Skidding and cage whirling of angular contact ball bearings: Kinematic-hertzian contact-thermal-elasto-hydrodynamic model with thermal expansion and experimental validation

https://doi.org/10.1016/j.ymssp.2021.108427Get rights and content

Highlights

  • Accurate modeling of the skidding phenomenon that occurs in angular contact ball bearings for low axial loads.

  • Introduction of a new model using a kinematic-Hertzian-thermo-elasto-hydro-dynamic model.

  • Experimental validation on data in different operating conditions.

  • Discussion of cage whirling as a consequence of bearing deformations owing to operating conditions.

Abstract

The occurrence of skidding and overskidding behaviour is likely to scratch rolling element raceways and lead to a significant temperature rise and thermal deformation. The ball-cage interaction and cage whirling motion are likely to oscillate as this skidding is aggravated. To address the damage problems introduced by skidding behaviour, a kinematic-Hertzian contact thermal-elasto-hydrodynamic (KH-TEHD) model with a 5-layer loop structure is proposed. Various factors are considered in the model, including the dynamic beahviour of bearing components, quasi-static analysis on bearing load distribution and deformation of bearing components, and the miscellaneous forces (unbalanced force, collision tangential friction, drag force) acting on the ball and cage. Ball-raceway lubrication, lubricant temperature rise, thermal expansion of shaft and bearing housings, and hydrodynamic pressure acting between the cage surface and guide-ring surface are established in this paper. A 7307AC bearing experimental study under varying operating speeds, axial loads and two lubricating conditions is conducted to validate the accuracy and effectiveness of the KH-TEHD model. The superiority of the developed model is illustrated by comparison with previous models, and the results suggest that the thermal expansion and cage-guide ring hydrodynamic pressure cannot be ignored. After comparisons with literature results, it is found that the overskidding degree is approximately proportional to the square of the bearing size under the same level of operating speed. The maximum thermal deformation at the bearing raceway under starvation lubrication reaches 4 times the ball-raceway clearance. The cage simulation results indicate that the cage whirling oscillation amplitude increases with increasing operating speed and unbalanced mass and generally decreases with increasing axial load. The proposed model and discussion of the results are useful for mitigating the skidding degree under certain conditions and avoiding cage instability and wear.

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 (ΣM=Jdωdt) 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 xo,yo,zo is fixed to describe the deformation of the outer ring, and the local reference systems xi,yi,zi and xj',yj',zj' rotating along the zo 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.

References (57)

  • S. Xi et al.

    Dynamic modeling of spindle bearing system and vibration response investigation

    Mech. Syst. Signal Process.

    (2019)
  • L. Niu et al.

    A systematic study of ball passing frequencies based on dynamic modeling of rolling ball bearings with localized surface defects

    J. Sound Vib.

    (2015)
  • Y. Wang et al.

    Investigation of skidding in angular contact ball bearings under high speed

    Tribol. Int.

    (2015)
  • J. Takabi et al.

    On the dynamic performance of roller bearings operating under low rotational speeds with consideration of surface roughness

    Tribol. Int.

    (2015)
  • Y. Liu et al.

    The effect of lubricant temperature on dynamic behavior in angular contact ball bearings

    Mech. Mach. Theory

    (2020)
  • S. Singh et al.

    Analyses of contact forces and vibration response for a defective rolling element bearing using an explicit dynamics finite element model

    J. Sound Vib.

    (2014)
  • F. Ma et al.

    Transient thermal analysis of grease-lubricated spherical roller bearings

    Tribol. Int.

    (2016)
  • S. Ai et al.

    Temperature rise of double-row tapered roller bearings analyzed with the thermal network method

    Tribol. Int.

    (2015)
  • W. Wang et al.

    Theoretical and experimental study on the static and dynamic characteristics of tilting-pad thrust bearing

    Tribol. Int.

    (2018)
  • P.V. Dang et al.

    Effect of the load direction on non-nominal five-pad tilting-pad journal bearings

    Tribol. Int.

    (2016)
  • N. Sawalhi et al.

    Vibration response of spalled rolling element bearings : Observations, simulations and signal processing techniques to track the spall size

    Mech. Syst. Signal Process.

    (2011)
  • L. Cui et al.

    HVSRMS localization formula and localization law : Localization diagnosis of a ball bearing outer ring fault

    Mech. Syst. Signal Process.

    (2019)
  • Q. Han et al.

    Nonlinear dynamic model for skidding behavior of angular contact ball bearings

    J. Sound Vib.

    (2015)
  • B. Choe et al.

    Experimental study on dynamic behavior of ball bearing cage in cryogenic environments, Part I: Effects of cage guidance and pocket clearances

    Mech. Syst. Signal Process.

    (2019)
  • B. Choe et al.

    Experimental study on dynamic behavior of ball bearing cage in cryogenic environments, Part II: Effects of cage mass imbalance

    Mech. Syst. Signal Process.

    (2019)
  • Z. Yang et al.

    Influence of cage clearance on the heating characteristics of high-speed ball bearings

    Tribol. Int.

    (2017)
  • Y. Liu et al.

    Skidding dynamic performance of rolling bearing with cage flexibility under accelerating conditions

    Mech. Syst. Signal Process.

    (2021)
  • Y. Cui et al.

    Vibration effect analysis of roller dynamic unbalance on the cage of high-speed cylindrical roller bearing

    J. Sound Vib.

    (2018)
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