## 2 Model and Formulation  
Compared to the clamped boundary, the spline interface allows the disc to move laterally along the shaft and tilt around the disc diameter [2]. Therefore, the spinning plane of the spline-guided disc can be flexible. Figure 2 shows the tilting behavior of a single spinning disc driven by a splined hub. Normally, when there is no continuous external excitation, the disc tilts shortly and aligns vertically. However, a few reasons could cause disc tilting: the disc has an initial angle when the machine starts to rotate, or the spinning disc is subjected to irregular external forces, such as mechanical contact between the parts or nonlinear oil flow in the wet clutch [28].  
Figure 3 shows a schematic of a spinning disc with an inner radius a and outer radius b splined to an arbor at an angular velocity of \(\omega\). Now, the coordinates include the following: (1) the local coordinate \((o-zr\theta)\) is set at the middle surface of the disc and rotates along with the disc, and its \(z\)-axis is always perpendicular to the middle plane of the disc; (2) \((O-XYZ)\) is a rotating intermediate frame that represents the rotation field of the spline shaft, and its \(Z\)-axis coincides with the rotating center of the shaft; and the (3) space-fixed coordinate system \((X^{f}Y^{f}Z^{f})\). One advantage is that, instead of being expressed by two angles between the \(X\)-axis and \(Y\)-axis, the tilting angle can be expressed by \(\theta_{Z}\), which is the angle between the two vertical axes of coordinate system (1) and coordinate system (2). From the perspective of the local coordinate, the disc exhibits the sole motion of tilting around the \(\theta=0(\pi)\) axis due to external forces and moments.  
Figure 2: Rigid-body tilting of the spinning disc.  
As the disc rotates, the tilting angle \(\theta_{Z}\) changes due to external forces and moments. Then, the motion in the fixed coordinate system can be determined by coupling the rotation and local tilting. By setting the disc at the origin of the space-fixed coordinate system, a random point \(p(r\cos\theta,r\sin\theta,z)\) on the disc frame can be transferred into the space-fixed coordinate system by introducing Euler rotation matrices:  
\[\left[\begin{array}{c}X^{f}\\ Y^{f}\\ Z^{f}\end{array}\right]=R_{r}R_{t}\left[\begin{array}{c}r\cos\theta\\ r\sin\theta\\ z\end{array}\right] \tag{1}\]  
where \(R_{t}\) and \(R_{r}\) are the rotation matrices due to tilting and shaft spinning, respectively. They are written as:  
\[R_{r}=\left[\begin{array}{ccc}\cos\omega t&-\sin\omega t&0\\ \sin\omega t&\cos\omega t&0\\ 0&0&1\end{array}\right],R_{t}=\left[\begin{array}{ccc}1&0&0\\ 0&\cos\theta_{Z}&-\sin\theta_{Z}\\ 0&\sin\theta_{Z}&\cos\theta_{Z}\end{array}\right] \tag{2}\]  
As the disc rotates, the angle that rotates around the shaft is \(\omega t\), and the space-fixed coordinates of any point during the rotation can be expressed as:  
\[\left[\begin{array}{c}X^{f}\\ Y^{f}\\ Z^{f}\end{array}\right]=\left[\begin{array}{ccc}\cos\omega t&-\sin\omega t \cos\theta_{Z}&\sin\omega t\sin\theta_{Z}\\ \sin\omega t&\cos\omega t\cos\theta_{Z}&-\cos\omega t\sin\theta_{Z}\\ 0&\sin\theta_{Z}&\cos\theta_{Z}\end{array}\right]\left[\begin{array}{c}r\cos \theta\\ r\sin\theta\\ z\end{array}\right] \tag{3}\]  
Knowing the space-fixed coordinates of the disc is necessary for model development and is important for calculating the impact force and axial displacement of the disc.