A dynamic model to predict the occurrence of skidding in wind-turbine bearings

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1 Journal of Phsics: Conference Series A dnamic model to predict the occurrence of skidding in wind-turbine bearings To cite this article: Sharad Jain and Hugh Hunt 2011 J. Phs.: Conf. Ser View the article online for updates and enhancements. Related content - Improved 2D model of a ball bearing for the simulation of vibrations due to faults during run-up Matej Tadina and Miha Boltežar - Finite line roller-to-race contact M Kushwaha and H Rahnejat - Modelling of Outer and Inner Film Oil Pressure for Floating Ring Bearing Clearance in Turbochargers Hao Zhang, Zhanqun Shi, Fengshou Gu et al. Recent citations - Overview of dnamic modelling and analsis of rolling element bearings with localied and distributed faults Jing Liu and Yimin Shao - Wind energ research: State-of-the-art and future research directions D.J. Willis et al - Methodolog for the phsics-based modelling of multiple rolling element bearing configurations Urko Leturiondo et al This content was downloaded from IP address on 08/04/2019 at 22:41

2 A dnamic model to predict the occurrence of skidding in wind-turbine bearings Sharad Jain and Hugh Hunt Universit of Cambridge, Department of Engineering, Trumpington Street, Cambridge, United Kingdom CB2 1PZ Abstract. Despite use of the best in current design practices, high-speed shaft (HSS) bearings, in a wind-turbine gearbox, continue to exhibit a high rate of premature failure. As HSS bearings operate under low loads and high speeds, these bearings are prone to skidding. However, most of the existing methods for analing skidding are quasi-static in nature and cannot be used to stud dnamic operating conditions. This paper proposes a dnamic model, which includes groscopic and centrifugal effects, to stud the skidding characteristics of angularcontact ball-bearings. Traction forces between rolling-elements and racewas are obtained using elastohdrodnamic (EHD) lubrication theor. Underling gross-sliding mechanisms for pure axial loads, and combined radial and axial loads are also studied. The proposed model will enable engineers to improve bearing reliabilit at the design stage, b estimating the amount of skidding. 1. Introduction Wind energ is the fastest growing renewable energ sector with an average annual growth rate of around 30% during last 10 ears. In order to harvest energ most efficientl and reliabl, various wind-turbine design concepts have been developed over the ears. Most of the modern wind-turbine designs utilie a gearbox which connects the rotor-shaft to high-speed shaft, and increases rotational speed from rpm (at blades) to rpm the speed required b most generators to produce electricit. As wind-turbines have grown larger, gearbox failure rates have gone up as well. Since gearbox is one of the most expensive components of a wind turbine, higher-than-expected failure rates increase the cost of energ production. For a tpical turbine, 20% of the downtime is due to gearbox failures and an average gearbox failure takes about 250 hours to repair [1]. Most of the problems in wind turbine gearboxes appear to emanate from bearings[2]. Bearings supporting the high speed shaft exhibit a high rate of premature failure and are identified as one of the most critical components [1, 2]. These bearings operate under low loads and high speeds, and therefore, are prone to skidding, i.e., gross-sliding of rolling-elements on racewas. Sliding can lead to rolling surface distress and eventuall to premature failure. Hence, skidding is an important design criterion for wind-turbine bearings. Both ball-bearings and roller-bearings are commonl used to support HSS. The focus of this work is on angular-contact ball-bearings. Researchers have developed numerous analtical and numerical models of varing complexit to understand the skidding behaviour of bearings. Jones [3, 4] developed the first mathematical theor to anale the motion of rolling-elements in ball bearings. He evaluated the frictional Published under licence b Ltd 1

3 forces resulting from interfacial slip at ball-race contacts using a dr friction model. One of the limitations of his theor is its dependence over racewa-control hpothesis to achieve a solution. According to this hpothesis, a ball is assumed to roll without spin on one race and roll with spin with respect to other race. Therefore, motion of the ball about its own axis and bearing axis is said to be controlled b the racewa at which no slip occurs. It is also further assumed that the groscopic moment acting on a ball is alwas resisted b frictional force acting at controlling racewa and no groscopic slippage takes place. Harris [5, 6] questioned the validit of racewa-control hpothesis b formulating an analtical model for axiall loaded angular contact ball bearings without using racewa control assumption. It was found that Harris model more closel approximated the measured data, reported b Poplawski and Mauriello [7], than the racewa-control method, which proves the inadequac of racewa-control hpothesis. Boness and Gentle [8] also developed a quasi-static force equilibrium model of a ball bearing b using an analtical traction equation, for EHD contacts, derived b Gentle and Cameron [9]. Some skidding threshold criteria, to predict the minimum load required to avoid skidding, are also available in the literature. Hirano [10] carried out an experimental investigation on the ball motion inside an axiall loaded angular contact ball bearing, and found that the gross-sliding of rolling-elements occurs when the value of the parameter F c /F a exceeds 0.1, where, F c and F a denote the number of balls in the bearing, centrifugal force acting on the ball and applied thrust load respectivel. However, it has been established b Poplawaski [7] and Boness [11] that this parameter alone is not sufficient to completel define the roll-slip behaviour of a ball bearing. Recentl, Liao and Lin [12] performed a geometric analsis of a ball bearing operating under combined axial and radial loads; and used a force balance approach to obtain axial and radial deformations. Since, the skidding maps produced b Liao and Lin [12] are based on the empirical criterion proposed b Hirano [10] for ball bearings under thrust load, their use can often be limited for combined loading conditions. Based on the experiments performed b Bujoreanu et al. [13], it was observed that the skidding damage in a bearing is related to the amount of heat generated inside a fluid film due to lubricant shearing. The onset of scuffing was estimated around W/m 3. The quasi-static analsis techniques and skidding threshold formulations described above provide a good insight into the frictional behaviour of ball bearings and also show the existence of gross-sliding. However, these methods cannot be used to anale combined radial and thrust loads or time-varing operating conditions, both of which are crucial for wind turbine applications. The work presented herein details a dnamic model formulation, which takes into account the centrifugal and groscopic effects. The frictional forces at contact interfaces are calculated for a Newtonian fluid using EHD lubrication theor. The model is used to investigate the skidding mechanisms for axial as well as combined axial and radial loading conditions, and to quantif the effect of operating parameters on gross-sliding behaviour. 2. Model Description The analsis approach consists of two stages. In the first stage, a quasi-static method is used to determine bearing internal load distribution; and during the second stage these loads are used in a dnamic model to anale rolling-element motion. The two methods are described in the following paragraphs Determination of Internal Load Distribution In bearings, load is transmitted from one racewa to another through rolling-elements. The magnitude of load carried b an individual rolling-element depends upon the internal geometr of a bearing, number of rolling-elements in contact and location of a rolling-element inside the load-one at a given time. In this stud, Hert elastic theor [14] is used to determine the contact 2

4 Outer race Y Outer race F out β Inner race θ i x F c F inn (a) (b) Inner race Figure 1: (a) Coordinate sstem for quasi-static analsis (b) Forces acting on a rolling element force between rolling-elements and racewas. According to Hert theor, the contact load (F ) between two elastic solids can be expressed in terms of maximum deformation (δ) at the centre of contact ellipse as F = Kδ 3/2 (1) Here, K is the stiffness parameter given b: K = πκe 3εR 3ξ ξ where, E is the effective modulus, R is the effective radius of curvature, κ is ellipticit parameter, and ξ and ε are the elliptical integrals of first and second kind respectivel. Simplified expressions for κ, ξ and ε, derived using linear regression, can be found in Brewe and Hamrock [15]. If K inn and K out are the stiffness parameters for inner and outer racewa contacts respectivel, defined b equation 2, then effective stiffness parameter for a rolling-element can be calculated as [16]: 1 K eff = { ( ) 2/3 ( ) } 2/3 3/2 (3) 1 K inn + 1 K out (2) Assuming that the racewas are rigid, for a rolling-element located at an angle θ i Figure 1a), the deformation along the contact-line can be calculated as: δ x δ i = [C θi C β, S θi C β, S β ] δ δ (see (4) where, C α = cos(α), S α = sin(α), β is the contact angle, and (δ x ĩ + δ j + δ k) is the inner-race displacement vector. Contact forces acting between rolling-elements and racewas can be obtained b the following expressions. F inn = F out = K eff δ 3/2 i (5) Element loads obtained from equations 5, are based on an assumption that contact angle between rolling-elements and racewas remain unaffected b bearing rotational velocit. However, at high rotational speeds, the centrifugal force acting on a rolling-element, forces contact angle to change from its nominal value, and this creates a differential between inner and 3

5 outer contact angles. Since, the bearings supporting the HSS of a wind-turbine, operate at fairl moderate speeds ( rpm), therefore, change in the contact angle due to centrifugal force would be ver small [10]. Resultant load acting on the inner-race can be determined b a vector summation of F inn, for all the rolling-elements which are in contact with inner-race. Considering centrifugal force, inner-race forces can be calculated as F x F F = i=1 ( K eff δ 3/2 i 1 ) 2 mr pωc 2 cos(β) [C θi C β, S θi C β, S β ] T (6) where, m is the mass of a rolling-element, r p is the pitch radius, ω c is the rotational speed of cage, and is the number of rolling-elements. Equations 4 to 6 are solved iterativel, using Newton-Raphson method, until calculated loads are equal to the applied ones. The method, outlined above, does not account for changes in the load distribution which can occur due to variation in numbers and positions of rolling-elements inside load-one. However, from a detailed experimental stud [17], it was concluded that the fluctuations in deflection and stiffness due to these factors can be less than 0.5% of the total value, for a given load Dnamic Model to Anale Rolling-Element Motion The model consists of two reference frames. The first reference frame Y Z is fixed at bearing centre with and Y axis ling in the bearing plane. The second reference frame, x, is a moving frame with its origin attached to the centre of a rolling-element (Figure 2). Each rolling-element has four degrees of freedom: three rotational degrees of freedom about its centre in moving reference frame (ω x, ω and ω ), and one rotational degree of freedom about bearing centre (ω c ). The equations governing the rolling-element motion are derived using Euler s equations and are given b: ΣM x [ ] 2 ω x 0 ω ΣM = diag ΣM 5 mr2 ω + Z ω [ ] Y ω Z 0 ω 2 diag ω ω Y ω 5 mr2 0 where, r is the ball radius, ΣM x ĩ + ΣM j + ΣM is the moment vector acting on a ball, k ω x ĩ + ω j + ω (= ωb) is the ball angular velocit vector in x-frame, and ω k i + ω Y j + ω Z k is the angular velocit vector of the frame x with respect to Y Z. For the sstem shown in the figure 2, reference frame x is constrained to rotate about Z axis with angular velocit ω c. Therefore, ω = ω Y = 0 and ω Z = ω c. Putting these values into equation 7 gives ω x ω ω (7) ΣM x = I( ω x ω c ω ) ΣM = I( ω + ω c ω x ) ΣM = I ω (8a) (8b) (8c) where, I = 2mr 2 /5. Calculation of moment terms in equation 8 is described in section 2.3. To determine the equation governing rolling-element motion about bearing axis, interaction between rolling-element and cage must be considered. A ver basic approach has been adopted here to define this interaction. Springs of ver high stiffness (k cage = 10 8 ) are inserted in between rolling-elements so that their motion about bearing axis can be coupled and cage 4

6 x ω b Y Moving reference frame ω ω x θ c x ω ω s β B r ω ω x ω ω c Z θ c Fixed reference frame A (a) (b) Figure 2: (a) Fixed and moving coordinate sstems used in dnamic model formulation (b) Angular velocit vectors in the moving reference frame x Figure 3: Illustration of ball-cage interaction model used to calculate cage force forces are determined b calculating the compression or elongation in these springs (figure 3). Deformations in the left and right springs (δ (+) and δ ( ) ) can be written as: δ (+) = r p {cos(θc) i cos(θ i+1 δ ( ) = r p {cos(θc) i cos(θ i 1 c )} 2 + {sin(θ i c) sin(θ i+1 c c ( π ) )} 2 2r p sin ( π ) )} 2 + {sin(θc) i sin(θc i 1 )} 2 2r p sin where, θc i is the position angle of the rolling-element, at time t, for which cage force is being calculated (= t 0 ω cdt), θc i+1 and θc i 1 are position angles of neighboring elements at time t. Now, the remaining differential equation governing the cage velocit can be formulated as: (9a) (9b) I c ω c = (f A x r i + f B x r o ) k cage ( δ( ) δ (+) ) rp F D (10) where, F D is rolling-element drag, r i and r o are inner and outer radii, and I c is the moment of inertia of the ball about Z axis and is given b: I c = 2 5 mr2 + mr 2 p. Equations 8 and 10 are the first-order non-linear differential equations and are solved numericall using Matlab s ODE solver. 5

7 Y Pressure distribution Z u 2 a b Exit u u 1 h Entr Figure 4: EHD film formation and contact patch geometr 2.3. Traction Equations for Elliptical Contact Traction forces in a lubricant film are generated due to shearing effect which also produces frictional heat (given b the product of shear stress and strain rate) inside the film. Crook [18, 19] investigated the effect of temperature rise due to film shearing on lubricant viscosit and traction properties. The investigation was based on a Newtonian fluid model according to which shear stress in a lubricant film is proportional to shear-strain rate. The dependenc of lubricant viscosit on pressure and temperature is described b well known Barus equation [20]: η = η 0 e cη P p cη T (T T R) (11) where η 0 is the reference viscosit at reference temperature T R, p is the hertian pressure, T is the lubricant temperature, and c ηp and c ηt are pressure and temperature coefficients respectivel. Values of c ηp and c ηt are generall determined from the viscosit-pressure and viscosit-temperature curves (measured experimentall) [21]. Consider a fluid film trapped between two contacting solids (figure 4). Thickness of the film is h and linear velocities of contacting surfaces are u 1 and u 2. Shear stress (τ) in the film can be expressed b the following expression (see Crook [18] for derivation). u(x, ) τ(x, ) = η m (x, ) h (12) where, u is the slip velocit and η m is the fluid vicosit at (x, ) and is described as η m (x, ) = η(x, ) ln ( ψ ψ ) ψ(ψ + 1) (13) η(x, ) = η 0 e cη P p(x,), ψ = η(x,)cη T u2 8K c and K c is the thermal conductivit of lubricant. At an point (x, ) on the contact patch, x-component of the spin velocit is ω s and - component is ω s x. If linear slip velocities are u x l and u l in x and direction respectivel then the slip velocit vector at (x, ) can be written as ũ(x, ) = ( u x l ω s )ĩ + ( u l + ω sx)j (14) For a contact patch shown in figure 4, the average traction force vector (f = f x ĩ + f j + f k) and traction moment vector = M k) can be calculated b integrating the equation 12. (M 6

8 Therefore, f = 1 h M = 1 h = 1 h a b a b a b a b a b a b η m (x, ) ũ(x, )dxd (15) η m (x, )(xĩ + j) ũ(x, )dxd η m (x, ) { u x l u x l + ω ( s x 2 + 2)} dxd k The film thickness, h, between the contacting surfaces is assumed to be constant throughout the contact patch, and is calculated using the central film-thickness formula provided b Hamrock and Dowson [16] (16) h R x = 2.69U 0.67 G 0.53 W ( e 0.73κ) (17) where, U (= η 0u avg E R x ), G (= E c ηp ), and W (= F ) are the dimensional parameters for speed, E Rx 2 material and load respectivel, R x is the effective radius along -axis, and u avg is the mean velocit of sliding surfaces. Now, the moment terms of equation 8 can be described in terms of traction forces as ΣM x = r ( f B f A ) (18a) ΣM = rsinβ ( fx B fx A ) ( + M A + M B ) cosβ (18b) ΣM = rcosβ ( fx B fx A ) ( M A + M B ) sinβ (18c) 3. Results and Discussion The skidding phenomenon in angular contact ball bearings is demonstrated using an example bearing, parameters of which are listed in the table 1a and 1b. The gross-sliding is represented b ( cage-slip, which is ) the deviation of actual cage speed from its corresponding theoretical value ω c ωc th 100%, where ω (ω c+ωc th c is the actual cage speed and ω th )/2 c is the one calculated from inner-race speed (ω i ), using pure-rolling condition (equation 19) ( ωc th = 1 cosβ r p /r ) (ωi 2 ). (19) 3.1. Skidding Under Constant Axial Load Figure 5a shows that at lower axial loads, high cage slip is present. As the applied load is increased, the value of cage-slip decreases. When the applied load is increased above a critical value, which is 1.3kN for this example bearing, cage-slip becomes less than 1% and almost no gross-sliding takes place. This critical load (1.3kN) can be considered as the minimum load required to prevent skidding; and corresponding rolling-element load is 125N. Note that the value of cage-slip in figure 5a never goes to ero, this is for the reason that in order to generate traction forces, some amount of relative sliding is required between two contacting solids. Ball orientation angles, β Y Z and β Z (figure 5b), are the angles which spin-axis of a ball (or vector ωb) makes with and x planes respectivel. In an angular contact ball bearing, ball 7

9 Table 1: Parameters defining the bearing geometr and lubricant properties (a) Bearing Parameter Value Number of rolling-elements () 16 Contact angle (β) 40 Ball radius (r) 12.5 mm Pitch radius (r p) 77.5 mm Ball mass (m) 64 grams Material Steel (b) Lubricant Parameter Value Dnamic viscosit (η 0) 0.04 Pa.s Reference temperature (T R) 30 C Viscosit-Pressure coefficient (c ηp ) Pa 1 Viscosit-Temperature coefficient (c ηt ) 0.05 C 1 Thermal conductivit (K c) J/(kgK) Densit (ρ) 860 kg/m 3 Cage slip (%) 1.3kN Gross-sliding Ball orientation angles (degrees) β Z β YZ β Z β YZ x ω b Applied axial load (N) Applied axial load (N) (a) (b) Figure 5: Simulation results for axial loading (a) Cage-slip variation with axial load (b) Ball orientation angles; Inner-race speed ω i = 1800rpm spins about an axis passing through its centre, at an angle β from the bearing axis. This spinning ball is also forced to rotate about the bearing axis. As the ball rotates around bearing axis, the direction of angular momentum vector changes continuousl, and this change in angular momentum generates a groscopic couple which is balanced b the traction forces acting at contact interfaces. At low loads, traction forces are not enough to provide required groscopic couple and ball spin-axis changes its orientation and becomes almost parallel to the bearing axis, thereb reducing the required gro-couple (figure 5b). As the applied load is increased, traction forces increase as well, and ball spin-axis approaches its theoretical orientation Skidding Under Combined Axial and Radial Loads Rolling-element motion in the presence of combined loads is remarkabl different from the axiall loaded bearings, because of the formation of loaded and unloaded ones. Figure 6 shows the angular velocit components and slip velocit of a rolling-element in the example bearing (table 1a) subjected to an axial load of 2.2kN and a radial load of 2kN. The maximum contact force between a rolling-element and racewa is 650N. The motion is also illustrated using a graphical representation (figure 7). During the unloaded one (C D), the angular momentum of the rolling-element remains nearl constant (in the absence of contact force) and the components of angular velocit var sinusoidall in the local coordinate sstem. As the rolling-element enters the load-one, slip velocit starts to decrease due to the application of 8

10 (a) Angular velocit components of a rolling-element (b) Relative slip-velocit between rolling-element and inner-racewa Figure 6: Simulation results for combined loading; F = 2.2kN, F = 2kN, ω i = 1800rpm tractive forces; and at point E, angular momentum vector flips its direction and aligns itself with the pure-rolling vector. The gross-slip doesn t occur between points E and B. Beond point B, the traction force between the rolling-element and racewas are not enough to avoid gross-sliding, and slip velocit starts to increase again. Note that for a rolling-element load of 650N, which is much higher than the skidding threshold of 125N determined for axial load (section 3.1), significant amount of gross-sliding is observed between points D and E Influence of Radial Load on Skidding Behaviour Firstl, in the presence of radial load maximum skidding damage takes place at the entr to load-one; whereas, damage is uniforml distributed under axial load. Secondl, a bearing with radial load requires a larger rolling-element force to minimie skidding than a bearing with pure axial load. Finall, if an unloaded-one is created inside a bearing (b appling radial load), then it is not possible to completel eliminate skidding, but the length of skid one (inside loaded region) can be reduced b increasing the applied load Measures to Prevent Skidding The most effective wa to avoid skidding in bearings is to provide a static preload. The amount of preload must be chosen such that the operating load on a rolling-element must be greater than the minimum load required to prevent skidding. Skidding can also be minimied b using a high-traction lubricant or b reducing the number of rolling-elements. 9

11 No grosssliding A E Maximum damage one Region without Region without gross-sliding gross-sliding B C Gross-sliding inside load-one Load-one (Exit from load-one) D (Entr into load-one) Angular momentum No No change change in in No change in angular angular momentum momentum angular momentum outside outside load-one load-one outside load-one Pure-rolling vector (a) (b) Figure 7: (a) Graphical representation of rolling-element motion under combined axial and radial loading; (b) Angular momentum vectors showing regions with and without gross-sliding 4. Conclusions and Future Work A dnamic model, considering EHD lubrication theor and groscopic effects, is formulated and used to anale the skidding characteristics of angular-contact ball bearings. The findings indicate that the gross-sliding mechanism for combined loading conditions is substantiall different from the one observed for pure-axial loads. Future work will include the implementation of a detailed cage interaction model, and consideration of cage clearance and frictional effects. 5. Acknowledgments Financial support from Romax Technolog Ltd. is gratefull acknowledged. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] Ribrant J and Bertling L 2007 IEEE Power Engineering Societ General Meeting, 2007 pp 1 8 Musial W, Butterfield S and McNiff B 2007 Proceedings of the European Wind Energ Conference (Citeseer) Jones A B 1959 ASME Trans Jones A B 1960 Journal of Basic Engineering Harris T A 1971 ASME Journal of Lubrication Technolog Harris T A 1971 Journal of Lubrication Technolog, Transactions of the ASME Poplawski J V and Mauriello A ASME Paper No 69-LubS Boness R J and Gentle C R 1975 Wear Gentle C R and Cameron A 1974 Wear Hirano F 1965 Tribolog Transactions Boness R J 1981 Journal of lubrication technolog Liao N T and Lin J F 2002 Mechanism and Machine Theor Bujoreanu C, Cret u S and Nelias D 2003 FASCICLE VIII, Tribolog ISSN Hert H 1881 J. Reine Angew. Mathematik Brewe D E and Hamrock B J 1977 ASME, Transactions, Series F-Journal of Lubrication Technolog Hamrock B J and Dowson D 1981 Ball bearing lubrication (Wile New York) While M F 1979 Journal of Applied Mechanics Crook A W 1961 Phil. Trans. Ro. Soc., London. Series A, Mathematical and Phsical Sciences Crook A W 1963 Phil. Trans. Ro. Soc., London. Series A, Mathematical and Phsical Sciences Barus C 1893 Am. J. Sci Evans C R and Johnson K L 1986 Proceedings of the Institution of Mechanical Engineers. Part C. Mechanical engineering science