Skidding and cage whirling of angular contact ball bearings: Kinematic-hertzian contact-thermal-elasto-hydrodynamic model with thermal expansion and experimental validation
Graphical abstract
Introduction
Rolling ball bearings are widely used in rotating machinery and equipment and play a key role in engineering since almost 40% of mechanical installation failures initiate from bearings. The kinematic and dynamic behaviours of the components inside a bearing play a crucial role in the formation of various bearing faults and in the assessment of equipment reliability and stability [1], [2], [3], [4]. Commonly, low-loaded and high-speed rolling element bearings tend to experience smearing wear and insufficient lubrication, which causes premature failure. One of the prominent reasons for such wear is rolling element skidding or overskidding behaviour. Skidding indicates that macrosliding occurs between the raceways and rolling elements; overskidding means that the rolling element orbit rotating speed exceeds the theoretical value obtained under a pure rolling state, which can also detect slipping [5], [6]. Both rolling element skidding and overskidding behaviours generate considerable heat that can damage the lubrication condition and cause bearing component thermal expansions [7].
Understanding bearing work mechanisms by a comprehensive model can be beneficial for bearing design and application. An accurate model for the prediction and simulation of skidding and overskidding behaviour can also significantly prevent equipment accidents and ensure bearing operating quality. Many related works have been developed in recent years [8]. For rolling element bearings, the common method initiates from popular quasistatic models since the bearing stiffness matrix can be obtained [9], [10], [11], [12], [13]. The raceway-control theory, which sets a rolling element without spinning on one raceway, is usually applied to the quasi-static model, which has been developed by various kinematic assumptions [14], [15]. The quasi-dynamic model improved the previous model by considering the time-varying motion state of bearing components, and differential equations could be integrated [16], [17]. Cage slipping [18], rolling element skidding and overskidding [6], and ball four-degree-of-freedom (DOF) motion [19] have been discussed. The lubricant temperature-viscous effect [20], [21], [22] in oil supply rolling bearings has clarified bearing dynamic behaviour, but this has not been fully elucidated. Gupta developed a dynamic model for rolling element bearings in which each element has 6 DOFs [23], [24]. Since the main interactions between ball raceways and cage balls have been considered, localized defective ball bearing dynamic behaviour has been developed [25], [26], [27]. Takabi [28] and Liu [29] discussed the effects of bearing heat generation and lubricant temperature based on a dynamic model. The results showed that the lubricant temperature and properties can tremendously influence the bearing skidding conditions. A finite element model (FEM) for defective bearing vibration prediction was proposed and shown to be effective [30]. A simple friction factor was adopted to consider the lubrication effect since it is difficult to assess lubricants in FEM analysis.
Although various models have been proposed for rolling element bearings to simulate the running state of a bearing in detail, there is very limited research on simulating the thermal expansion of bearing components when heat is generated due to friction. Since the scale of thermal expansion of metal is one order of magnitude higher than that of force deformation, this research is necessary for cases where considerable heat is produced due to rolling element skidding or overskidding. Tarawneh [31] investigated bearing assembly temperature scenarios and heat generation and the bearing house surface temperature by applying the FEM. Ma [32] and Ai [33] discussed the reasons for the temperature rise of grease-lubricated roller bearings and found that a higher grease filling ratio and rotating speed led to a high temperature. Since a journal bearing usually bears heavy loads and the speed gradient of the lubricant in the thickness direction of the oil film generates considerable heat, the simulation of journal bearing mechanical and thermal deformation is more common [34], [35], [36], [37], [38]. Junho [34] adopted a 3D FEM on the Reynolds equation and a 3D energy equation coupled with lubricant viscosity-temperature correlation to investigate shaft and bearing pad thermal deformation. Laukiavich [35] conducted a study on how the oil film thickness changes with temperature and pressure and whether a complete thermal seizure occurs at high temperature. Guo [36] developed a transient tribology model to investigate the correlation of the mixed thermal-hydrodynamic behaviour and the dynamic performance of journal bearings, and the results illustrated that a smaller radial clearance tends to generate a higher temperature and thermal expansion, which is helpful for the thermal analysis of roller bearings. In addition, for lubricant supplied rolling bearings, the temperature distribution can be affected significantly by a lubricant mixing model at the lubricant feed port [6], [39], [40]. A precision estimation of oil inlet temperatures under bearing full-flood and reduced flow conditions was conducted [39], and the results showed that the model has advantages for bearing thermal analysis.
The existing literature on bearing dynamic models has typically focused on the vibration response of the ball interaction with a defect [41], [42], [43], and ball-cage collisions and cage lateral motion behaviours have received less attention [6], [44], [17], [45]. The cage whirling state has not been fully considered when discussing the skidding behaviour. Wen [46], [47] discussed the effect of external loads, operating speeds and unbalanced forces on the cage motion state. The inherent relationship between cage motion and cage-guide ring wear was revealed by applying an unbalanced cage mass. The effect of cage-pocket clearance and cage imbalance [48], [49] on cage whirling state and stability was assessed by cryogenic experiments. The results found that the cage whirling amplitude increased for the whole speed range as the mass imbalance increased, and the cage whirling radius decreased in all ball-pocket clearances as the speed increased. Yang et al. [50] conducted an experiment on the temperature rise under varying cage clearances. Their results showed that the temperature rise was less significant as the cage-guide ring and ball-cage pocket clearances increased to a certain value.
Although current studies on rolling bearing kinematic and dynamic behaviour have been systematically developed, the majority of these studies have focused on bearing vibration responses due to localized or distributed faults. Comprehensive studies on healthy bearing skidding and cage behaviour analysis are limited, especially regarding bearings lubricated with supplied cooling lubricant. In addition, most of the thermal expansion and viscosity analyses on bearing components and lubricant oil have been conducted on journal bearings, and rolling bearings have not been fully described in this field. Thermal expansion and temperature rise induced by skidding could aggravate the skidding degree and shrink the ball-raceway clearance, and seizure failure may occur. To address the above questions, a kinematic-Hertzian contact-thermal-elasto-hydrodynamic (KH-TEHD) model is proposed. The innovative contributions are listed below:
(1) Comparing with the KH-THD model without considering thermal deformation and cage whirling motion analysis which proposed in Ref. [6], an improved bearing model with a 2*Nb + 7 + 2 DOF quasistatic and deformation model and a 4*Nb + 3 DOF ball-cage motion dynamic model is established. The structural thermal deformation, detailed lubricant mixing model, and cage dynamics driven by the miscellaneous ball-cage interaction force and hydrodynamic pressure are introduced in this paper. The load distribution analysis and ball-raceway traction dynamic model are based on the previous KH-THD model [6]. The comparison between the KH-TEHD and KH-THD and experimental results validate the superiority of the improved model.
(2) Combined with an experimental case study and model calculation results, a mechanistic explanation of the parameters that affect the degree of skidding and overskidding is proposed, which provides a theoretical basis for avoiding the occurrence of skidding.
(3) According to the calculation results of the model, the influence of temperature rise and deformation caused by skidding is discussed. The cage whirling state and the stability of the cage are discussed under three-dimensional operating conditions (rotating speed, applied load, unbalanced mass).
This paper is organized as follows. The detailed KH-TEHD model is presented in Section 2. The experimental results for model validation and a discussion of the results are presented in Section 3. The cage whirling state and tendency discussion are presented in Section 4. Some conclusions and future prospects are illustrated in Section 5.
Section snippets
Description of bearing static configuration under mechanical and thermal deformation
The KH-TEHD model with cage whirling motion presented in this paper is composed of quasi-static mechanical-thermal analysis and ball-cage dynamic motion analysis. As shown in Fig. 1, four Cartesian reference systems are presented. The global reference system is fixed to describe the deformation of the outer ring, and the local reference systems and rotating along the axis are set to describe ball motion and inner ring motion, respectively. The coordinate system
Experimental validation
The experiment focused on the cage rotating speed of a 7307 AC bearing under variable operating conditions [44], including the axial load, bearing rotating speed and oil supply flow rate. The test bearing was lubricated by two horizontal symmetrically placed oil spray nozzles that connect with an oil pump, as shown in the blue area of Fig. 3. Two oil flow rates (1.28 and 0.43 L/min) were tested, which were denoted as full flood and starvation conditions. The bearing was mounted on a rotating
Cage whirling discussion
After the calculation and experimental comparison of the overall skidding of the bearing cage in Section 3.1, the validity and accuracy of the KH-TEHD model (including cage whirl analysis) are verified. This section discusses the influence of operating conditions and the unbalanced mass of the cage on the stability of the cage by analysing the radius and divergence of the cage centre whirl trajectory.
Fig. 15 and Fig. 16 present the cage centre whirling orbit for inner ring rotating speeds of
Conclusions
An improved bearing dynamic model, the “KH-TEHD”, is developed based on the basic “KH-THD” model. The thermal deformation of the bearing components, the improved oil mixing model, the hydrodynamic oil film pressure acting on the cage, and the miscellaneous forces (unbalanced force, collision tangential friction, etc.) acting on the ball and cage are introduced in the model to investigate the bearing skidding and overskidding behaviour, as well as the 3-DOF motion of the cage. The following
CRediT authorship contribution statement
Shuai Gao: Investigation, Visualization, Conceptualization, Methodology, Software, Formal analysis. Steven Chatterton: Conceptualization, Software, Investigation. Paolo Qinkai Pennacchi Han: Supervision, Conceptualization, Project administration, Resources, Funding acquisition, Project administration, Funding acquisition. Fulei Chu: Supervision, Funding acquisition.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
This research is supported in part by the scholarship from the China Scholarship Council (CSC) under Grant CSC N° 201806880007. The Italian Ministry of Education, University and Research is acknowledged for the support provided by the Project “Department of Excellence LIS4.0 - Lightweight and Smart Structures for Industry 4.0”. The National Science Foundation of China contributed under Grant No. 11872222 and the State Key Laboratory of Tribology contributed under Grant No. SKLT2021D11.
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