# Mesh Stiffness and Dynamic Modeling and Analysis of Modified Straight Bevel Gears
## 1 Introduction  
The internal structure of a gear system includes numerous mechanical components, forming an elastic mechanical system where the components are coupled and connected [1]. The dynamic characteristics of vibration and noise have always been major challenges in gear system research. The dynamic excitations of a gear system are mainly divided into two parts: external and internal excitations. External excitation is similar to that of general mechanical systems, such as the primary torque of the prime mover and load resistance, while internal excitation is generated within the system during the gear meshing process. Internal excitation is a key issue in gear system dynamics, and its correct establishment ensures the rationality of dynamic response predictions for the system. SBGs are used to transmit motion and power between intersecting shafts. They offer advantages such as compact structure, ease of manufacture, easy maintenance, and low cost. As a result, they are widely used in industrial transmission equipment and automotive differentials. Because the gear teeth are straight, each pair of teeth engages suddenly along their full length, resulting in impact loads and unstable operation during high-speed transmission, which also generates considerable noise. Therefore, gear modifications are necessary to achieve vibration and noise reduction [2, 3]. The precise calculation of time-varying meshing stiffness has been a research hotspot for many scholars. [PERSON] [4] proposed a method for calculating the meshing stiffness of spur gears based on the potential energy method, but it did not account for the fillets and gearbodies of typical gears. [PERSON] et al. [5] developed a meshing stiffness calculation model for gear blanks, providing a more detailed consideration of the overall stiffness during the gear meshing process. [PERSON] et al. [6] proposed a time-varying meshing stiffness model for modified gears based on actual machining, which showed excellent agreement with finite element results. [PERSON] et al. [7] investigated meshing stiffness in spur gears with spalling. [PERSON] et al. [8] proposed a finite element method for calculating the meshing stiffness of spiral bevel gears, which has general applicability but slower computational efficiency. [PERSON] et al. [9] fully considered the coupling effect between gear pairs and provided an analytical solution for the meshing stiffness of straight bevel gears based on the slicing method. [PERSON] et al. [10] divided spiral bevel gears and treated them as spur gears to further calculate the deformation of the gear teeth and obtain the mesh stiffness of the spiral bevel gears. In addition to time-varying meshing stiffness and backlash, many other nonlinear excitations affect the dynamic response of gear pairs. [PERSON] et al. [11] considered multi-state meshing, load distribution rates, time-varying meshing stiffness, and backlash to establish a dynamic model of a straight bevel gear system, investigating the effects of meshing frequency and transmission error on the system's dynamic characteristics; however, their mesh stiffness model employed a calculation model based on the gear blank stiffness of spur gears and did not consider the influence of contact stiffness. Some researchers have shown interest in establishing more comprehensive gear system dynamics. [PERSON] et al. [12] proposed a method for calculating static transmission error based on measured tooth surfaces, which can determine the effects of assembly errors and eccentricity. [PERSON] [13] established a dynamic model of hypoid gears and studied the influence of assembly errors, gyroscopic effects, gear eccentricity, external excitation, and dry friction on the dynamic performance of gears. [PERSON] [14; 15] proposed a multi-point coupled meshing model using multiple springs to simulate the meshing characteristics of multiple pairs of teeth in multi-tooth regions. [PERSON] et al.