiffness is of greater concern than the material strength [17].