# Dynamic Simulation Model-Driven Fault Diagnosis Method for Bearing under Missing Fault-Type Samples
## 2 Theoretical Background  
### Construction and Acquisition of Simulation Signal  
Some types of fault samples may be missing during the fault diagnosis of mechanical equipment. To simulate this type of situation and obtain the lacked fault samples, the concentrate quality standard was established with the rolling bearing fault of the rotor-bearing-casing coupling system dynamics model. This was followed by the establishment of the dynamic model of the rolling bearing fault and the construction of the lack of fault samples. As shown in Figure 1, a simulation model made up of a rotor, bearing, bearing seat, shaft, and other parts are depicted. The bearing inner ring remains connected to the bearing housing while the shaft moves. Suppose the stiffness and damping of the primary components are not changed numbers and the left bearing occurs faults, six differential equations of motion pertaining to the left support bearing can be listed [17]:  
\[m_{bL}\ddot{x}_{bL}+k_{fLH}(x_{bL}-x_{c})+c_{fLH}\big{(}\dot{x}_{bL}-\dot{x}_{ c}\big{)}+k_{tLH}(x_{bL}-x_{wL})+c_{tLH}\big{(}\dot{x}_{bL}-\dot{x}_{wL}\big{)}=0 \tag{1}\]  
\[m_{bL}\ddot{y}_{bL}+k_{fLV}(y_{bL}-y_{c})+c_{fLV}\big{(}\dot{y}_{bL}-\dot{y}_{ c}\big{)}+k_{tLV}(y_{bL}-y_{wL})+c_{tLV}\big{(}\dot{y}_{bL}-\dot{y}_{wL}\big{)}=-m _{bL}g \tag{2}\]  
\[m_{tL}\ddot{x}_{rL}+k\big{(}x_{rL}-x_{rp}\big{)}+c_{rb}\dot{x}_{rL}-F_{xbL}=0 \tag{3}\]  
\[m_{rL}\ddot{y}_{rL}+k\big{(}y_{rL}-y_{rp}\big{)}+c_{rb}\dot{y}_{rL}-F_{ybL}=-m _{rL}g \tag{4}\]  
\[m_{wL}\ddot{x}_{wL}+k_{tLH}(x_{wL}-x_{bL})+c_{tLH}\big{(}\dot{x}_{wL}-\dot{x}_ {bL}\big{)}+F_{xbL}=0 \tag{5}\]  
\[m_{wL}\ddot{y}_{wL}+k_{tLH}(y_{wL}-y_{bL})+c_{tLH}\big{(}\dot{y}_{wL}-\dot{y}_ {bL}\big{)}+F_{ybL}=-m_{wL}g \tag{6}\]  
where, \(m_{bL}\) is the mass of bearing support; \(k_{fLH}\) and \(k_{fLV}\) are the transverse and longitudinal support stiffness between the shell and the bearing support respectively. \(c_{fLH}\) and \(c_{fLV}\) are the transverse and longitudinal support damping between the shell and the bearing support respectively. \(k_{tLH}\) and \(k_{tLV}\) are the transverse and longitudinal support stiffness between the bearing outer ring and the bearing support respectively. \(c_{tLH}\) and \(c_{tLV}\) are the transverse and longitudinal extrusion film damping between the bearing outer ring and the bearing support. \(m_{rL}\) is the equivalent mass of rotor. \(k\) and \(c_{rh}\) are the shaft stiffness and damping of the rotor at the bearing. \(F_{xbl}\) and \(F_{ybl}\) are the supporting reaction forces of the bearing, and \(m_{wL}\) is the mass of the outer ring of the bearing. A schematic illustration of a bearing with a localized defect area is shown in Figure 2 on the ball, inner ring, and outer ring components. The cyclically varying contact forces \(F_{xbl}\) and \(F_{vbl}\) are the sum of the contact forces of the balls at different angles:  
\[F_{xbl}=\sum_{j=1}^{Z}f_{j}cos\theta_{j},F_{ybl}=\sum_{j=1}^{Z}f_{j}sin\theta _{j} \tag{7}\]  
where \(\theta_{j}\) is the angular position of the jth ball, and \(f_{j}\) is the contact force between the jth ball and the raceway calculated according to the following formula:  
\[f_{j}=C_{b}\big{[}\delta_{j}\big{]}^{n}=C_{b}\big{(}xcos\theta_{j}+ysint\theta _{j}-\delta_{D}\big{)}^{n}\times H\big{(}xcos\theta_{j}+ysint\theta_{j}-\delta _{0}-\delta_{D}\big{)} \tag{8}\]  
where \(C_{b}\) is the Hertzian contact stiffness, \(H(\cdot)\) is the Heaviside function, \(\delta_{j}\) is the clearance between the ball and the raceway, \(\delta_{0}\) is the initial clearance, \(\delta_{D}\) is the clearance caused by the local defect area, and \(n\) is the load-deformation coefficient, which is \(2/3\) for the ball bearing. The simulation model adopts a SKF6203 bearing and its main parameters are listed in Table 1. The damage diameter \(\mathrm{L_{D}}\) is set to 0.3556 mm, the damage depth a is set to 2.794 mm, the rotational speed is 1797r/min, and the sampling frequency is set to 10 kHz.