## 1 Introduction
[15] investigated the impact of tooth profile modification length on the dynamic characteristics of the transmission system by using the modified mesh stiffness calculation formula. Studies have shown that tooth surface modification and wear are the main sources of deviation. In order to further investigate the impact of these two key parameters on mesh stiffness, [PERSON] et al. [16] conducted a sensitivity analysis based on the potential energy method. Many studies on the calculation of mesh stiffness for modified gears have primarily concentrated on tooth profile modifications or flank line crowning. However, this single modification has not sufficiently improved the mesh stiffness of gears. To this end, [PERSON] et al. [17] proposed a 'fully analytical calculation method' that does not rely on finite element simulation. This method can calculate the TVMS of the spur gear pair under the combined effects of tooth profile modification and flank line crowning. Although many scholars have employed analytical methods to calculate the TVMS of external gear pairs, it is equally important to accurately determine the mesh stiffness of internal gear pairs to ensure the stable operation of the planetary gear system. [PERSON] et al. [18] developed an analytical model for the TVMS mesh characteristics of the Addendum modification internal spur gear, and it was verified that the model still had a high computational accuracy under a loaded state. [PERSON] et al. [19] analyzed the phenomenon that excessive load may lead to extended meshing of the internal spur gear pair and then studied the impact of this phenomenon on TVMS and LTE. [PERSON] et al. [20] proposed a calculation method for the TVMS of internal and external gear pairs by combining the potential energy method and the tooth wear equation. It was confirmed that the impact of wear on mesh stiffness is mainly reflected in the gear tooth part. [PERSON] et al. [21] found through finite element simulation that as the rim thickness increased, the bending stress and meshing stiffness of the internal gear ring decreased significantly. [PERSON] et al. [22] calculated the mesh stiffness of an internal gear pair with a small tooth number difference using both the analytical method and finite element analytical approach. The results showed that the finite element analytical method was closer to the finite element results, and the computational efficiency was higher. In summary, the development of an analytical model for calculating the TVMS of the internal gear pair, while considering the influence of the ring gear flexibility under load conditions, is not yet perfect. Moreover, the current analytical model does not consider the nonlinear contact caused by tooth surface modification. Although finite element simulation can compensate for the shortcomings of the current analytical models, it necessitates longer solution times. The precise and rapid calculation of the TVMS of the internal mesh transmission is an important part of the dynamic analysis of the planetary gear system. Therefore, this paper proposes an analytical model of mesh stiffness for nonlinear contact by integrating the slice method, the potential energy method, and deformation coordination principle. This model can accurately and efficiently calculate the TVMS of the internal mesh transmission under load, and then provide a reliable theoretical basis for the dynamic characterization of various internal mesh transmission systems. The rest of this paper is organized as follows. In Section 2, a calculation model for the meshing stiffness of a slice element pair is developed, which accounts for the influence of gear ring flexibility. In Section 3, the calculation of mesh stiffness for the tooth surface nonlinear contact under modification is completed based on the deformation coordination principle. In Section 4, the accuracy of the analytical model is verified by finite element method, and the influence of different factors on TVMS and LTE is analyzed.