## 5 Conclusions  
An analytical modeling method, based on dynamic of the SRB system which considers the bearing nonlinearity, is presented in this paper, so as to study the formative mechanism of dynamic rotational error and predict the rotational performance for the SRB system. Core conclusions of the study as a whole are as follows. The presented modeling method for dynamic rational error, based on the dynamic model of the SRB system, is feasible and valid to analyze the formative mechanism of the rotational error motion. Surface waviness, dynamic unbalance force and disturbance force are involved in the established model. The surface waviness hardly causes synchronous error motion. However, when the difference between the order and the number of rolling elements is one, the SRB system would produce a very obvious asynchronous error motion. The effects of dynamic unbalance and disturbance force are verified by the comparison experiments. Dynamic unbalance force has little effect on the synchronous error; however, the TIR and asynchronous error increases with the increasing of rotational speed and unbalance mass. Both synchronous and asynchronous error motion values increase when the amplitude of disturbance force and its frequency increases. Besides, the frequency of the radial synchronous error is the same as that of the disturbance force. The merit of this modeling method is that it can provide designers and/or field engineers with an informative guideline to improve the dynamic rotational accuracy. **Study prospects:**  
Based on the modeling method in this paper, investigations about the effect of temperature and lubrication on the dynamic rotational error during the spindle operation will be emphasized in future studies. **Author Contributions:** Conceptualization, Y.T., H.Q., and G.H. ; Funding acquisition, G.J., L.H. and T.W. ; Investigation, G.H. and Y.C. ; Methodology, G.H. and Y.C. ; Project administration, Y.T., H.Q. ; Supervision, L.C. and L.H. ; Validation, L.C. and L.H. ; Writing--original draft, G.H. ; Writing--review and editing, G.H. and Y.C. All authors have read and agreed to the published version of the manuscript. **Funding:** The authors gratefully acknowledge the support of the National Science Foundation of China No.51705364, the fund of Tianjin Natural Science Foundation (18J[CZD]C10050, 20J[CQN]JC01060 and 18J[CQN]JC4900), the fund of Tianjin Municipal Education Commission (Grant No. JWK1601). **Conflicts of Interest:** The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results. **Nomenclature**  
\(\delta_{i}\): The displacement vector of the inner ring \(\phi_{lk}\): The position angle of the kth ball \(N_{b}\): Number of the rolling element of the bearing \(\omega_{0}\): The rotation angular speed of the inner ring (rad/s) \(\omega_{lk}\): The orbital angular speed of the rolling element (rad/s) \(\omega_{sk}\): The self-rotation angular speed of the rolling element (rad/s) \(\omega_{o}\): Initial contact angle of the bearing (\({}^{\circ}\))  
Figure 21: Spindle synchronous errors induced by different amplitude of disturbance force.