# Dynamic Modeling of Motorized Spindle System with Unbalanced Mass and Spindle Inclination
## 4 Conclusions  
In this work, focusing on the unbalanced mass fault and inclined axis fault, we improved the traditional WTMM. Then we used it to analyze the mode shape of motorized spindle under the fault at different rotation speeds. We also studied which part of the motorized spindle is more sensitive to the unbalanced mass fault, and the influence of different tilt angles on vibration of motorized spindle. The findings in this work will have different applications in many different types of machinery in which there are rotating cylinders, shafts, or spindles. The main conclusions are as follows:  
1. If the unbalanced mass fault occurs, when the spindle speed is close to the first-order or second-order critical speed, the deflection at the initial end of the spindle is small; when the spindle speed is close to the fourth-order critical speed, the deflection is large. The opposite is true when the spindle inclination fault occurs. It can be seen that the motorized spindle is more sensitive at high rotating speeds when the unbalanced mass fault occurs, and it is more sensitive at low rotating speed when the spindle inclination fault occurs.
2. CNC machine tools are vibration systems with high stiffness, and the support system is flexible or nearly flexible. The rotor may exhibit rigid behavior, manifested as a smaller rotor vibration and larger bearing seat vibration. So, regardless of whether the spindle has an unbalanced mass fault or spindle inclination fault, when the spindle speed reaches the third-order critical speed, the deflection at the initial end of the spindle is the largest.
3. When the spindle speed approaches the first-order critical speed, an unbalanced mass fault will make the middle part of the spindle bend upward, while the spindle inclination fault will make the middle part of the spindle bend downward.
4. When the unbalanced mass fault occurs, the deflection will be increased first, then suddenly decrease until the direction change, and the deflection will continue to increase along this direction. When the spindle inclination fault occurs, the deflection will be increased first in the opposite direction, then gradually decreases after reaching the maximum value, and then gradually continues to increase along the original direction when it returns to the original direction.  
Conceptualization, S.L. and Y.W.; methodology, D.G. and Y.X.; software, Y.X.; validation, J.S.; formal analysis, D.G. and Y.X.; investigation, D.G. and Y.X.; resources, Y.W.; data curation, Y.X.; writing--original draft preparation, D.G. and Y.X.; writing--review and editing, S.L. and Y.W.; supervision, S.L. and Y.W; project administration, S.L.; funding acquisition, S.L. All authors have read and agreed to the published version of the manuscript.  
Figure 10: Mode shape in the case of spindle inclination fault: (**a**) mode shape in working speed range; (**b**) mode shape around the first-order critical speed.  
**Funding:** This research was funded by the National Natural Science Foundation of China (No. 11802168, No. 52075310).  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** The data presented in this study are available upon request from the corresponding author.  
**Conflicts of Interest:** The authors declare no conflict of interest.