## 3 3D Dynamics Deduction
### Dynamic Equilibrium Equations  
3.3.1. Tri-Axial Force Balance of \(<xyz>_{O}\) Frame  
The rotor system equipped with a skew-mounted ABB has a constant spindle speed \(\omega\). In the rotating reference frame \(<xyz>_{O}\), the angular velocity of the system is denoted as \(\Omega_{O}= \begin{bmatrix}0&0&\omega\end{bmatrix} ^{T}\), and the position of the offset centroid \(C\) after the deformations of the support springs can be expressed as the vector \(r_{OC}= \begin{bmatrix}x_{c}&y_{c}&z_{c}\end{bmatrix} ^{T}\). The system is in a dynamic balance, the equilibrium position of the rotor centroid \(C\) is expressed as the vector  
\[\hat{r}_{OC}= \begin{bmatrix}\hat{x}_{c}\\ \hat{y}_{c}\\ \hat{z}_{c}\end{bmatrix} . \tag{2}\]  
Additionally, \(\hat{\gamma}_{1}\), \(\hat{\gamma}_{2}\), and \(\hat{\beta}_{i}(i=1\sim n)\) are the equilibrium pitch angle, equilibrium yaw angle, and \(n\) balls' equilibrium angular positions, respectively. From the definition of coordinate systems (Figure 2), the position vector of mass center \(G\) of the rotor in \(<x^{\prime\prime}y^{\prime\prime}z^{\prime\prime}>_{C}\) as  
\[\hat{r}^{\prime\prime}_{m}= \begin{bmatrix}\varepsilon\cos\psi\\ \varepsilon\sin\psi\\ 0\end{bmatrix} . \tag{3}\]  
The equilibrium position vector of the mass center of the \(i\)-th ball in \(<x^{\prime\prime}y^{\prime\prime}z^{\prime\prime}>_{C}\) is \(\hat{r}^{\prime\prime}_{Bi}\), the reverse transformation is in \(<x^{\prime\prime}y^{\prime\prime}z^{\prime\prime}>_{C}\) as \(\hat{r}^{\prime\prime}_{Bi}\), where  
\[\hat{r}^{\prime\prime}_{Bi}= \begin{bmatrix}R\cos\hat{\beta}_{i}\\ R\sin\hat{\beta}_{i}\\ 0\end{bmatrix} ,i=1\sim n\;\;\text{and} \tag{4}\]  
\[\hat{r}^{\prime\prime}_{Bi}=\mathbf{T}_{3}^{-1}\hat{r}^{\prime\prime}_{Bi}= \begin{bmatrix}R\cos\hat{\beta}_{i}\cos\theta\\ R\sin\hat{\beta}_{i}\\ -R\cos\hat{\beta}_{i}\sin\theta\end{bmatrix} ,i=1\sim n. \tag{5}\]  
Figure 3: Each ball and the mass center of the rotor rotate around the spin axis in different parallel planes, respectively, at steady operation. Therefore, the equilibrium position vectors (Equations (3) and (5)) are inverted to the rotating reference frame \(<xyz>_{O}\), which is  
\[\hat{\mathbf{r}}_{Om}=\hat{\mathbf{r}}_{OC}+\mathbf{T}_{1}^{-1}\mathbf{T}_{2}^{-1}\hat{\mathbf{r}}_{m} ^{\prime\prime}=\] \[ \begin{bmatrix}\varepsilon\cos\psi\cos\hat{\gamma}_{2}+\hat{x}_{c} \\ \varepsilon\sin\psi\cos\hat{\gamma}_{1}+\varepsilon\cos\psi\sin\hat{\gamma}_{1} \sin\hat{\gamma}_{2}+\hat{y}_{c}\\ \varepsilon\sin\psi\sin\hat{\gamma}_{1}-\varepsilon\cos\psi\cos\hat{\gamma}_{1 }\sin\hat{\gamma}_{2}+\hat{z}_{c}\end{bmatrix} \text{ and } \tag{6}\]  
\[\hat{\mathbf{r}}_{OBi}=\hat{\mathbf{r}}_{OC}+\mathbf{T}_{1}^{-1}\mathbf{T}_{2}^{-1}\hat{\mathbf{r}}_ {Bi}^{\prime\prime}= \begin{bmatrix}\text{Rcos}\,\hat{\beta}_{i}\text{cos}( \hat{\gamma}_{2}+\theta)+\hat{x}_{c}\\ \text{Rsin}\,\hat{\beta}_{i}\text{cos}\,\hat{\gamma}_{1}+\text{Rcos}\,\hat{ \beta}_{i}\text{sin}\,\hat{\gamma}_{1}\text{sin}(\hat{\gamma}_{2}+\theta)+ \hat{y}_{c}\\ \text{Rsin}\,\hat{\beta}_{i}\text{sin}\,\hat{\gamma}_{1}-\text{Rcos}\,\hat{ \beta}_{i}\text{cos}\,\hat{\gamma}_{1}\text{sin}(\hat{\gamma}_{2}+\theta)+ \hat{z}_{c}\end{bmatrix} . \tag{7}\]  
The action forces of the steadily balanced system should include the centripetal force \(\mathbf{F}_{Om}\) and \(\mathbf{F}_{OBi}(i=1\sim n)\) acting on the system when the rotor mass center and \(n\) balls revolve around the spin axis, as well as the supporting spring force \(\mathbf{F}_{k}\) and the supporting damping force \(\mathbf{F}_{d}\). \(P_{Om}\) and \(F_{OBi}(i=1\sim n)\) are all on parallel planes perpendicular to the spin axis (ref. Figure 3). By 3D dynamics [25], the above forces in the \(<xyz>_{O}\) frame can be obtained from the following equations, respectively. \[\mathbf{F}_{Om}=m\mathbf{\Omega}_{O}\times(\mathbf{\Omega}_{O}\times\hat{\mathbf{r}}_{Om})=-m \omega^{2}