## 2 Problem Formulation  
In the fault diagnosis of gearboxes, it is known that modulation phenomena will be observed in vibration signals when there is a gear fault in a gearbox [22; 23]; e.g., amplitude modulation (AM) and frequency modulation (FM). In order to improve the success rate of the fault diagnosis method, it is necessary to determine the physical mechanism of the modulation phenomena caused by gear faults. Figure 1a presents a schematic diagram of a gear pair with a local tooth fault; except for the faulty tooth, the other teeth of the pinion correspond to normal meshing. Figure 1b compares the stiffnesses of the normal and faulty gear mesh, which are calculated based on an analytical method [6; 7; 8]. It can be seen that there is a sudden reduction of the gear mesh stiffness when the faulty tooth is engaged; thus, a periodic reduction of gear mesh stiffness will be observed in each revolution of the pinion. In order to study the modulation mechanism of the vibration signals caused by the periodic reduction of gear mesh stiffness, a six degrees-of-freedom (6 DOFs) lumped parameter model for a gear system is established; the schematic diagram of the spur gear system is shown in Figure 2. \(J_{i}\) means the inertial moment; \(mi\) is the mass; \(K_{iBx}\) and \(K_{iBy}\) stand for the stiffness of the support bearings in the \(x\) direction and \(y\) direction, respectively; \(C_{iBx}\) and \(C_{iBy}\) are the damping of the support bearings in the \(x\) direction and \(y\) direction, respectively; \(F_{f}\) is the inter-tooth friction force; \(T_{i}\) is the torque; \(\Omega_{i}\) is the rotating speed; \(k_{t}\) represents the time-varying mesh stiffness; \(C_{m}\) is the mesh damping between gear teeth; and \(i=p\), \(g\) for the pinion and gear, respectively. The governing equations of the gear system can be obtained as follows:  
\[\left\{ \begin{aligned} & J_{p}\delta_{p}^{i}=T_{p}-M_{pN}+M_{pf}\\ & J_{g}\delta_{g}^{i}=-T_{g}+M_{gN}-M_{gf}\\ & m_{p}\dot{x_{p}}=F_{f}-K_{pBx}x_{p}-C_{pBx}x_{p}\\ & m_{g}\dot{x_{g}}=-F_{f}-K_{gBx}x_{g}-C_{gBx}x_{g}\\ & m_{p}\dot{y_{p}}=-N-K_{pBx}y_{p}-C_{pBx}y_{p}\\ & m_{g}\dot{y_{g}}=N-K_{gBx}y_{g}-C_{gBx}y_{g}\end{aligned} \right. \tag{1}\]  
where \(\dot{\theta_{i}}\) denotes the angular acceleration; \(x_{i}\), \(\dot{x_{i}}\) and \(\dot{x_{i}}\) stand for displacement, velocity and acceleration in the \(x\) direction, \(y_{i}\); \(y_{i}\) and \(\dot{y_{i}}\) refer to the displacement, velocity and acceleration in the \(y\) direction; \(M_{iN}\) means the normal mesh force moment; \(M_{if}\) is the friction force moment; and \(N\) is the net contact force due to the elasticity. The subscript \(i=p\), \(g\) for the pinion and gear, respectively. Figure 1: Schematic diagram of (**a**) a gear pair with a local tooth fault and (**b**) comparison between normal and fault gear mesh stiffnesses. Figure 3 illustrates the simulated waveform of the dynamic transmission error (DTE) of the gear system with a local tooth fault and its corresponding spectrum. It is observed that there is a periodic impulse caused by the tooth fault, and the cycle equals \(2\pi\). Furthermore, abundant modulation sidebands are found around the meshing frequency and its harmonics, as shown in Figure 3b. Then, the variation of the order spectrum corresponding to the rotating speeds is simulated and plotted, as shown in Figure 4. It is found that there are many modulated sidebands around the dotted line in each rotating speed, and the frequency range of the sub-bands are time-varying with respect to the rotating speeds. However, there are also some sub-bands that contain the fault-related components, as shown by the oval area in Figure 4, which are defined as multi-scale modulation phenomena in this paper. Figure 4: Variation of the order spectrum corresponding to the rotating speeds. Figure 3: (**a**) Waveform of the dynamic transmission error (DTE) of the gear system with a local tooth fault and (**b**) its corresponding order spectrum at a speed condition of \(30\pi\) rad/s.