# Research on Dynamic Modeling and Vibration Characterization of Integrated Bearings
## 1 Dynamic Modeling of Integrated Bearing Systems
### Dynamical Differential Equations of Balls
\begin{array}{c}T_{\eta ij}\cos\alpha_{ij}-T_{\eta ej}\cos\alpha_{ej}-Q_{ij}\sin \alpha_{ij}-Q_{ej}\sin\alpha_{ej}+\\ F_{\rm R\eta ij}\cos\alpha_{ij}+F_{\rm R\eta ej}\cos\alpha_{ej}+F_{\rm H\eta ij} \cos\alpha_{ij}-F_{\rm H\eta ej}\cos\alpha_{ej}+\\ Q_{cjxz}\sin\gamma_{je}-P_{\rm S\bar{\gamma}jz}\cos\gamma_{jz}-P_{\rm R\bar{ \gamma}j}\cos\gamma_{jz}=m_{\rm b}\bar{x}_{\rm bj}\\ T_{\eta ej}\sin\alpha_{ej}-T_{\eta ij}\sin\alpha_{ij}+Q_{ij}\cos\alpha_{ij}-Q_ {ej}\cos\alpha_{ej}+\\ F_{\rm R\eta ij}\sin\alpha_{ij}-F_{\rm R\eta ej}\sin\alpha_{ej}-F_{\rm H\eta ij }\sin\alpha_{ij}+F_{\rm H\eta ej}\sin\alpha_{ej}+\\ F_{\eta j}+Q_{cjy}+P_{\rm S\eta ij}\sin\gamma_{jy}+P_{\rm R\eta ij}\sin\gamma_ {jy}+\\ m_{\rm b}\bar{\rm g}\cos\varphi_{j}=m_{\rm b}\bar{y}_{bj}\\ F_{R\bar{\gamma}j}-F_{H\bar{\gamma}j}+F_{H\bar{\gamma}ej}-F_{R\bar{\gamma}ej }+T_{\bar{\gamma}ej}-T_{\bar{\gamma}ij}-P_{S\eta j}\cos\gamma_{jy}-\\ P_{R\eta j}\cos\gamma_{jy}-P_{Dj}-F_{\bar{\gamma}j}+Q_{cjxz}\cos\gamma_{jz}+P_ {S\xi j}\sin\gamma_{jz}+\\ P_{R\bar{\gamma}j}\sin\gamma_{jz}-m_{\rm b}\bar{\rm g}\sin\varphi_{j}=m_{\rm b }\bar{z}_{bj}\\ \left(T_{\bar{\gamma}ij}-F_{R\bar{\gamma}ij}\right)\frac{D_{\rm B}}{2}\cos \alpha_{ij}+\left(T_{\bar{\gamma}ej}-F_{R\bar{\gamma}ej}\right)\frac{D_{\rm B }}{2}\cos\alpha_{ej}+\\ J_{x}\dot{\omega}_{xj}-\left(P_{\rm S\eta j}+P_{\rm R\eta j}\right)R_{jz}/\sin \gamma_{jy}+\\ \left(P_{\rm S\bar{\gamma}j}\sin\gamma_{jz}+P_{\rm R\bar{\gamma}j}\sin\gamma_ {jz}\right)R_{jz}\cot\gamma_{jy}=I_{\rm b}\dot{\omega}_{\rm bjx}\\ \left(F_{\rm R\bar{\gamma}ij}-T_{\bar{\gamma}ij}\right)\frac{D_{\rm B}}{2}\sin \alpha_{ij}+\left(T_{\bar{\gamma}ej}-F_{R\bar{\gamma}ej}\right)\frac{D_{\rm B }}{2}\sin\alpha_{ej}-G_{yj}-\\ J_{y}\dot{\omega}_{yj}+\left(P_{\rm S\bar{\gamma}j}\cos\gamma_{jy}+P_{\rm R \eta j}\right)\cos\gamma_{jy}R_{jz}\tan\gamma_{jz}-\\ \left(P_{\rm S\bar{\gamma}j}+P_{\rm R\bar{\gamma}j}\right)R_{jz}/\cos\gamma_{ jz}=I_{\rm b}\dot{\omega}_{\rm bjy}-I_{\rm b}\omega_{\rm bjz}\dot{\theta}_{bj}\\ \left(T_{\eta ej}-F_{R\eta ej}\right)\frac{D_{\rm B}}{2}+\left(T_{\eta ij}-F_ {R\eta ij}\right)\frac{D_{\rm B}}{2}-G_{zj}-J_{z}\dot{\omega}_{zj}+\\ \left(P_{S\eta j}\sin\gamma_{jy}+P_{R\eta j}\sin\gamma_{jy}\right)R_{jz}\tan \gamma_{jz}+\left(P_{R\bar{\gamma}j}\cos\gamma_{jz}+\right.\\ \left.P_{S\xi j}\cos\gamma_{jz}\right)R_{jz}\cot\gamma_{jy}=I_{\rm b}\dot{ \omega}_{\rm bjz}+I_{\rm b}\omega_{\rm bjy}\dot{\theta}_{bj}\\ \end{array}  \tag{6}\]  
In the equation, (\(m_{b}\)) represents the mass of the ball; (\(\bar{x}_{bj}\)), (\(\bar{y}_{bj}\)), and (\(\bar{z}_{bj}\)) denote the accelerations at the center of mass of the ball; and (\(I_{bj}\)) indicates the moment of inertia of the ball. Additionally, (\(\omega_{bjx}\)), (\(\omega_{bjy}\)), and (\(\omega_{bjz}\)) are the components of angular velocity, while (\(\dot{\omega}_{bjx}\)), (\(\dot{\omega}_{bjy}\)), and (\(\dot{\omega}_{bjz}\)) are the components of angular acceleration. Lastly, (\(\dot{\theta}_{bj}\)) signifies the translational velocity of the ball, and (\(D_{w}\)) represents its diameter.