## 1 Introduction  
Gear systems commonly exist in mechanical equipment, such as wind turbines, shearer, and the motor vehicle [1]. Due to the excessive load, poor lubrication, and other harsh operating conditions, faults often occur in the gear systems. The faults may generate serious economic losses and even casualties. Specifically, damages (e.g., pitting, cracking, and spalling) on the gear teeth are the main causes of system failure, and they usually impact the dynamic characteristics of gear devices [2; 3; 4; 5]. Background noise often occurs in actual fault detection. It is difficult to extract fault feature accurately when background noise is involved [6]. Therefore, it is essential to effectively obtain the dynamic characteristics of gear systems and eliminate the noise for identifying faults because they play an important role in improving reliability and reducing the maintenance cost of gear systems. Dynamical modeling is a useful way to obtain the dynamic characteristics of gear systems, and there are many studies about the dynamic modeling [7; 8; 9; 10; 11; 12]. To effectively obtain the dynamic characteristics of gear systems, researchers either increase the degrees of freedom from single to multi degrees, or consider various practical phenomena such as backlash, tooth friction, manufacturing eccentric errors, etc. [PERSON] et al. [13] proposed a simplified purely torsional mechanical model of a gear pair, which was supported by rigid mounts with backlash and time-varying mesh stiffness. [PERSON] et al. [14] modelled a one-stage reduction gear by using three different dynamic models, which consider the gear mesh friction. [PERSON] et al. [15] considered manufacturing eccentric errors and tooth profile errors, and generalized a bending-torsional-axial coupled dynamical model to investigate the dynamic floating performances. Dynamical modeling of gear systems with faults is also an important topic for understanding the dynamic characteristics of the gear fault. Spalling is one of the most destructive surface failure models of gear faults [16]. Until now, many researchers have modeled the tooth spalling and investigated its effect on dynamic characteristics of the gear systems [17]. According to their research, tooth spalling was regarded as different shapes (e.g., rectangle, circular, and V-shape), and the influence of the spalling on time-vary mesh stiffness (TVMS) of gear was explored [18; 19; 20]. [PERSON] et al. [21] proposed a curved bottom shape method to model the geometric features of gear tooth spalls and studied the effect of different shapes and severity conditions time-vary mesh stiffness by finite element analysis. [PERSON] et al. [22] presented a new two-dimensional Gaussian distribution model to depict pitting distribution. In their analysis, the overlaps of the pits and pits in the boundary of the tooth surface were considered in the computation of TVMS. Furthermore, [PERSON] et al. [23] built a dynamic model that includes tooth surface roughness changes and geometric deviations because of the pitting and spalling of gear teeth. This model was validated by comparison with experiments under different rotational speed and fault conditions. [PERSON] et al. [24] proposed a 12-degree-of-freedom lumped parameter model to simulate the effects of tooth crack, breakage, and spalling on the dynamic response of wind turbine gearbox with spur gears. [PERSON] et al. [25] proposed a 6-degree-of-freedom approach to investigate the dynamic characteristics of a pair of helical gears under sliding friction with spalling faults. The previously mentioned studies only focus on a single tooth spalling fault. In fact, tooth spalling, which is caused by excessive local Hertzian contact fatigue stress, may distribute over multiple teeth [26; 27]. In term of this phenomenon, [PERSON] et al. [28] modeled a gear system with double-teeth spalling fault, and simulated its effect on vibration characteristics. However [28], only the phenomenon where the tooth spalling occurs on two adjacent teeth is considered.