## 3 3D Dynamics Deduction  
### Coordinate System  
In order to understand and analyze the spatial position, attitude, and motion mechanism of the rotor system equipped with skew-mounted ABB, the coordinate systems of the ABB mechanism, eccentric rotor, and support device are constructed as shown in Figure 2. The rectangular coordinate \(<\mathbf{X}\mathbf{Y}\mathbf{Z}>_{O}\) is a space-fixed reference frame of the planar rotor, the origin \(O\) is defined at the centroid position of the eccentric rotor when the spindle speed is zero, the system is stationary, and the support springs are not deformed. The \(\mathbf{Y}\)-axis is the deflection axis where mounting deviation occurs (refer to Figure 1). Moreover, the rotating reference frame \(<\mathbf{xyz}>_{O}\) is defined with the origin located at point \(O\) and \(\mathbf{z}=\mathbf{Z}\) axis of the \(<\mathbf{X}\mathbf{Y}\mathbf{Z}>_{O}\) frame. The \(<\mathbf{xyz}>_{O}\) coordinate system rotates around the spindle \(\mathbf{Z}\)-axis of the system with a constant angular velocity \(\omega\). The reference frame can be transformed to the rotating frame by a rotation matrix \(\mathbf{\Upsilon}_{\omega}\). Figure 1: The support configurations of the rotor system equipped with skew-mounted ABB. When the support spring is deformed, the position of the offset center \(C\) is set to the origin of the \(<xyz>_{C}\) coordinate system, which axes are always parallel to and in the same direction as the axes of \(<xyz>_{O}\), except for the different position of the origin. Then rotating the pitch angle \(\gamma_{1}\) around the \(x\)-axis, which is in the \(<xyz>_{C}\) coordinate system, and the rotating coordinate system \(<x^{\prime}y^{\prime}z^{\prime}>_{C}\), where \(x^{\prime}=x\) can be obtained. Next, rotate the yaw angle \(\gamma_{2}\) around the \(y^{\prime}\)-axis of \(<x^{\prime}y^{\prime}z^{\prime}>_{C}\) coordinate system to obtain the rotating coordinate system \(<x^{\prime}y^{\prime}z^{\prime\prime}>_{C}\), where \(y^{\prime\prime}=y^{\prime}\) and the \(z^{\prime\prime}\)-axis passes through the rotor centroid and is perpendicular to the plane of the rotor (\(x^{\prime\prime}y^{\prime\prime}\) plane). Converting \(<xyz>_{C}\) to \(<x^{\prime}y^{\prime}z^{\prime}>_{C}\) and \(<x^{\prime\prime}y^{\prime\prime}z^{\prime\prime}>_{C}\) by rotation matrixes \(\mathbf{T}_{1}\) and \(\mathbf{T}_{2}\) in sequence. The mass center \(G\) of the eccentric rotor with mass \(m\) is located on the plane of rotor and the eccentricity between \(G\) and its centroid \(C\) is \(\varepsilon\). The angle between \(\overline{CG}\) and the \(x^{\prime\prime}\)-axis is \(\psi\). The body-fixed coordinate system \(<x^{\prime\prime}y^{\prime\prime}z^{\prime\prime}>_{C}\) follows the rotor system in 3D space such that the \(y^{\prime\prime}\)-axis is also the deflection axis of skew-mounted ABB. Rotating the deflection angle \(\theta\) around the \(y^{\prime\prime}\)-axis of the \(<x^{\prime\prime}y^{\prime\prime}z^{\prime\prime}>_{C}\) to obtain the coordinate system \(<x^{\prime\prime}y^{\prime\prime}z^{\prime\prime}>_{C}\) of the orbital plane, where \(y^{\prime\prime\prime}=y^{\prime\prime}=y^{\prime}\). It can convert \(<x^{\prime\prime}y^{\prime\prime}z^{\prime\prime}>_{C}\) to \(<x^{\prime\prime}y^{\prime\prime}z^{\prime\prime}>_{C}\) by rotation matrix \(\mathbf{T}_{3}\). The radius of the circular orbit of the ABB is \(R\), there are \(n\) balls moving in the orbit, and the mass of each ball is \(m_{b}\). The angular position of the \(i\)-th ball from the \(x^{\prime\prime}\)-axis is denoted \(\beta_{i}(i=1\sim n)\). The rotation matrices between the coordinates as  
\[\mathbf{T}_{\omega}= \begin{bmatrix}\cos\omega t&\sin\omega t&0\\ -\sin\omega t&\cos\omega t&0\\ 0&0&1\end{bmatrix} ,\,\mathbf{T}_{1}= \begin{bmatrix}1&0&0\\ 0&\cos\gamma_{1}&\sin\gamma_{1}\\ 0&-\sin\gamma_{1}&\cos\gamma_{1}\\ \end{bmatrix} , \tag{1}\] \[\mathbf{T}_{2}= \begin{bmatrix}\cos\gamma_{2}&0&-\sin\gamma_{2}\\ 0&1&0\\ \sin\gamma_{2}&0&\cos\gamma_{2}\end{bmatrix} \text{and }\mathbf{T}_{3}=