# Research on Dynamic Modeling and Vibration Characterization of Integrated Bearings
## 1 Dynamic Modeling of Integrated Bearing Systems
### Dynamical Differential Equations of Cage  
The integrated bearing cage is a crown-shaped cage. During the operation of the bearing, the relative positions of the steel balls and the pockets of the cage are quite complex. Therefore, a diagram illustrating the relative positional relationships has been established, as shown in Figure 3. Point (\(o_{cbj}\)) represents the contact point between the steel ball and the pocket, while (\(R_{cjx}\)), (\(R_{cjy}\)), and (\(R_{cjz}\)) are the projections of the contact points between the steel ball and the cage, as well as the distance to the center of mass of the cage in the cage coordinate system. A set of dynamic differential equations for the cage has been established, as follows:  
\[\sum\limits_{j=1}^{Z}\left(Q_{cjxz}\sin\gamma_{jz}-P_{S\bar{\gamma}j}\cos\gamma _{jz}-P_{R\bar{\gamma}j}\cos\gamma_{jz}\right)=m_{c}\bar{x}_{cx}\] (7  
\[\begin{array}{c}\sum\limits_{j=1}^{Z}\big{[}\big{(}Q_{cjxz}\cos\gamma_{jz}-P_{ Snj}\cos\gamma_{jy}-P_{Rqj}\cos\gamma_{jy}+\\ P_{S\xi j}\sin\gamma_{jz}+P_{R\xi j}\sin\gamma_{jz}\big{)}\cos\varphi_{j}\big{]} +\sum\limits_{j=1}^{Z}\big{[}\big{(}P_{Snj}\sin\gamma_{jy}+\\ P_{R\eta j}\sin\gamma_{jy}+Q_{cjy}\big{)}\sin\varphi_{j}\big{]}=m_{c}\tilde{z}_ {cz}\\ \\ \sum\limits_{j=1}^{Z}\big{[}\big{(}Q_{cjxz}\cos\gamma_{jz}-P_{S\eta j}\cos \gamma_{jy}-P_{R\eta j}\cos\gamma_{jy}+\\ P_{S\xi j}\sin\gamma_{jz}+P_{R\xi j}\sin\gamma_{jz}\big{)}R_{cjy}\cos\varphi_{ j}\big{]}+\\ \\ \sum\limits_{j=1}^{Z}\big{[}\big{(}P_{S\eta j}\sin\gamma_{jy}+P_{R\eta j}\sin \gamma_{jy}+Q_{cjy}\big{)}R_{cjy}\sin\varphi_{j}\big{]}-\\ \\ \sum\limits_{j=1}^{Z}\big{[}\big{(}-P_{S\eta j}\cos\gamma_{jy}-P_{R\eta j}\cos \gamma_{jy}+Q_{cjxz}\cos\gamma_{jz}+\\ P_{S\xi j}\sin\gamma_{jz}+P_{R\xi j}\sin\gamma_{jz}\big{)}R_{cjz}\sin\varphi_{ j}\big{]}+\\ \\ \sum\limits_{j=1}^{Z}\big{[}\big{(}P_{S\eta j}\sin\gamma_{jy}+P_{R\eta j}\sin \gamma_{jy}+Q_{cjy}\big{)}R_{cjz}\cos\varphi_{j}\big{]}=\\ \\ \sum\limits_{j=1}^{Z}\big{[}\big{(}Q_{cjxz}\sin\gamma_{jz}-P_{S\xi j}\cos \gamma_{jz}-P_{R\xi j}\cos\gamma_{jz}\big{)}R_{cjz}\big{]}-\\ \\ \sum\limits_{j=1}^{Z}\big{[}\big{(}Q_{cjxz}\cos\gamma_{jz}-P_{S\eta j}\cos \gamma_{jy}-P_{R\eta j}\cos\gamma_{jy}+\\ P_{S\xi j}\sin\gamma_{jz}+P_{R\xi j}\sin\gamma_{jz}\big{)}R_{cjx}\cos\varphi_{ j}\big{]}-\\ \\ \sum\limits_{j=1}^{Z}\big{[}\big{(}P_{S\eta j}\sin\gamma_{jy}+P_{R\eta j}\sin \gamma_{jy}+Q_{cjy}\big{)}R_{cjx}\sin\varphi_{j}\big{]}=\\ \\ \sum\limits_{j=1}^{Z}\big{[}\big{(}Q_{cjxz}\cos\gamma_{jx}-P_{S\eta j}\cos \gamma_{jy}-P_{R\eta j}\cos\gamma_{jy}+\\ P_{S\xi j}\sin\gamma_{jz}+P_{R\xi j}\sin\gamma_{jz}\big{)}R_{cjx}\cos\varphi_{ j}\big{]}-\\ \\ \sum\limits_{j=1}^{Z}\big{[}\big{(}P_{S\eta j}\sin\gamma_{jy}+P_{R\eta j}\sin \gamma_{jy}+Q_{cjy}\big{)}R_{cjx}\cos\varphi_{j}\big{]}-\\ \\ \sum\limits_{j=1}^{Z}\big{[}\big{(}Q_{cjxz}\sin\gamma_{jz}-P_{S\xi j}\cos \gamma_{jz}-P_{R\xi j}\cos\gamma_{jz}\big{)}.\\ \\ R_{cjy}\big{]}+m_{c}gL=I_{cz}\omega_{cz}-\big{(}I_{cx}-I_{cy}\big{)}\omega_{ cxt}\omega_{cy}\end{array} \tag{11}\]  
In the equation, (\(m_{c}\)) represents the mass of the cage; (\(\tilde{x}_{\rm{c}}\)), (\(\tilde{y}_{\rm{cy}}\)), and (\(\tilde{z}_{cz}\)) denote the acceleration of the cage's center of mass; (\(I_{cx}\)), (\(I_{cy}\)), and (\(I_{cz}\)) indicate the moment of inertia of the cage; (\(\omega_{cx}\)), (\(\omega_{cy}\)), and (\(\omega_{cz}\)) refer to the components of the cage's angular velocity; (\(\dot{\omega}_{cz}\)), (\(\dot{\omega}_{cy}\)), and (\(\dot{\omega}_{cz}\)) signify the components of the cage's angular acceleration; (\(Z\)) represents the number of steel balls; and (\(L\)) is the axial distance between the center plane of the cage's holes and the position of the cage's center of mass.