# Dynamic Modeling of Motorized Spindle System with Unbalanced Mass and Spindle Inclination
## 1 Introduction  
Aerospace products are mostly thin-walled parts with complex shapes, needed to suit the specific working environment and performance demands, and are typically processed using high-speed milling. In the field of aerospace processing, there are strict requirements for the precision of machine tool processing, in which the precision of the motorized spindle plays a vital role in the whole processing. The common faults of motorized spindles include spindle fault [1; 2; 3], bearing fault [4; 5], lubrication system fault [6], stator and rotor fault [7; 8], spindle housing fault [9], and cooling system fault [10]; the bearing fault is the most common and analyzed fault. However, compared with other faults, the spindle fault is more difficult to find and more likely to affect the progress of this machining. For aerospace equipment, a small error may lead to the failure of the whole mechanism, so the faults of motorized spindles cannot be ignored in the processing of aerospace equipment. In recent years, the fault dynamics of motorized spindles have become an important issue. [PERSON] et al. (2018) [11] established a dynamic model of a dual-rotor system to solve its unbalanced misalignment coupling faults. [PERSON] et al. (2016) [12] obtained the unbalanced magnetic pull by numerical method and discussed the effects of initial static eccentricity on displacement spectra. [PERSON] et al. (2019) [13] established a dynamic model of the permanent magnet synchronous motors' rotor-bearing system which considers unbalanced magnetic pull and the nonlinear restoring force. They concluded that unbalanced magnetic pull causes a negative stiffness effect on the natural frequencies. [PERSON] (2017) [14] established an analytical rotor dynamic model which considers mass eccentricity, bent rotor deflection,and magnetic eccentricity. [PERSON] et al. (2018) [15] analyzed and modeled the longitudinal dynamics of the motorized spindle under the excitation of torque ripple. Based on the Jeffcott rotor model, [PERSON] et al. (2016) [16] analyzed the stiffness characteristics of the rotor system of the permanent magnet synchronous motors. [PERSON] (2017) [17] gave the reason of unbalanced magnetic pull. Based on the law of energy conservation, [PERSON] and [PERSON] (2001) [18] presented a power flow model of high-speed motorized spindles. [PERSON] et al. (2004) [19; 20] established a comprehensive integrated thermo-dynamic model for high-speed spindles. [PERSON] et al. (2017) [21] built the dynamic model of the motorized spindle system. They used this dynamic model to show that the vibrations are proportional to the mass of the unbalance. [PERSON] et al. (2016) [22] presented an improved dynamic model for unbalanced high-speed motorized spindles, and obtained the sensitivities of the motorized spindles. Although a huge number of scholars have contributed to this research, the fault dynamics of motorized spindles still have some unsolved problems, such as the vibration of motorized spindles caused by the unbalanced mass and inclined axis. In practical engineering, the two kinds of faults will occur due to the uneven distribution of rotor material density during the manufacturing process or errors caused by the installation of a motorized spindle. Although the faults in these two cases can be quickly solved after discovery, they will still affect the normal operation of the motorized spindle before discovery. Moreover, it is also very important to judge whether these two faults occur. If they are found as early as possible, their influence on the normal operation of the whole machinery can be avoided. Therefore, it is necessary to analyze the influence of these two kinds of faults on the vibration of the spindle, so as to help the operator find out whether the faults happen as soon as possible. In this work, a model of motorized spindles based on the whole transfer matrix method (WTMM) is established and a model of a spindle with fault caused by unbalanced mass or spindle inclination is given in Section 2.