Strojniški vestnik - Journal of Mechanical Engineering 62(2016)2, 86-94 © 2016 Journal of Mechanical Engineering. All rights reserved. D0l:10.5545/sv-jme.2015.2848 Original Scientific Paper Received for review: 2015-07-13 Received revised form: 2015-12-14 Accepted for publication: 2015-12-17 An Improved Quasi-Dynamic Analytical Method to Predict Skidding in Roller Bearings under Conditions of Extremely Light Loads and Whirling Junning Li12* - Wei Chen2 - Libo Zhang2 - Taofeng Wang2 1 Xi'an Technological University, School of Mechatronic Engineering, China 2 Xi'an Jiaotong University, Key Laboratory of Education Ministry for Modern Design and Rotor-Bearing System, China Slip often occurs in high-speed and light-load roller bearings (HSLLRBs) when the frictional drive force is inadequate to overcome the drag forces between the rolling elements and the raceway. Formerly, skidding analysis of HSLLRBs considering bearing whirling based on a simplified method using the Dowson-Higginson empirical model, although the analytical results of the cage slip fraction show significant discrepancies with the experimental data, as for extremely light radial loads. One of the main reasons is the inaccuracy of the evaluation of oil drag forces using the empirical equations. In this study, the elastohydrodynamic lubrication (EHL) method was adopted to calculate the oil film thickness and pressure distribution of HSLLRBs, so as to obtain more accurate oil drag forces. The cage speed and cage slip fraction were obtained by combining the whirl orbits, drag forces, load, kinematic equations and other related equations and then solved using the Newton-Raphson method. The skidding mechanism was investigated in terms of various operating parameters such as whirl orbit radii, radial load and viscosity. The results showed that the cage slip fraction and cage speed oscillate over time because of the whirl, which leads to an increase in the risk of bearing skidding damage. Under the extremely light load and high speed, the slip and influence of the whirl on bearing skidding increases as the whirl radius and radial load increases, while the viscosity shows a reverse trend. Therefore, in order to reduce slip and skidding damage, the whirl radius and radial load should be decreased suitably, while the viscosity should be increased moderately. A comparison between the calculated and experimental results shows that the proposed method is both feasible and valid. Keywords: skid, whirl, roller bearing, squeeze film damper, EHL, operating parameters Highlights • Extremely light load and whirling are considered in the skidding analysis of roller bearings. • An improved quasi-dynamic skidding analysis method coupled with EHL is proposed. • The proposed method proved to be feasible and useful for predicting skidding in HSLLRBs. • The influence of the whirl orbit, radial load and viscosity on HSLLRBs skidding are analyzed. • Suggestions for preventing skidding are summarized. 0 INTRODUCTION Rolling-element bearings are key precision components used for rotor support in nearly all machinery. In high-speed and light-load roller bearings (HSLLRBs), such as the typical main shaft bearings of an aircraft engine, the centrifugal forces on the rollers are considered to play a major role in its mechanics. Under these conditions, the tractive forces between the inner raceway and the rollers are frequently insufficient to overcome the drag on the rolling element assembly, which results in the phenomenon of slip [1]. Extreme slip between the roller and the raceway can cause wear on the rolling contact surfaces and subsequently result in a smearing type of surface damage [2]. Several analytical models have been developed for the prediction of slip in roller bearings under different conditions [1] to [7]. Dowson and Higginson analyzed the effects of film thickness and frictional forces on cage slip and derived equations for calculating the various forces acting on a roller under rigid and elastohydrodynamic lubrication (EHL) regimes [3], but did not include the effect of centrifugal forces. Harris [4] and Harris and Kotzalas [5] proposed an analytical method for predicting skid in high-speed roller bearings; this method yielded good results under conditions of medium and excess loading. However, analytical data on the cage speed show significant discrepancies with the experimental data at extremely light radial loads. Poplawski developed a roller bearing model to estimate the cage slip and cage forces, whose analytical predictions of slip correlated well with his test results [6], however they were not in agreement with Harris's cage speed predictions and experimental results for extremely light radial loads. Since the slip velocities at the rolling element-cage contact are typically large, a constant friction coefficient is used at the rolling element-cage pocket contact [7]. Tu et al. [8] and Shao et al. [9] presented an analytical model to investigate skidding during acceleration of a rolling element bearing, which 86 *Corr. Author's Address: Xi'an Technological University, 4 Jinhua Road,, Xi'an, China, junningli@outlook.com Strojniski vestnik - Journal of Mechanical Engineering 62(2016)2, 86-94 takes into consideration the contact force and friction force between the rolling elements races and the cage, as well as the gravity and centrifugal force of the rolling elements, and analyzed the effects of a localized surface defect on the vibration response of the cylindrical roller bearing. In addition, based on the computed tomography, Zbrowski and Matecki analyzed the grinding smudges and subsurface defects in roller bearing rings [10]. Zhang [11] and Li et al. [12] analyzed various factors that influence slip and developed skidding analysis software for high-speed roller bearings. Most of the above mentioned theoretical analyses assume that roller bearings are installed directly in the bearing house. In practice, squeeze-film damper bearings (SFDBs) are widely used to inhibit the effects of vibration in a rotor-bearing system [13] and [14]. The inner ring of SFDBs is often mounted on the outer ring of the roller bearing with an interference fit that makes all of them whirl together, and thus inevitably influencing skidding damage of a roller bearing. Severe skidding of HSLLRBs in a whirling squeeze film damper often leads to vibration and failure of rotor-bearing systems or even entire machines. Many studies have focused on HSLLRBs skidding analysis without considering bearing whirl. Based on a simplified method using the Dowson-Higginson empirical model, Li and Chen analyzed the effects of different structure parameters on skidding of high-speed roller bearings in SFDBs considering bearing whirling [12], in which the radial load exceeds 500 N. As for the extremely light radial loads, the analytical data on slip have significant discrepancies with the experimental data. One of the main reasons is the inaccuracy of the evaluation of oil drag forces with the Dowson-Higginson empirical equations, which have a significant effect on the bearing skidding analysis, especially under conditions of extremely light radial loads. In this manuscript, HSLLRBs in a whirling squeeze film damper were taken as an example for skidding analysis. In order to obtain more accurate oil drag forces to assist in the skidding analysis of HSLLRBs, the EHL equations were solved using multigrid methods to obtain the values of the oil film thickness and pressure distribution, and then to acquire the fluid frictional drag forces. The skidding mechanism was investigated systematically in terms of various operating parameters such as whirl orbit radii, radial load and lubricating oil viscosity. 1 SQUEEZE-FILM DAMPER BEARINGS Squeeze-film damper bearings (SFDBs) are widely used to inhibit the effects of vibration in a rotor-bearing system. It has been shown that correctly designed SFDBs are a very effective means for reducing both the amplitude of rotor motion and the force transmitted to the bearing support [13] and [14]. In general, the rotating shaft carries a roller bearing, whose outer ring whirls with the inner ring of the SFDBs in the oil-filled clearance space between the inner and outer rings of the SFDBs. The outer ring of the bearing forms a damper so that rotation is inhibited by an anti-rotation pin [15] and [16]. Therefore, the damping effect of SFDBs mainly relies on an inner ring whirl, which squeezes the oil film in the clearance and forms resistance from the film. The inner ring of SFDBs is often mounted on the outer ring of roller bearing with an interference fit that makes all of them whirl together, thus inevitably influencing skidding damage to the roller bearing. The HSLLRBs of a whirling squeeze film damper is taken as an example for skidding analysis. The contact between roller and raceway is viewed as a rigid contact, the roller purely rolls along the outer ring raceway for the unloaded zone, and the cage normal forces are equal for every roller over all contact areas of the roller bearing. 2 MATHEMATICAL MODELS 2.1 Whirl Orbits Bearing whirl is majorly stimulated by shaft rotation. Therefore, the whirling frequency is relevant to shaft speed. In this manuscript, the authors considered the example of a roller and assumed its whirl orbits to be a circle with radius e with simple harmonic vibration in the x and y directions [17] and [18]. Taking the time of the maximum amplitude as the initial moment, the coordinates of orbit center Ow can be expressed as follows: So, ex = ecosmj, ey = esinmjt. er = -em cos m t, e„ =—em sin m t. (1) (2) (3) (4) An Improved Quasi-Dynamic Analytical Method to Predict Skidding in Roller Bearings under Conditions of Extremely Light Loads and Whirling 87 Strojniski vestnik - Journal of Mechanical Engineering 62(2016)2, 86-94 2.2 Fluid Frictional Drag Forces 2.3 EHL Formulas As for the extremely light radial loads, the cage slip shows significant discrepancies with the experimental results. One of the main reasons is the inaccuracy of the evaluation of oil drag forces using the Dowson-Higginson empirical equations, which have a significant effect on the bearing skidding analysis, especially under conditions of extremely light radial loads [19]. The oil drag forces relate to the film thickness and pressure. The fluid frictional drag forces model of the HSLLRBs is shown in Fig. 1. Fig. 1. Model of the fluid frictional drag forces The fluid frictional drag forces T can be expressed as follows [20]: T = —°h^Ldx + (u2 - ul) — dx. (5) Jxi 2 dx Jxi h Here, xi and xo are the coordinates of the inlet and outlet. Eq. (5) can be expressed in terms of dimensionless quantities as follows: rXoHb dpx , ' Jx Wo J?3 PiY J. X, V R n ' ' ' dX, !X 8leR3 dX Jx Hieb F=-! x Hß er , çxo v0R0n dx+rx X -dX. (6) X 8leR3 dX JX Holeb The solution to the fluid frictional drag forces equations are related to the oil film thickness and pressure distribution, which are the main challenge and the emphasis of this study. Here, the EHL method is adopted to calculate the oil film thickness and pressure distribution of the roller bearing, and then the fluid frictional drag forces can be obtained using Eq. (6) to replace the inaccuracy values obtained using the Dowson-Higginson empirical equations. As for the isothermal EHL in the line contact, the basic equations and their dimensionless forms can be expressed as follows: Reynolds equation: d ( dP ^ d(pH) dX dX I dX 0, (7) where s = PHI Z = 3n U 4W2 ' The Reynolds boundary conditions of the equation are: in the area of the oil inlet X=Xin , P = 0; and in the area of oil outlet X=Xout , P=dP / dX= 0. (1) Film thickness equation: X2 1 rX0Ut 2 H = Ho + — -- f P (S ) ln( X - S)2dS. (8) 2 2n Xin (2) Viscosity equation varying with pressure (Roelands equation): n= exp{(lnn0 + 9.67)[(1 + 5.1x10-9pHP)z -1]}. (9) (3) Density equation: P 1 + ■ 0.6 xlQ"9 pHP 1 +1.7 x10-9 pHP (4) Load balancing condition equation: f X"u' PdX = n. ¡xin 2 (10) (11) Eqs. (7) to (11) can be expressed in terms of x o dimensionless quantities as follows: X = —, p=—, b Po H = R hR n, P=P U = no Ph lou E ' R' W = w ER ' Ph Here, the EHL equations are solved using multigrid methods to obtain the values of the oil film thickness and pressure distribution, and then to acquire the fluid frictional drag forces using Eq. (6) to assist in the skidding analysis [19] and [20]. 2.4 Kinematics and Mechanical Models of the Roller The roller whirls along with the bearing. The kinematics model of a roller is shown in Fig. 2, and 2 b 88 Li, J. - Chen, W. - Zhang, L. - Wang, T. Strojniski vestnik - Journal of Mechanical Engineering 62(2016)2, 86-94 the forces acting on the roller at angular location qj are shown in Fig. 3 [12]. Fig. 2. Kinematics model Consider a separate roller as an analysis object; its kinematics model is shown in Fig. 2. Thus, the sliding velocities can be expressed as follows: Vj= \(dm - Db ) - 2 Dbrnmj, (12) Vj= \(dm + Db)wc -2Dbwaj. (13) Additionally, the fluid entrainment velocities are defined as follows: Uj = \(dm -Db) + 2DbW0]j, (14) Uj= \(dm + Db H + 2 DbWmj. (15) The sliding and the fluid entrainment velocities can be expressed in terms of dimensionless quantities as follows: V - " ER '' V = - ER U = 'J ER U = j-0 ER ' (16) (17) (18) (19) In the loaded zone, the dynamic balance equations are expressed as follows: Qyj + Fm- Qj = me, (20) Fj+ Qxij - Qxoj- F0j- Fdj= mrex, (21) F+ F0j- fdj= 0. (22) In the unloaded zone, the dynamic balance equations are expressed as follows: F — Q = m e , m ^yoju r y > F.- - Q - - F - = m 'e dju ^xoju oju r x Foju + fdj = 0. (23) (24) (25) Strojniski vestnik - Journal of Mechanical Engineering 62(2016)2, 86-94 Eq. (20) to (25) can be expressed in terms of dimensionless quantities. In the loaded zone, - Ro - - mr ey Qy'i + ( R F Qy°i ) = iß t R (26) F + Qzi} -(f)(QZ0} + F01 + Fdi) = Ä (27) IE' R F + (R )Foj- fd = o. In the unloaded zone, _ _ me F -Q - rn y^-yoju IE' R Fd- - Q - - F- = dju xoju oju IE' R„ Foju + fdj = 0. For entire rollers, ±F}= 0. (28) (29) (30) (31) (32) Qyij and Qyoj denote normal forces transmitted by the raceways to the roller owing to bearing radial load and geometry. The new fluid frictional drag forces equations are adopted here: F = _f dPdx +fxVRn ' JX, 8leR dX y F S. x, 8leR3 dX' ~ ' Jx, H ¡leb X Hob dP , rX VRn dX +i Jx X- 8leR3o dX JX Holeb Qxj = 18.4(1 - D )G dX, (33) dX, (34) (35) In addition, the cage drag normal force and tangential force acting on a roller are expressed as follows: _ 'r 2.447n/ • (Dbx /2)2 d (37) Fdj = 2J-h-^ (37) 0 ^¡"o fdj= 2 J-h-dx (3 8) o Vho In this study, the EHL method is adopted to calculate the oil film thickness and pressure distribution of the roller bearing, which allows the fluid frictional drag forces to be determined. Eqs. (27) and (30) can be further substituted into Eq. (32), and then Eqs. (28), (31) and (32) form an equation set with Z+1 equations. Then, the cage speed and cage slip fraction can be obtained by combining Eqs. (3) and (4) with the kinematic equations, load equations, Dowson-Higginson formulae, and other related equations and solved using the Newton-Raphson method [4] and [12]. 3 ANALYSIS AND DISCUSSION A single-row cylinder roller bearing is taken as an example. The default parameters are listed in Table 1. The effects of various operating parameters on the skidding of HSLLRBs taking into consideration bearing whirl are investigated. Table 1. Details of the bearing and lubrication Type d [m] dm [m] G E' [Pa] pr [kg/m3] NU214 0.07 0.0975 5000 2.28 e11 7800 Cdyn db [m] l [m] Z Pd [m] n [Pas] 1.37 e5 0.013 0.013 18 2.5 e-5 0.08 The cage slip fraction Sf can be expressed as follows: Sf = 1--, nm n = - n (1 - D). 2 ' d' (39) (40) 3.1 Effect of Different Whirl Radii on Skidding Bearing whirl induces the generation of additional forces in roller-cage and roller-rings pairs. More importantly, the oil film forces magnitudes of the roller-rings and roller-cage to oscillate as well. These forces oscillate over time, which often effect fatigue life and skidding damage to the bearing. Fig. 4 shows that the cage slip fraction and cage speed vary with time as the result of the whirl. In addition, the amplitudes of the cage speed and cage slip fraction increase with an increase in the whirl radius. So the degree of skidding can be reduced by suitably decreasing the whirl radius. 3.2 Effect of Different Radial Loads on Skidding In contrast to the previous results, Fig. 5 shows that the cage slip fraction increases as the radial load increases, but that the cage speed shows a reverse trend. On the other hand, Table 2 shows that the amplitudes of cage speed and cage slip fraction decrease with an increase in the radial load. In other words, the influence of whirl on bearing skidding increases as the radial load increases. Therefore, the degree of skidding can be 90 Li, J. - Chen, W. - Zhang, L. - Wang, T. Strojniski vestnik - Journal of Mechanical Engineering 62(2016)2, 86-94 a) b) Fig. 4. Skidding analysis under different whirl radii (ni = 5000 r/min; Fr = 200 N; Cdyn/P=675); a) cage speed vs. whirl radius, and b) cage slip fraction vs. whirl radius ,_, 1300 'I 1280 K 1260 13