# Mesh Stiffness and Dynamic Modeling and Analysis of Modified Straight Bevel Gears
## 1 Introduction
[16] established a system dynamics model for differential structures and studied the vibration characteristics of SBGs in differentials. [PERSON] et al. [17] established a dynamic model for spur gear transmission systems with different modifications and sought optimal modification parameters. [PERSON] et al. [18] studied the vibration characteristics of bevel gears under tooth profile modification but did not consider the time-varying spatial meshing characteristics of bevel gears. [PERSON] et al. [19] studied the calculation method of mesh stiffness for SBGs, but did not consider Hertz contact stiffness under different loads. [PERSON] et al. [20] considered the complex internal excitations of spiral bevel gears and conducted a detailed analysis of their intricate dynamic behaviors, providing an approach for the parameter design and dynamic characteristic control of spiral bevel gear drive systems. [PERSON] et al. [21] studied the effects of shaft stiffness and elastic deformation on vibration response by considering different support positions and types of torque (constant torque and periodic torque). [PERSON] et al. [22] studied the impact of time-varying uncertain parameters on the system response in a two-stage straight bevel gear system. However, the calculation of meshing stiffness was not validated, and the dynamic model established using the lumped mass method could not comprehensively consider the transmission system. [PERSON] et al. [23] developed a dynamic model of a transmission system composed of spiral bevel gear pairs, shafts, and bearings to investigate the influence of coupling effects on system dynamics. [PERSON] et al. [24] modeled the shaft as a rotating cantilever beam to analyze the vibration characteristics of helical gear transmission systems. [PERSON] et al. [25] used the finite element method to calculate mesh stiffness and established a dynamic model for bevel gear-coupled planetary gear transmission systems. However, they did not address the actual spatial meshing characteristics of spiral bevel gears, using equivalent meshing points to establish the meshing pair model. Despite the substantial amount of research conducted, the aforementioned dynamic modeling does not take into account the time-varying nature of the gear meshing position and the influence of the degree of overlap. Therefore, the effects of gear modification on the dynamic character istics of the gear system, as well as associated issues of vibration and noise, still require further investigation. Tooth Contact Analysis (TCA) is a basic means of studying gear transmission systems, with detailed explanations provided by [PERSON] [26; 27]. This paper addresses the contact analysis of SBGs by determining the TVMS under various modification conditions and performing a comparative analysis with the finite element method. The findings demonstrate that computational efficiency is significantly enhanced while maintaining accuracy, facilitating the resolution of practical engineering problems. By considering the meshing characteristics of SBGs, a comprehensive dynamic model of a system comprising two shafts, bearings, and gears is constructed using the finite element node method. The Newmark-beta method is employed to iteratively solve the system's response, allowing for a comparison of the impact of meshing stiffness under different modification conditions on the gear system. This approach offers a dynamic perspective for selecting gear modification techniques and extents.