## 1 Introduction
[34] presented a theoretical model for spherical roller bearings by considering the angular and centroidal misalignments of bearing rings. [PERSON] et al. [35] proposed a five degree-of freedom quasi-static model for double-row self-aligning ball bearings and investigated the effect of angular misalignment on the dynamic characteristic of the bearing. [PERSON] and [PERSON] [36; 37; 38] completed a series works on the tapered roller bearings subjected to angular misalignment. Their studies suggested that angular misalignment greatly influences the fatigue life, contact state, running torque, and stiffness of tapered roller bearings. Up to now, only a few articles on double-row or combined ball bearings have been reported. [PERSON] et al. [39] developed a unified analytical model for various types of double-row bearings, such as tapered, spherical, cylindrical roller, and angular contact ball bearings. Based on the efforts of [PERSON] and [PERSON] [40] and [PERSON] and [PERSON] [41], [PERSON] [42] proposed a comprehensive mathematical model of double row angular contact ball bearings for back-to-back, face-to-face, and tandem configurations. Double row bearings were considered as an integrated unit, and the detailed stiffness expression of DACBB was also formulated. [PERSON] et al. [43] developed a new iterative algorithm to evaluate the preload and stiffness coefficients of combined ACBBs. [PERSON] et al. [44] estimated the performance of the spindle supported by the ACBB with different configurations. [PERSON] and [PERSON] [45; 46] established a dynamic model for double row ball bearings with different defects. [PERSON] et al. [47] established a dynamic stiffness model of DACBB with back-to-back configuration according to the dynamic analysis of the bearing. On this basis, the influences of bearing structural and load condition on the radial, axial, and angular stiffness coefficients of DACBB were analyzed. However, the dynamic analysis of bearing was not clearly given. [PERSON] et al. [48] established a quasi-static model of DACBB to study the effects of speed, external load, and preload on the characteristics of DACBB. Based on [PERSON] theory, [PERSON] and [PERSON] [49] proposed a five degree-of-freedom quasi-static model for preloaded DACBB. The stiffness of DACCB with three configurations was simulated under different rotating speeds and unloaded contact angles. Subsequently, they further investigated the dynamic performance of the bearing-rotor system supported by the DACBB with tandem configuration [50]. From the above literature survey, it can be found that, unlike single-row ACBB that has been extensively studied, only a few studies on DACBB have been published. Therefore, the theoretical research on the DACBB should be further enriched. Furthermore, most previous researches related to ring misalignment are mainly aimed at roller bearing, self-aligning bearing and single row angular contact ball bearing, very few studies on the ring misalignment phenomenon in the DACBB have been reported. [PERSON] and [PERSON] [51] carried out some meaningful study and proposed a quasi-static mechanical model for DACBB according to [PERSON]'s theory [7]. The stiffness and fatigue life of DACBB under angular misalignment were studied. Generally, owing to main shaft deflection, mounting error, and geometric inaccuracy, DACBB commonly runs with angular misalignment [35; 36; 37; 52]. In order to control the magnitude of angular misalignment of inner ring, it is necessary to improve the machining and assembly accuracy of bearing, shaft, and bearing seat. This would lead to a sharp increase of manufacturing costs. In addition, the ignorance of misalignment phenomenon in the study of DACBB may lead to large deviation between actual performance and simulated results of DACBB. Therefore, it is necessary to study the DACBB under misalignment operating condition in theory so as to meet the urgent requirements of engineering.