# Dynamic Modeling of Motorized Spindle System with Unbalanced Mass and Spindle Inclination
## 3 Results and Discussion
### Dynamic Characteristics of Motorized Spindle with Spindle Inclination Fault  
From Section 2.3, when the spindle inclination fault occurs, the general equation from the first node to the last node can be written as  
\[\mathbf{Z}_{n+1}=\mathbf{T}_{n}\mathbf{Z}_{n}+\mathbf{B}_{n}=\mathbf{T}_{n}( \mathbf{T}_{n-1}\mathbf{Z}_{n-1}+\mathbf{B}_{n-1})+\mathbf{B}_{n}=\cdots=\mathbf{ T}_{all}\mathbf{Z}_{1}+\sum_{k=1}^{n}\left(\prod_{j=k+1}^{n}\mathbf{T}_{j} \mathbf{B}_{k}\right)= \mathbf{T}_{all}\mathbf{Z}_{1}+\mathbf{B}_{all} \tag{24}\]  
where \(\mathbf{B}_{all}\) is the sum of the unbalance vector of all units from left to right. Both ends of the spindle are free ends. The equation of angular velocity is  
\[ \begin{bmatrix}y\\ \theta\end{bmatrix} _{1}=- \begin{bmatrix}a_{31}&a_{32}\\ a_{41}&a_{42}\end{bmatrix} ^{-1} \begin{bmatrix}b_{3}\\ b_{4}\end{bmatrix}  \tag{25}\]  
where \(b_{3}\) and \(b_{4}\) are the third and fourth elements of \(b_{all}\). All of them contain a parameter angular velocity of \(\omega\). \(y\), \(\theta\) indicate the section deflection and rotation angle of the initial end of the spindle. Assuming that the inclination angles are 0.1-0.5\({}^{\circ}\), the section de-flection and rotation angle of different initial ends can be obtained under different angular velocities. Within the whole specified rotating speed, when the spindle inclines, the deflection of the spindle initial end is shown in Figure 9a. Around the second-order critical rotating speed, the spindle inclination fault influence on the deflection of the spindle initial end when the spindle incline at different inclination angles is shown in the schematic diagram, Figure 9b. As can be seen from Figure 9a, the deflection of the initial spindle end increases gradually at first when the spindle inclination fault occurs, then the deflection suddenly changes direction and increases gradually along this direction, when approaching the first-order critical speed. At the first-order critical speed, the deflection decreases suddenly. Until the deflection equals zero, the deflection increases gradually along the opposite direction. The same characteristics are shown at the second-order and fourth-order critical speeds. However, at the fourth-order critical speed, the local maximum is smaller than it is at the second-order critical speed. Differently, when the rotational speed reaches the third-order critical speed, the deflection suddenly jumps to the maximum value in the upward direction. Then the direction suddenly changes, and decreases gradually. As shown in Figure 9b, the bigger the angle, the higher the deflection of the spindle initial end. However, the changing trend of the motorized spindle initial end deflection is the same at different angles. That is to say, in the case of spindle inclination fault at the small angle, the inclination angle only changes the size of the deflection, but does not influence the trend of the deflection changing with the rotating speed. As shown in Figure 10a, different from the unbalanced mass fault, the spindle inclination fault affects the whole motorized spindle around the first-order critical speed, while with the increase in the speed, the influence of the spindle inclination fault on the spindle initial end begins to be greater than that on the middle part of the spindle, around the second-order and third-order critical speed. In the third order, the influence on the spindle initial end reaches the maximum value. After the speed continues to increase, the influence of fault on the motorized spindle begins to increase. At the fourth-order critical speed, the deflection of the spindle end directly shows an up direction and the other end shows a down direction. Although the deflection at this time is small, if we do not pay attention to this situation, the spindle inclination fault may further develop into a membrane collision fault, which will cause serious damage to the motorized spindle.