## 1 Introduction  
In gear transmission systems, the internal and external gear pairs are two common meshing forms. Different from the external gear pair that contains two external gears, the internal gear pair includes an external gear and an internal gear. The internal gear pair is usually used in planetary gears, which are widely applied in the gear transmissions of helicopters. The transmission system is a critical life-deciding part of the helicopter, containing dozens of planetary gear pairs. However, when the planetary gear transmission system operates for long-term operation or short-term dry operation, gear wear unavoidably occurs and aggravates the internal gear pairs. As a common form of early failures, tooth wear is a critical factor leading to more serious faults such as the reduction of transmission accuracy and the aggravation of vibration and noise. Helicopter health and usage monitoring systems (HUMS) mainly use the condition indicators to monitor the wear status of the gear components of the transmission system, which needs a clear fault mechanism for the gear wear. Therefore, it is of great significance to accurately evaluate the influence of tooth wear on gear transmission systems. This study intends to establish an analytical model for internal spur gears with tooth wear that can reveal the effects of tooth wear on gear mesh relationship and mesh stiffness and provide instructive information for gear transmission fault diagnosis. Some studies have in fact focused on the fault mechanism of gear wear. [PERSON] developed a numerical model based on the [PERSON]'s wear equation to predict mild tooth wear in external and helical gears [1; 2]. [PERSON] et al. proposed a wear depth prediction methodology of parallel axis gear pairs by combining a FEM-based gear contact model with the [PERSON]'s wear equation [3]. Subsequently, according to the variation of dynamic contact pressures, [PERSON] established a wear evaluation model of gear tooth considering the time-varying wear coefficient [4]. [PERSON] predicted the steady state adhesive wear of spur gears under the elastohydrodynamic lubrication (EHL) [5]. Moreover, Osmansimulated mild abrasive wear in wide-faced solid spur and helical gears by considering the [PERSON]'s law and the influence of the lubrication regime [6]. [PERSON] adapted [PERSON]'s wear equation to investigate wear theoretically [7], and the result was verified experimentally. As can be seen from the aforementioned studies, the [PERSON]'s wear equation has been successfully used for the evaluation of gear tooth wear distribution. Based on the results of the tooth wear distribution obtained from the above models, some scholars have studied the influences of tooth wear on gear ratio, backlash, tooth profile deviation, mesh stiffness, and transmission error. [PERSON] et al. investigated the variation of gear ratio caused by tooth wear [8]. Inspired by the investigation in this work, the mesh relationship of internal spur gears with tooth wear is investigated, which is described in Figure 1. [PERSON] et al. introduced gear tooth wear into the dynamic model by directly turning the gear tooth wear into the tooth profile deviation [9]. [PERSON] et al. employed a computational model to study the effects of tooth wear on the dynamic behavior of planetary gears, in which contact theory and [PERSON]'s wear model were used to obtain the wear distribution [10]. In addition, [PERSON] et al. investigated the increasing backlash caused by gear wear [11]. In their work, the tooth wear was regarded as backlash and then incorporated into the dynamic model. As for the effect of tooth wear on the transmission error, [PERSON] proposed a finite element procedure to study the development of transmission error with gear tooth wear [12]. [PERSON] et al. assumed the wear depth as profile error and took the dynamic contact load into account to analyze the transmission error of spur gears [13]. In addition, [PERSON] et al. and [PERSON] et al.