In this paper, a thermo-mechanical coupling analysis model of the spindle-bearing system based on Hertz's contact theory and a point contact non-Newtonian thermal elastohydrodynamic lubrication (EHL) theory are developed. In this model, the effect of preload, centrifugal force, the gyroscopic moment, and the lubrication state of the spindle-bearing system are considered. According to the heat transfer theory, the mathematical model for the temperature field of the spindle system is developed

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... the effect of the spindle cooling system on the spindle temperature distribution is analyzed. The theoretical simulations and the experimental results indicate that the bearing preload has great effect on the frictional heat generation; the cooling fluid has great effect on the heat balance of the spindle system. If a steady-state heat balance between the friction heat generation and the cooling system cannot be reached, thermally-induced preload will lead to a further increase of the frictional heat generation and then cause the thermal failure of the spindle. Entropy 2016, 18, 271 2 of 25 by an interfacial dissipative process. They formulated the entropy generation theorem relating any form of degradation to the irreversible dissipative processes germane to the degradation mechanism. Entropy generation as related to tribology and thermal effects has been studied [13] [14] [15] . For certain combinations of thermal variables, the total entropy generated is minimized, so the model can be optimized. Bossmanns and Tu [16] developed a heat transfer model among the spindle units based on the finite difference method. Holkup et al. [17] developed a thermo-mechanical coupling model with axisymmetric structure of the high-speed spindle by using the finite element method to predict the temperature distribution of the spindle, the changes of the bearing stiffness, and the contact load with time. However, the influence of the spindle cooling system on the heat balance of the spindle was not considered. Through theoretical analysis and experimental study, Mizuta et al. [18] developed an accurate heat transfer model of the spindle bearing. In their model, the authors considered the changes of the thermal resistance with the rotated speed and the axial force, but the model was only suitable for the selection of the type of spindle bearing at the research and development phase and has its limitations when applied to the analysis of the whole spindle system. The above studies calculated frictional moment and heat generation through empirical formulas; without considering the effect of the lubricant on the frictional stress and heat transfer, there is large error between the calculated results and the actual frictional heat generation. Frictional heat generation can be accurately calculated by a detailed analysis of the frictional shear stress, contact deflection, and rotation speed of the rolling elements and raceway with considering EHL. Previous studies did not consider the cooling system, or analyzed a cooling system with only one heat-transfer coefficient [19] . In fact, the parameters of the cooling system have a great effect on the temperature distribution of the spindle-bearing system, so it is very useful to analyze the cooling system in detail. In this paper, first, contact deflection and load distribution between ball and inner (outer) raceway are calculated and analyzed under the effect of external load, preload and inertial loads. Second, based on the analysis and calculations of lubricating oil film, the frictional heat generation of the ball and raceways are calculated; moreover, a heat transfer model of the spindle system is developed and the temperature distribution of the spindle system is calculated. Finally, after the experimental verification of the temperature distribution model, the influence of different parameters on the thermal characteristic of the spindle system is further analyzed, to verify whether the proposed calculation method can be widely applied to research and development of the precise spindle under different operating conditions. Thermo-Mechanical Coupling Model of the Spindle-Bearing System This paper used the precise spindle of a new type of high-power horizontal machining center as the prototype to study the thermal characteristic of a spindle-bearing system, as shown in Figure 1 . The spindle bearing (71928CD/P4ATBTA) (SKF, Goteborg, Sweden) is a triple angular contact ball bearings; the suffix TBT represents tandem duplex and back-to-back combination mode. The supporting bearing (71924CD/P4ADBA) (SKF, Goteborg, Sweden) is a back-to-back combination bearing; the suffix DB represents back-to-back combination mode. The bearing was preloaded using the axial sleeve, the adjustable collar, and the lock nut. The spindle cooling liquid was ISO VG32, whose temperature and flow were controlled by the oil cooler. The cooling jacket had a rectangular cross section. The spindle (rotating) and the cooling jacket were made of 15CrMo and 45 steel, respectively. The spindle system was equipped with a complete detection system used for real-time monitoring of the transient temperature field and the thermal displacement, and then for predicting the thermal failure.