## 1 Introduction  
Rotating machinery with brush seals is an effective and efficient equipment where the brush seal can reduce leakage and improve the vibration performance. Brush seals do not require the shaft and the seal rings to be centrally symmetrical, so the brush seals not only increase the system's thermal efficiency but also its steadiness and stability. It is one of the most economic and practical seals, and has various applications [1; 2; 3; 4]. To continue to improve the design and efficiency of the brush seal, it is important to figure out the impacts of nonlinear seal forces on the rotor dynamics. One determines the seal force by a combination of experiments and theoretical analysis. [PERSON] et al. [5; 6; 7] constructed a brush seal rotor experimental bench to examine the bristle deformation and motions of the brush seal. They found that the motion primarily occurred in the area where the bristle density was increased when the pressure was increased, and the free end of the bristle usually oscillated. A 3D heat transfer model of the brush seal was developed in order to conduct the investigation of the thermodynamic properties of the brush seal rotor system. It was difficult to produce an exact model of the seal force that could be used in the test since the brush bristles had complex working phases [8]. Researchers suggestedapproximate theoretical models to determine the seal forces [9; 10; 11; 12]. Due to nonlinearities, the influence of the brush seals on the rotor system has to be studied. [PERSON] et al. [13] performed a study of the brush seal leakage rate and established a theoretical model of the brush seal rotor system considering flow-thermal-structure coupling. [PERSON] et al. [14; 15] constructed a multi-stage rotor system with radial offset of the disk. Nonlinear seal forces were derived analytically. The Runge-Kutta method was employed to study the nonlinear response of the brush seal rotor system. They optimized the parameters, and this resulted in a more stable rotor system. [PERSON] et al. [16] adopted the perturbation method of multiple scales to deal with a nonlinear rotating system and obtained nonlinear solutions with good accuracy. [PERSON] et al. [17] applied numerical simulation to study the effects of the installation position of the porous brush bristle model on the dynamics of the brush seal rotor system. They found that the direct stiffness, cross-coupling stiffness, and damping of the brush seal installed upstream were lower than those downstream. [PERSON] et al. [18] built a nonlinear brush seal rotor system based on a nonlinear Darcian porous media. The rotor dynamics with various pressure ratios and inlet spinning rates were evaluated.  
In this paper, an analytical brush seal force model is established and the nonlinear dynamic characteristics of brush seal rotor system are studied by a semi-analytical method for accurate stability and bifurcations. This paper is structured as follows: the first section is the introduction. The second section establishes the dynamic model of the brush seal rotor system, which includes the nonlinear brush seal force of a single bristle. The resultant seal force model of the full ring and the brush seal rotor system are built then. The nonlinear motions and bifurcations are achieved in the third section by a semi-analytical technique. Nonlinear motion illustrations of the brush seal rotor system are presented. Finally, the stabilities are evaluated and discussed.