# A Corrected Adaptive Balancing Approach of Motorized Spindle Considering Air Gap Unbalance
## 5 Conclusions  
The original intention of this work was to improve the existing rotor adaptive balancing method by considering the effect of air gap unbalance on the balancing scheme, which is an interesting topic for motorized spindles. In order to achieve this goal, we made the following assumptions and simplifications: (i) a single-plane balancing strategy was adopted to mainly reduce the tool unbalance in the machining process, which is an approximation to the tool-holder-shaft combined rotor; (ii) the influence coefficient method (ICM) and the frequency-domain analysis of vibration signals were adopted, which are based on the assumption that the spindle system runs smoothly; however, in practice, the non-stationary factors affect the balancing effect; (iii) in the calculation of unbalanced magnetic pull (UMP), the 1\(\times\) rotation frequency component with small order and large amplitude was retained, and the correction coefficient was approximately treated. Since the electromagnetic vibration of a motorized spindle is complex, and the rotor balancing cannot be reduced to 0 in practice, the balancing strategy can be accepted as long as its accuracy meets the requirements of the user. Therefore, the above approximations and simplifications are feasible, and the tested results prove this point. This idea is an exploration, and further research will be carried out in the future. The valuable conclusions of this paper are as follows:  
(1) The mechanical and electromagnetic effects caused by rotor unbalance for a motorized spindle were introduced. Mechanical unbalance causes an inertia force, while electromagnetic unbalance induces an unbalanced magnetic pull (UMP). The UMP contains many components, whereas this work mainly focused on the 1\(\times\) rotation frequency component affecting the frequency-domain adaptive balancing strategy. The derivation of UMP for a three-phase asynchronous squirrel-cage spindle motor was given, and, for a specific motorized spindle, the UMP and ratio of the air gap unbalance-induced 1\(\times\) rotation frequency force component and the total 1\(\times\) rotation frequency force were determined.  
(2) Based on the study on UMP, a newly defined corrected coefficient for the improvement of the active balancing effect was defined. A frequency-domain adaptive balancing algorithm based on the corrected estimation scheme of the influence coefficient was built for motorized spindles.  
(3) In order to validate the proposed balancing approach, a vibration test bed of a motorized spindle with a single electromagnetic ring balancer was developed. In the tests, two typical unbalance cases were designed, where displacement and acceleration sensors were used and the balancing tests were done at four different speeds. Finally, the balancing data under no balancing, conventional adaptive balancing, and the corrected adaptive balancing schemes were obtained. According to the analysis results, the proposed adaptive balancing approach was surely proven effective for the reduction of residual vibration after balancing for motorized spindles.