## 4 Dynamic Simulation of the Tooth Breakage Fault  
In this section, the dynamic simulation of a healthy planetary gear and seven fault gears is conducted. The breakage sizes \(p\) are 0, 0.1, 0.2, 0.5, 0.7, 0.8, 0.9 and 1.0, respectively. In this paper, the gear with \(p=0\) is called the healthy gear; a fault gear with \(0<p<1\) is called the half tooth breakage fault gear; and the fault gear with \(p=1\) is called the whole tooth breakage fault gear. The driving torque applied on the carrier is constant in time,  
\[T_{1}(t)=C\times\frac{1 Nm}{10\text{ rpm}}, \tag{15}\]  
and the load torque on the sun gear is set to,  
\[T_{2}(t)=\left\{ \begin{array}{ll}0,&t<t_{0}\\ (T_{1}/ratio)\times(\dot{\theta}_{cZ}/C)^{2},&t\geq t_{0}\end{array} \right., \tag{16}\]  
where \(ratio\) represents the transmission ratio of the planetary gear; \(C\) is the setting speed of the carrier and its unit is \(rpm\); the initial state of all of the DOFs is zero; and \(t_{0}\) is the time when the carrier rotational speed reaches the speed \(C\) for the first time. Based on the given loads of Equations (15) and (16), it can be found by qualitative analysis that the rotational speeds of the carrier, the sun gear and the planet gears will increase almost linearly until time \(t_{0}\), at which the load on the sun gear starts to join and the rotational speed of the carrier will ultimately fluctuate around speed \(C\), approximately. The time \(t_{0}\) can be estimated by,  
\[t_{0}=\pi\times J_{Iss}/3, \tag{17}\]  
Figure 5: Partial enlarged views of the mesh stiffness with different breakage sizes: (**a**) \(p=0\), (**b**) \(p=0.2\), (**c**) \(p=0.5\) and (**d**) \(p=1\). where \(J_{lss}\) is the total inertia of the planetary gear cast on the carrier side. The time \(t_{0}\) can be used for the simulation result checking. For the gearbox in Table 1, \(J_{lss}=0.1242\) kg \(\cdot\) m\({}^{2}\). Seven setting speeds are simulated for each gearbox; hence, 56 simulations in total are conducted in this paper. These setting speeds are listed in Table 3, where \(f_{c}\), \(f_{s}\) and \(f_{p}\) represent the rotational frequency of the carrier, the sun gear and the planet gear respectively, and \(f_{m}\) represents the mesh frequency. \(f_{p}\) is also the fault characteristics frequency in this paper. The damping matrix \(C\) in Equation (1) is built by the Rayleigh damping model, and the coefficients of Rayleigh damping are chosen as \(\alpha=0.01256\), \(\beta=2.12~{}\times~{}10^{-7}\) in this paper. The implicit Newmark algorithm is adopted to solve the nonlinear dynamic model [29]; the time step is \(1~{}\times~{}10^{-6}\) s. A virtual acceleration sensor is installed on the ring to collect its vertical vibration, and the data sampling rate is 40 kHz. In this paper, the signal is synthesized by,  
\[s=\bar{u}_{rY}+r_{r}\bar{\theta}_{rZ}, \tag{18}\]  
where \(\bar{u}_{rY}\) is the vertical acceleration of the ring gear, \(\bar{\theta}_{rZ}\) is the rotational acceleration of the ring gear and \(r_{r}\) is the radius of the ring gear. Noises are ignored in this paper. The time length of each simulation is five seconds. The data at the last four seconds are used for analysis. Figure 6 shows the changes of the root mean square (RMS) of the ring vertical acceleration versus breakage size and rotational speed. From Figure 6a, it can be found that, under some rotational speeds, the RMS does not increase monotonously with the increasing of the breakage size, and at some speeds, the RMS corresponding to the whole tooth breakage fault may be less than the RMS corresponding to a half tooth breakage fault. For example, when rotational speed is set to 197 rpm, the half tooth breakage fault gear with \(p=0.5\) has the largest vibration RMS. It can be found from Figure 6b that, for the healthy gear, the RMS is almost linearly increasing with the rotational speed due to the increasing of the external load, while for the fault gears, the RMSs are nonlinearly increasing with the rotational speed.