# Dynamic Modeling of Motorized Spindle System with Unbalanced Mass and Spindle Inclination
## 3 Results and Discussion
### Dynamic Characteristics of Motorized Spindle with Unbalanced Mass Fault  
From Section 2.2, when the unbalanced mass fault happens, the general equation from the first node to the last node can be written as  
\[\mathbf{Z}_{n+1}=\mathbf{T}_{n}\mathbf{Z}_{n}+\mathbf{H}_{n}=\mathbf{T}_{n}( \mathbf{T}_{n-1}\mathbf{Z}_{n-1}+\mathbf{H}_{n-1})+\mathbf{H}_{n}=\cdots= \mathbf{T}_{all}\mathbf{Z}_{1}+\sum_{k=1}^{n}(\prod_{j=k+1}^{n}\mathbf{T}_{j} \mathbf{H}_{k})= \mathbf{T}_{all}\mathbf{Z}_{1}+\mathbf{H}_{all} \tag{22}\]  
where \(\mathbf{H}_{all}\) is the sum of the unbalanced vector of all units from left to right. It is the same as the condition of the motorized spindle without fault; both ends of the spindle are free ends. The equation of angular velocity can be obtained:  
\[ \begin{bmatrix}y\\ \theta\end{bmatrix} _{1}=- \begin{bmatrix}a_{31}&a_{32}\\ a_{41}&a_{42}\end{bmatrix} ^{-1} \begin{bmatrix}h_{3}\\ h_{4}\end{bmatrix}  \tag{23}\]  
where \(a_{31}\), \(a_{32}\), \(a_{41}\) and \(a_{42}\) are the elements in the first or second columns of the third row and the first or second columns of the fourth row of the total transfer matrix, respectively. \(h_{3}\) and \(h_{4}\) are the third and fourth elements of \(h_{all}\). All of them contain the parameter angular velocity of \(\omega\). \(y\) and \(\theta\) indicate the section deflection and rotation angle of the initial end of the spindle, respectively. Assuming that the amount of unbalance is 1-5 g\(\cdot\)mm, the section deflection and rotation angle of different initial ends can be obtained under different angular velocities. Within the whole specified rotating speed, when the whole spindle is unbalanced, the deflection of the spindle initial end is shown in Figure 7a. Around the second-order critical rotating speed, the schematic diagram of the unbalanced mass fault influence on the deflection of the spindle intimal end when the amount of unbalance exists in different units is shown in Figure 7b. As can be seen from Figure 7a, the deflection of the spindle initial end increases gradually at first when the unbalanced mass fault occurs, and reaches a local maximum when it reaches the first-order critical speed. At this time, the deflection direction of the shaft end suddenly changes. With the further increase in rotational speed, the deflection of the spindle initial end continues to increase along this direction and the direction suddenly changes again when the second-order critical rotational speed is reached. The same characteristics are shown at the third-order and fourth-order critical speeds. Moreover, the maximum value is reached at the third-order critical speed. Figure 6: The first fourth-order mode shapes of the motorized spindle: (**a**) first-order mode shape; (**b**) second-order mode shape; (**c**) third-order mode shape; (**d**) fourth-order mode shape. It can be seen from Figure 7b that the direction and magnitude of the spindle initial end deflection are different in different units when the unbalanced mass faults occur. The spindle initial end deflection is upward when a fault occurs in the 6 th, 9 th, 12 th, or 15 th unit. When the speed reaches the second-order critical speed, the direction suddenly changes, and the deflection value of the 12 th unit is the largest, which may be more sensitive to the unbalanced fault. On the other hand, the spindle initial end deflection of the 18 th, 21 st, and 24 th units are downward, and the direction suddenly changes when it reaches the second-order critical speed too. The vibration mode of the whole motorized spindle can be further solved when the unbalanced mass fault occurs, as shown in Figure 8a. Around the first-order critical speed, the influence of the unbalanced mass fault on the middle part of the spindle is slightly greater than that on the spindle ends. While with the increase in speed, the influence of the unbalanced mass fault on the spindle ends begins to be greater than that on the middle part of the spindle around the second-order and third-order critical speeds.