## 1 Introduction  
The planetary gearbox is used in wind turbines because of its advantages, including high power density, compactness, etc. However, it suffers a high failure rate due to its complex structure and the time-varying working load [1]. The common gear failure modes include crack, pitting, wear, spalling and breakage. Figuring out the dynamic characteristics of these faults is of great importance for reducing equipment downtime and maintenance expenditure and improving the safety by providing an early warning of potential catastrophic failure [2]. Many research works have been done on the dynamics behaviors of the planetary gearbox. [PERSON] et al. developed a hybrid method of the analytical solution and the finite element model for the contact force calculation [3], and they studied the effect of manufacturing errors such as eccentricity and planet pin positioning errors on the load sharing behavior of a three-planet planetary transmission [4]. [PERSON] et al. built a lumped-parameter model of a two-stage helical planetary gear transmission system and studied the dynamic sensitivity of the load-sharing coefficient to errors [5]. Gravity has an important effect on the dynamics of the wind turbine gearbox because of its great size. [PERSON] et al. numerically studied the effects of gravity, ring support stiffness and bed plate tilt angle on the load-sharing characteristics of the planetary gear in horizontal axis wind turbines [6]. [PERSON] and [PERSON] used two different models, the phenomenological model and the lumped-parameter model, to study the frequency content of planetary gearbox vibrations under non-fault and different fault conditions [7]. [PERSON] et al. developed a two-dimensional lumped mass model to simulate the vibration signals of a planetary gear set in the perfect and crack situations; amplitude modulation can be found in the resultant vibration signals; sidebands appeared in the vibration spectrum and can used for fault detection. [PERSON] and [PERSON] studied the influences of cracks on the mesh stiffness and the dynamic performance of the planetary gear set [8]. Some other research works on the dynamics of planetary gearbox are [9; 10; 11]. Gearbox dynamic models vary from considering only torsional vibration [12] to the multibody model with lateral-torsional vibration coupled [13; 14; 15]. This paper focus on the vibration characteristics of the planetary gear with a planet gear tooth broken. Tooth breakage leads to the reduction of the time-varying gear mesh stiffness and the incensement of mass unbalance, and these excitations will affect the gears dynamic behavior and then the generated vibration [16; 17]. Compared with the flexible multibody model, the rigid multibody model can obtain an acceptable result with a much lower time cost [18], so it was most widely used in the tooth breakage fault research. [PERSON] et al. [19] proposed an analytical method to quantify the reduction of gear mesh stiffness due to tooth breakage fault. Their model considered bending, fillet-foundation and contact deflection, but it was only used for a single stage spur gear transmission. [PERSON] et al. [20] derived an expression of the effective mesh stiffness of the mating gears with tooth breakage fault, based on the fact that tooth breakage causes a great drop and rise at the transition points of single/double-tooth-pair mesh. Combining the fault model with a coupled lateral and torsional vibration model of a one-stage spur gear pair, they found that the properties of the vibration response of a gear pair with tooth faults were consistent with the variations of mesh stiffness. [PERSON] and [PERSON]. [21] developed a three-dimensional analytical helical gear pair model considering the mass eccentricity and mesh stiffness induced by tooth breakage with sliding friction. The effects of the tooth breakage shape and size on the mesh stiffness and dynamic responses of a helical gear pair were calculated.