## 2 Dynamical Mode of AMB Rigid Rotor System During Accelerated Motion  
The structure of the AMB rigid rotor system proposed in this study is shown in Figure 1. The axisymmetric rotor is supported axially by AMBs. In order to describe the kinematic state of the AMB rotor system, the relevant planes and coordinate systems are also defined in Figure 1. It is assumed that the center planes of AMB-A and AMB-B are \(\Pi_{A}\) and \(\Pi_{B}\), respectively. The center of mass of the unbalanced rotor is \(C\). Due to symmetry, point \(C\) is located on the geometric centerline of the rotor. The distances from point \(O\) to the planes \(\Pi_{A}\) and \(\Pi_{B}\) are \(l_{\rm a}\) and \(l_{\rm b}\), respectively, and the distance between planes \(\Pi_{A}\) and \(\Pi_{B}\) is \(l\). A fixed coordinate system \(Oxyz\) is established, where the \(z\) axis is the axis of rotation, and a right-handed system is formed between \(x\), \(y\), and \(z\). Then, the motion state of the rotor can be described by \((x,y,\theta_{x},\theta_{y})\).  
For an unbalanced rotor, the unbalance of the rotor is equivalently generated by an additional unbalanced mass that is not at the center of mass \(C\) of the balanced rotor. Assuming that the unbalanced mass of the additional rotor is at the point position, the projection in the plane of the geometric center of the balanced rotor is \(\varepsilon_{z}\), the length of the projection on the \(O_{Z}\) axis is \(\varepsilon_{z}\), and the eccentricity distance from point to point is \(\varepsilon\). In order to describe the motion of the rotor geometric center, the \(Cx_{r}y_{r}\) rotating coordinate system is established with \(C\) as the origin, when the \(Cx_{r}\) axis of the rotating coordinate system is parallel to the \(Ox\) axis, where the transient rotation angle of \(G\) is \(\phi\). As shown in Figure 2, when the rotor is running steadily, \(\phi=\omega\) and \(\phi=\omega t\).  
Using the theory of rotor dynamics and considering the effects of rotor unbalance and rotor acceleration, the matrix equation of the AMB rigid rotor system during the accelerated operation is as follows:  
\[\mathbf{M}\ddot{q}+\mathbf{G}\dot{q}=\mathbf{L}_{f}\mathbf{F}_{AMB}+\mathbf{F}_{u}, \tag{1}\]  
where \(\mathbf{M}\) and \(\mathbf{G}\) are the generalized mass matrix and gyro matrix of the rotor system, respectively, \(\mathbf{L}_{f}\) and \(\mathbf{F}_{u}\) are the force arm coefficient matrix of the rotor system and the generalized imbalance vector applied to the system, respectively, and \(\mathbf{F}_{AMB}\) is the vector of electromagnetic forces generated by the radial AMBs at the A/B end in the x and y directions.  
During acceleration, the rapid changes in speed and force can increase systemic instability, particularly in the presence of rotor imbalance-induced vibrations. These vibrations may be further amplified during acceleration, potentially leading to systemic instability or even failure. Based on these facts, a dynamic model of the AMB rigid rotor system for accelerating conditions is established. The aim of this study is to address the suppression of imbalance-induced vibrations during acceleration through theoretical analysis and control strategy design, providing theoretical support for the stable operation of the system.