## 1 Introduction  
The motorized spindle unit is crucial to numerical control (NC) machine tools and the key component to guarantee the precision of machines. Its performance directly influences the technology used and determines the development of the machine tool [1]. The motorized spindle unit is indispensable to high performance computer numerical control (CNC) machine tools. Several studies have focused on improving the spindle [2; 3; 4; 5]. The motorized spindle incurs mechanical and electromagnetic losses because of its high speed and non-sinusoidal power supply. Most of these losses are transformed into heat and transferred to the surrounding air, cooling fluid, and machine parts. This results in uneven heat deformation of the parts of the machine, which directly affects the accuracy of the spindle and the bearing preload. As many as 75% of the overall geometrical errors in workpieces manufactured by machines are considered to be induced by the effects of temperature [6]. The temperature of the motorized spindle is an important index to evaluate its performance at high speeds [7]. High-power and high-speed motorized spindles feature a larger rise in temperature than surroundings, which is therefore important to control. There are two methods for controlling the rise in temperature of the motorized spindle. One involves actively controlling the cooling and lubrication systems to avoid high temperatures, and the other consists of optimizing the design of the structure of the motorized spindle [8; 9]. These two methods are different, but both require the development of a thermal model for the motorized spindle to predict the temperature field. The Ellesmere Company has been developing CAD software for motor design software for the last 10 years, which includes modules for thermal error calculation [10]. However, it is challenging to effectively calculate thermal error in the motorized spindle as it is fundamentally different from an ordinary motor. The motorized spindle system contains a few subsystems, such as water cooling, lubrication, and the inverter. Thus, its heat generation and transfer processes are more complex. Finite element analysis is often used in a simplified thermal design of the motorized spindle unit [11; 12]. Such thermal boundary conditions as the processing conditions of the heating and cooling environments are loaded into the model to simulate the temperature distribution of the motorized spindle. [PERSON] et al. [13; 14; 15; 16; 17] researched finite element thermal modeling of NC machine tools and created a hybrid model of a high-speed machining center spindle box using finite element modeling. This method was used to model the components of the symmetrical axis. A comparison of the results between the hybrid prediction model and the finite difference method shows that the prediction error in the hybrid model was less than 2 \({}^{\circ}\)C. [PERSON] et al. [18] proposed a method that allows the permanent adaptation of the heat transmission coefficient to better apply convective heat transmission to time-variable thermal simulations, especially for mechanical finite element models. [PERSON] [19] presented a thermo-mechanical model of spindles that predicts temperature distribution and thermal growth by considering transient changes in the bearing stiffness and contact loads. [PERSON] et al. [20] built a model for the thermal characteristics of the high-speed motorized spindle using finite element analysis and calculated the static/transient temperature field as well as the thermal coupled field. [PERSON] et al. [21] proposed a bearing thermo-mechanical dynamic model that considers preload methods and thermal responses, and analyzed frictional losses and the support stiffness of the bearings. The electromagnetic loss of the built-in motor with input power was investigated using electromagnetism.