## 3 3D Dynamics Deduction
### Dynamic Equilibrium Equations
#### 3.3.2 Moment Balance around the \(\mathbf{x}\)-Axis of the \(<xyz>_{C}\) Coordinate System
\begin{array}{c}\frac{m^{2}\omega^{2}}{2}\big{\{}-\text{sin}(2\psi)\text{ cos}\,(2\hat{\gamma}_{1})\text{sin}\,\hat{\gamma}_{2}+\text{sin}(2\hat{ \gamma}_{1})\big{[}\text{sin}^{2}\,\psi-\text{cos}^{2}\,\psi\text{sin}^{2}\, \hat{\gamma}_{2}\big{]}\big{\}}\\ \hskip 14.226378 pt+m\epsilon\omega^{2}(\text{sin}\,\psi\text{sin}\,\hat{\gamma }_{1}-\text{cos}\,\psi\text{cos}\,\hat{\gamma}_{1}\text{sin}\,\hat{\gamma}_{2} )\hat{y}_{c}+\omega^{2}\text{cos}^{2}\,\hat{\gamma}_{2}\text{sin}\,\hat{\gamma }_{1}\text{cos}\,\hat{\gamma}_{1}\text{(J}_{P}-\text{J}_{T})\\ \hskip 14.226378 pt+\frac{m_{b}R^{2}\omega^{2}}{2}\sum_{i=1}^{n}\big{\{}-\text{ sin}\big{(}2\hat{\beta}_{i}\big{)}\text{cos}(2\hat{\gamma}_{1})\text{sin}(\hat{ \gamma}_{2}+\theta)+\text{sin}(2\hat{\gamma}_{1})\big{[}\text{sin}^{2}\,\hat{ \beta}_{i}-\text{cos}^{2}\,\hat{\beta}_{i}\text{sin}^{2}(\hat{\gamma}_{2}+ \theta)\big{]}\,\big{\}}\\ \hskip 14.226378 pt+m_{b}R\omega^{2}\sum_{i=1}^{n}\big{[}\text{sin}\,\hat{ \beta}_{i}\text{sin}\,\hat{\gamma}_{1}-\text{cos}\,\hat{\beta}_{i}\text{cos}\, \hat{\gamma}_{1}\text{sin}(\hat{\gamma}_{2}+\theta)\big{]}\hat{y}_{c}+k_{T} \hat{\gamma}_{1}=0.\end{array}  \tag{20}\]  
3.3 Moment Balance around the \(\mathbf{y}^{\prime}\)-Axis of the \(<\mathbf{x}^{\prime}\mathbf{y}^{\prime}z^{\prime}>_{C}\) Coordinate System  
\(F_{m}\) and \(F_{Bi}(i=1\sim n)\) in \(<\mathbf{xyz}>_{C}\) are transformed to \(<\mathbf{x}^{\prime}\mathbf{y}^{\prime}z^{\prime}>_{C}\) by the rotation matrix \(\mathbf{T}_{1}\), described as follows, respectively:  
\[\mathbf{F}_{m}^{\prime}=\mathbf{T}_{1}\mathbf{F}_{m}=-m\omega^{2} \begin{bmatrix}\cos\psi\text{ cos}\,\hat{\gamma}_{2}+\hat{x}_{c}\\ \cos\,\hat{\gamma}_{1}(\text{sin}\,\psi\text{cos}\,\hat{\gamma}_{1}+\epsilon\cos \psi\text{sin}\,\hat{\gamma}_{1}\text{sin}\,\hat{\gamma}_{2}+\hat{y}_{c})\\ -\text{sin}\,\hat{\gamma}_{1}(\text{sin}\,\psi\text{cos}\,\hat{\gamma}_{1}+\text{ cos}\,\psi\text{sin}\,\hat{\gamma}_{1}\text{sin}\,\hat{\gamma}_{2}+\hat{y}_{c}) \end{bmatrix} \text{ and } \tag{21}\]  
\[\mathbf{F}_{Bi}^{\prime}=\mathbf{T}_{1}\mathbf{F}_{Bi}=-m_{b}\omega^{2} \begin{bmatrix}R \cos\hat{\beta}_{i}\text{cos}(\hat{\gamma}_{2}+\theta)+\hat{x}_{c}\\ \cos\,\hat{\gamma}_{1}(R\sin\hat{\beta}_{i}\text{cos}\,\hat{\gamma}_{1}+R\cos\hat{ \beta}_{i}\text{sin}\,\hat{\gamma}_{1}\text{sin}(\hat{\gamma}_{2}+\theta)+\hat{y}_{ c})\\ -\text{sin}\,\hat{\gamma}_{1}(R\sin\hat{\beta}_{i}\text{cos}\,\hat{\gamma}_{1}+R \cos\hat{\beta}_{i}\text{sin}\,\hat{\gamma}_{1}\text{sin}(\hat{\gamma}_{2}+\theta)+ \hat{y}_{c})\end{bmatrix} ,i=1\sim n. \tag{22}\]The equilibrium position vector of the mass center of the rotor and the \(i\)-th ball in \(<x^{\prime}y^{\prime}z^{\prime}>_{C}\) is \(\hat{\textbf{\emph{r}}}_{m}^{\prime}\) and \(\hat{\textbf{\emph{r}}}_{Bi}^{\prime}(i=1\sim n)\). The inverse transformation of \(\hat{\textbf{\emph{r}}}_{m}^{\prime\prime}\) and \(\hat{\textbf{\emph{r}}}_{Bi}^{\prime\prime}\) (Equations (3) and (5)) or the transformation of \(\hat{\textbf{\emph{r}}}_{m}\) and \(\hat{\textbf{\emph{r}}}_{Bi}\) (Equations (13) and (14)) can be obtained. \[\hat{\textbf{\emph{r}}}_{m}^{\prime}=\textbf{T}_{2}^{-1}\hat{\textbf{\emph{r} }}_{m}^{\prime\prime}=\textbf{T}_{1}\hat{\textbf{\emph{r}}}_{m}=\left[ \begin{array} []{c}\varepsilon\cos\psi\cos\hat{\textbf{\emph{\emph{\gamma}}}_{2}}\\ \varepsilon\sin\psi\\ -\varepsilon\cos\psi\sin\hat{\textbf{\emph{\gamma}}}_{2}\end{array} \right]\text {and} \tag{23}\]  
\[\hat{\textbf{\emph{r}}}_{Bi}^{\prime}=\textbf{T}_{2}^{-1}\hat{\textbf{\emph{r }}}_{Bi}^{\prime\prime}=\textbf{T}_{1}\hat{\textbf{\emph{r}}}_{Bi}=\left[ \begin{array}{c}R\cos\hat{\textbf{\emph{\emph{\beta}}}}_{i}\cos(\hat{ \textbf{\emph{\gamma}}}_{2}+\theta)\\ R\sin\hat{\textbf{\emph{\beta}}}_{i}\\ -R\cos\hat{\textbf{\emph{\beta}}}_{i}\sin(\hat{\textbf{\gamma}}_{2}+\theta) \end{array}