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Determination of the precise static load-carrying capacity of pitch bearings based on static models considering clearance
International Journal of Mechanical Sciences 100 (2015) 209–215
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International Journal of Mechanical Sciences journal homepage: www.elsevier.com/locate/ijmecsci
Determination of the precise static load-carrying capacity of pitch bearings based on static models considering clearance Yanshuang Wang n, Jiawei Cao Tianjin University of Technology and Education, Tianjin 300222, China
art ic l e i nf o
a b s t r a c t
Article history: Received 6 December 2013 Received in revised form 12 June 2015 Accepted 28 June 2015 Available online 11 July 2015
Static models for a single row and a double row four-point contact pitch bearing, taking into account the clearances therein, were presented. A precise computation method was proposed for the static loadcarrying capacity curves of such pitch bearings, which can be used to pre-select pitch bearings and slewing bearings supposing rigid bearing rings. The effects on the static load-carrying capacity induced by changing the clearance, the raceway groove radius of curvature, and initial contact angle were analysed. The clearance has a significant effect on the static load-carrying capacity of the bearing only in load cases with a rather small axial force and a large tilting moment. When the coefficient of raceway groove curvature radius increases, the static load-carrying capacity decreases. The smaller the radial load, the more significant the effect of the coefficient of raceway groove curvature radius on the static load-carrying capacity of the bearing. When radial loads range from 0 to 800 kN, the load-carrying capacity increases with increasing initial contact angle. When the radial loads are above 800 kN, the load-carrying capacity decreases with increasing initial contact angle. & 2015 Elsevier Ltd. All rights reserved.
Keywords: Pitch bearing Negative clearance Static model Static load-carrying capacity
1. Introduction The pitch bearing of a wind-power generator is basically a single row four-point contact ball bearing or a double row fourpoint contact ball bearing. The running speed of a pitch bearing is usually very small, therefore, the static model and static loadcarrying capacity, which can be described by the static loadcarrying curves, are mainly considered in pitch bearing design. The impact load acting on the pitch bearing is very large, so a zero, or negative, clearance is considered to reduce fretting wear. Negative clearance not only affects the start-up torque, rotation precision, and stiffness of the bearing, but also affects the load distribution, load capability, and the life of the bearing. It is worth establishing a static model considering the effect of clearance and using it to study the static load-carrying capacity of pitch bearings. Göncz [1] analysed the contact force distribution for a threerow roller slewing bearing considering no clearances and no ring deformations. Aguirrebeitia [2–4] presented the derivation of the general static load-capacity of three-row roller slewing bearings and four-contact point slewing bearings, based on Rumbarger's method. Their calculations assume zero clearance in the contact and rigid rings. A procedure for obtaining the load distribution in a four-contact point slewing bearing considering the effect of the
n
Corresponding author. Tel.: þ 86 37964238583. E-mail address: [email protected] (Y. Wang).
http://dx.doi.org/10.1016/j.ijmecsci.2015.06.023 0020-7403/& 2015 Elsevier Ltd. All rights reserved.
structural elasticity was provided elsewhere [5]. Kunc [6,7] studied the actual carrying capacity of the rolling contact in single-row ball bearings using a computation model considering the material properties, which includes isotropic and kinematic hardening, and cumulative damage. Amasorrain [8] provided a static model taking into account the clearance for a single-row slewing ball bearing and analysed the load distribution; however, the effects induced by the supporting surface of the bearing were not considered. Zupan [9] discussed the carrying angle and carrying capacity of a large single-row ball bearing taking into account the osculation, actual carrying angle, and clearance. Kania [10] presented an FE method for computing the catalogue capacity of a slewing bearing taking into account the flexibility of its rings and clamping bolts. Kania also discussed the problem of calculating the real carrying capacity of a single-row roller slewing bearing by FEM, but did not give the analytical expressions for the proposed static model [11]. Göncz [12] provided a new computational model which considered ring deformations and clearances for determination of the internal contact force distribution and the determination of static load curves for a three-row roller slewing bearing. Glodež [13] presented a similar model for a large slewing ball bearing. However, the vector approach was used in the two reports [12,13] to give a mathematical description of the bearing geometry and the static force and moment equilibrium calculation. Potočnik [14] developed a design method for assessing the static capacity of a large double-row slewing ball bearing, but only gave the vector expression of the static model for a single-row slewing ball
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bearing because it is thought that the contact deformation and contact force of the upper row are same as those of the lower row for the double-row slewing ball bearing, which does not suit actual loading conditions. In the aforementioned papers discussing a double-row slewing ball bearing, the static model in the case of a single-row ball bearing has been described and the differences in contact formation and contact force of each row have not been pointed out [14]. In this paper, the proposed static model for a double-row fourpoint contact ball bearing was felt to be more reasonable than existing models as it could differ from the contact deformation and contact force between the upper row and the lower row. The determination of the contact deformation in existing static models is based on the changes in the initial and final coordinates of the centres of curvature of the raceways, which are complicated, especially for double-row ball bearings. In this research, new static models for single- and double-row ball bearings are presented, in which the determination of the contact deformation was based on the changes in the initial and final distances between the diagonally opposed centres of curvature of the inner and outer raceways. The new static models are as precise as the existing models taking the clearances into account, but they are easier to describe and realise than existing static models. In the aforementioned literatures and elsewhere [15], the static load-carrying capacity curves were presented in the form of a moment-axial-force diagram and the radial force was not taken into consideration; the method of establishing a static loadcarrying curve considering the radial force was not provided. In this research, the method used to establish the precise static loadcarrying curve, considering the effect of radial load, was presented on the basis of the static force model considering the effect of clearance. The effects of structure parameters on the load-carrying capacity of pitch bearings were also analysed. Here, the new static models are based on the supposition that the bearing rings were ideally stiff and elastic contact deformations only existed on the contact area between balls and their raceways. The study of the static models and static load-carrying capacity curves has significance with regards the pre-selection of slewing bearings.
2. Static model considering clearance Published research results [15] show that clearance greatly affects the magnitude of the contact force and its distribution. Accurate calculation of the contact force is essential when establishing the precise static load-carrying curve. So the clearance must to be taken into account when the static model used to compute the contact force is established. In this article, the static models considering clearance for a single-row four-point contact bearing, or a double-row four-point contact bearing, are provided, and a procedure for establishing the static model is not given as it may be found elsewhere [15]. The zero or negative clearance is always used in pitch bearings. Negative clearance not only affects the load distribution and load capability, but also affects the friction torque of the bearing. Taking a double-row four-point ball bearing with the geometry shown in Table 1 as an example, the start-up friction torques measured by the spring balance scale were 2.8 kN m and 3.5 kN m when the clearances were 0 and 0.01 mm respectively, with no external loads. So it was necessary to take the clearance into consideration in the static model of such a pitch bearing. The flexibility of rings and the flexibility of their fastening also affect the distribution of the contact force on the balls in a fourpoint contact ball bearing. Kania [10] has already discussed this
problem. In this research, the deformed states of the rings and support structures of the bearing are not taken into account. The procedure described here supposes that the rings are ideally stiff and only assumes elastic Hertzian contact deformations on the contact area between the balls and their raceways. The outer ring was supposed to be fixed and the external loads, which are an axial load Fa, a radial load Fr, and a tilting moment load M, were applied on the inner ring. Before loading, the distance between the diagonally opposed centres of curvature of the inner and outer raceways, with clearance, can be written as: 1 A ¼ f i þ f e −1 DW − ua cos α0 2
ð1Þ
The initial distance between diagonally opposed centres of curvature with zero clearance is: A0 ¼ ðf i þf e 1ÞDw
ð2Þ
where fi ¼ri/Dw is the coefficient of the inner raceway groove curvature radius, fe ¼re/Dw is the coefficient of the outer raceway groove curvature radius, Dw is the ball nominal diameter, ri and re are the inner and outer raceway groove curvature radii, respectively, ua is the bearing axial clearance, and α0 is the initial contact angle. After loading, the distances between diagonally opposed centres of curvature along the direction of contact pairs k (k ¼1, 2) for a single row four-point contact bearing were defined as Skφj , and can be computed thus: 8 1 < S1φj ¼ ½ðA sin α0 þ δa þ Ri θ cos φj Þ2 þ ðA cos α0 þ δr cos φj Þ2 2 : S2φ ¼ ½ðA sin α0 δa Ri θ cos φj Þ2 þ ðA cos α0 þ δr cos φj Þ2 12 j ð3Þ where the angular position of each ball inside the bearing is φj ¼ 2π ðj 1Þ=Z (j¼1, 2, 3…Z ), Z is the number of balls in a singlerow of the bearing, δa , δr and θ are the axial, radial, and angular displacements, respectively generated in the inner ring relative to the outer ring, and Ri is the radius of the track of the raceway groove curvature centre of the inner ring. Ri is given by: 1 1 Ri ¼ dm þ f i −0:5 Dw cos α0 − ua ð cos α0 Þ2 2 4
ð4Þ
where dm is the pitch diameter of the bearing. The distances between diagonally opposed centres of curvature along the direction of contact pairs k (k ¼1, 2, 3, 4) for a doublerow four-point contact bearing are defined as Akφj , they are: A1φj ¼
A sin α0 þ δa þRi θ cos φj
2
þ ðA cos α0 þ δr cos φj þ 0:5dc θ cos φj
2 12
ð5Þ A2φj ¼
A sin α0 δa Ri θ cos φj
2
þ ðA cos α0 þ δr cos φj þ 0:5dc θ cos φj
2 12
ð6Þ A3φj ¼
A sin α0 þ δa þRi θ cos φj
2
þðA cos α0 þ δr cos φj 0:5dc θ cos φj Þ2
12
ð7Þ A4φj ¼
A sin α0 δa Ri θ cos φj
2
þðA cos α0 þ δr cos φj 0:5dc θ cos φj Þ2
12
ð8Þ where dc is the distance between the centres of the two balls in the upper and lower rows. Under load, the contact angles at angular position φj along the directions of contact pairs k (k ¼1, 2) for a single-row four-point
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Table 1 Pitch bearing parameters. Parameter
Value
Pitch diameter of the bearing dm/mm Ball nominal diameter DW/mm Initial contact angle α0/1 Distance of the centres of the two balls between the upper row and the lower row dc/mm Coefficient of inner raceway groove curvature radius fi Coefficient of outer raceway groove curvature radius fe Number of balls Z Poisson's ratio of balls and rings v Elastic modulus of balls and rings E=GPa
2215 44.45 45 69 0.525 0.525 128 2 0.3 207
contact bearing are given by:
α1φj ¼ arcsin
A sin α0 þ δa þRi θ cos φj S1φj
α2φj ¼ arcsin
A sin α0 δa Ri θ cos φj S2φj
! ð9Þ ! ð10Þ
The contact angles at angular position φj along the directions of contact pairs k (k¼ 1, 2, 3, 4) for a double-row four-point contact bearing are given by: ! A sin α0 þ δa þRi θ cos φj α1φj ¼ arcsin ð11Þ A1φj
α2φj ¼ arcsin
A sin α0 δa Ri θ cos φj A2φj
α3φj ¼ arcsin
A sin α0 þ δa þRi θ cos φj A3φj
α4φj ¼ arcsin
A sin α0 δa Ri θ cos φj A4φj
! ð12Þ ! ð13Þ ! ð14Þ
The inner ring equilibrium equations for a single-row fourpoint contact bearing are: 8 2π X > > > ðQ 1φj sin α1φj Q 2φj sin α2φj Þ F a ¼ 0 > > > > > φj ¼ 0 > > > > 2π < X ðQ 1φj cos α1φj þ Q 2φj cos α2φj Þ cos φj F r ¼ 0 ð15Þ > > φj ¼ 0 > > > > 2π > X > > 1 > d ðQ 1φj sin α1φj Q 2φj sin α2φj Þ cos φj M ¼ 0 m > 2 > : φj ¼ 0
where Q kφj is the normal contact force at angular position φj along the directions of contact pairs k (k ¼1, 2). The inner ring equilibrium equations for a double-row fourpoint contact bearing are: 8 2π X > > > ðQ 1φj sin α1φj Q 2φj sin α2φj þ Q 3φj sin α3φj Q 4φj sin α4φj Þ F a ¼ 0 > > > > φj ¼ 0 > > > > > 2π > X > > > ðQ 1φj cos α1φj þ Q 2φj cos α2φj þ Q 3φj cos α3φj þ Q 4φj cos α4φj Þ cos φj F r ¼ 0 > > > < φj ¼ 0 2π X > > 1 > > d ðQ 1φj sin α1φj Q 2φj sin > >2 m > > φj ¼ 0 > > > > 2π > X > > > 1d c ðQ 1φj cos α1φj þ Q 2φj cos > 2 > > : φj ¼ 0
α2φj þ Q 3φj sin α3φj Q 4φj sin α4φj Þ cos φj þ α2φj Q 3φj cos α3φj Q 4φj cos α4φj Þ cos φj M ¼ 0
ð16Þ
where Q kφj is the normal contact force at angular position φj along the directions of contact pairs k (k ¼1, 2, 3, 4). At angular position φj along the direction of contact pair k, contact deformation δkφj due to normal contact force acting on the balls and their raceways can be calculated by using the following formulae: For a single-row four-point contact bearing:
δkφj ¼ Skφj A0 ðk ¼ 1; 2Þ
ð17Þ
For a double-row four-point contact bearing:
δkφj ¼ Akφj A0 ðk ¼ 1; 2; 3; 4Þ
ð18Þ
According to Hertz theory, at angular position φj along the direction of contact pair k, the relationship between the deformation of each ball, and the normal contact force which produces it, is given by [16]: 8 < K n δ1:5 kφj ; 8 δkφj Z 0 ð19Þ Q kφj ¼ : 0; 8 δkφ o 0 j where Kn is the load-induced deflection: for a steel ball–steel raceway contact, it is given by: h i 3=2 X X K n ¼ 2:15 105 nδi ð ρ Þ1=3 þ nδe ð ρ Þ1=3 ð20Þ i
e
P P where ρi and ρe are the total curvatures of the ball-inner and ball-outer raceway contacts respectively, nδi and nδe are dimensionless quantities relating to the functions of the curvature differences for the ball-inner and ball-outer raceway contacts respectively.
3. Determination of precise static load-carrying capacity curve 3.1. Coefficient of safety of the pitch bearing For a point contact, the maximum contact stress at the centre of the most heavily loaded ball–raceway contact is given by:
σ max ¼
3 Q max 2 πab
ð21Þ
where Qmax is the maximum contact force, a is the semi-major axis of the projected contact ellipse, b is the semi-minor axis of the projected contact ellipse: a and b can be calculated by [16]: 1 3Q max 3 η a ¼ na P ð22Þ 2 ρ b ¼ nb
1 3Q max 3 P η 2 ρ
ð23Þ
where na is the dimensionless semi-major axis of the projected contact ellipse, nb is the dimensionless semi-minor axis of the P projected contact ellipse, ρ is the total curvature of the ball–
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raceway contact, η ¼
ð1 ν21 Þ E1
þ
ð1 ν22 Þ E2
is the equivalent modulus of
elasticity, E1 and E2 are the moduli of elasticity of the contact pair, and ν1 and ν2 are Poisson's ratios of the contact pair. Substituting Eqs. (22) and (23) into Eq. (21) gives: " P #1=3 1 3 ρ 2 σ max ¼ Q max ð24Þ π na nb 2 η The allowable stress ½σ max is defined as the contact stress applied to a non-rotating bearing that will result in a total permanent deformation of 0.0001 of the ball diameter DW at the centre of the most heavily loaded element–raceway contact. If the ring is made of 42CrMo steel, its hardness is 55HRC and its hardened depth is larger than 0.1 DW, the allowable stress is suggested to be 4200 MPa [17]. The coefficient of safety of the pitch bearing according to the industry standard [17] can be expressed by: 3 f s ¼ ½σ max =σ max ð25Þ 3 1=3 when f s 4 ½σ max =σ max or σ max o ½σ max =f s , the bearing is in a safe operating condition with regard to the static load-carrying capacity. 3.2. The method used to establish the precise static load-carrying capacity surface or curves The points making up the static load-carrying capacity curve indicated the load combinations that result in static failure of the most loaded element. The method used to establish the precise static load-carrying capacity surface, or curves, is as follows: (1) In Eq. (15), for a single-row pitch ball bearing, or Eq. (16) for a double-row pitch ball bearing, let the radial load be zero and let axial force and tilting moment continuously change within a certain range, the maximum contact forces acting at the centre of the most heavily loaded ball–raceway contact corresponding to different axial forces and tilting moments can then be obtained. The maximum contact stresses corresponding to the maximum contact forces can be obtained by using Eq. (21). (2) If the value of the maximum contact stress approaches ½σ max =f 1=3 s , the corresponding axial forces, tilting moments, and radial forces are chosen as the points used to draw the static load-capacity curves or surface. (3) Change the value of radial load and repeat steps (1) and (2), the static load-carrying capacity surface or curves can then be obtained.
3.3. The method used to establish the simplified static load-carrying capacity curve as used widely in engineering The external axial load on a bearing can be calculated by: F a ¼ ½Q max z sin α0 J a ðε1 Þ for a single row ball bearing
ð26Þ
F a ¼ ½Q max z sin α0 J a ðε1 ; ε2 Þ for a double row ball bearing
ð27Þ
The external tilting moment on a bearing can be calculated by: M ¼ ½Q max zdm sin α0 J m ðε1 Þ for a single row ball bearing
ð28Þ
M ¼ ½Q max zdm sin α0 J m ðε1 ; ε2 Þ for a double row ball bearing ð29Þ where z is the number of bearing balls, J a ðε1 Þ and J a ðε1 ; ε2 Þ are the integral coefficients of the axial load, J m ðε1 Þ and J m ðε1 ; ε2 Þ are the integral coefficients of the tilting moment, and ε1 and ε2 are load
distribution factors: they satisfy the following relationship: ( ε1 þ ε2 ¼ 1 ε1 r1
ε2 ¼ 0
ε1 41
ð30Þ
½Q max is the allowable maximum contact force, which can be derived from Eq. (24) and given by: 2 η 2 P ½Q max ¼ ðπ na nb ½σ max Þ3 ð31Þ 3 ρ The method used to establish the simple static load-carrying capacity curve is as follows: (1) Let ε1 be any value, the corresponding value of ε2 may be obtained from Eq. (30). (2) Compute the allowable maximum contact force ½Q max . (3) The integral coefficients J a ðε1 Þ, J a ðε1 ; ε2 Þ, J m ðε1 Þ, and J m ðε1 ; ε2 Þ can be obtained by consulting the literature [16] or other work according to the values of ε1 and ε2 . (4) Substitute the values of ½Q max , J a ðε1 Þ, and J m ðε1 Þ into Eqs. (26) and (28) to obtain a series of values of axial load F a and tilting moment M for a single-row ball bearing. Substitute the values of ½Q max , J a ðε1 ; ε2 Þ, and J m ðε1 ; ε2 Þ into Eqs. (27) and (29) to obtain a series of values of axial load F a and tilting moment M for a double-row ball bearing. (5) The above axial and tilting moment loads are chosen as the points used to draw the simple static load-capacity curve. 4. Results and discussion 4.1. Static load-carrying capacity surface and curves of a pitch bearing A practical example was made of a double-row four-point contact ball bearing with the geometry shown in Table 1. The rings and balls in this bearing were made of 42CrMo steel. The raceways were induction quenched, their hardness was between 55 and 60 HRC and their depths were greater than 4 mm. The allowable contact stress was 4200 MPa, and the value of f s was taken as 1.5. The precise static load-carrying capacity curve at zero radial load is shown in Fig. 1. The corresponding simplified static loadcarrying capacity curve is shown in Fig. 2. Figs. 1 and 2 show that the scopes of the axial load and tilting moment that the bearing can carry, as obtained by the computing method used to find the precise static load-carrying capacity, are bigger than that found by the simplified static load-carrying capacity method as used widely in engineering practice. If the load-carrying capacity of the bearing is judged using the simplified static load-carrying capacity curve, an acceptable product may be considered unacceptable. The precise static load-carrying capacity surface of a double-row four-point ball bearing is shown in Fig. 3. 4.2. Effects of geometry parameters of bearing on the static loadcarrying capacity Some geometric pitch bearing parameters such as: the bearing basic bore diameter, bearing basic outside diameter, ball nominal diameter, and bearing pitch diameter have been standardised; however, there are some geometric parameters such as: clearance, coefficient of raceway groove curvature radius, and contact angle, which are designed by experience. The values of clearance, coefficient of raceway groove curvature radius, and contact angle affect the load-carrying capacity of such bearings. So it is necessary to know the effects of these parameters on the bearing performance.
Y. Wang, J. Cao / International Journal of Mechanical Sciences 100 (2015) 209–215
× 10
3
3
12 10 8 6 4 2 0
×10
14
Tilting moment ( kN • m )
Tilting moment ( kN • m )
14
0
213
5
10 15 20 Axial load(kN)
25
ur=-0.08mm
10
ur=-0.1mm
8
ur=-0.2mm
6 4 2 0
30 × 10
ur=0mm
12
3
0
5
10
15
20
25
30 ×10
Axial load (kN)
3
Fig. 1. Precise static load-carrying capacity curve. Fig. 4. Static load-carrying capacity curves for different clearances.
× 10
3
10
14
8 6 4 2 0
0
5
10 15 Axial load(kN)
20
25 × 10
3
Fig. 2. Simplified static load-carrying capacity curve.
3 kN m) Tilting moment(×10
14
Tilting moment ( kN • m )
Tilting moment (kN • m)
12
3
12 10
×10
Fr=0 Fr =200
ur=0mm ur=-0.08mm
8 Fr =400 6 F =600 r 4
Fr =800
2 0 Fr=1000 0 5
10 15 20 Axial load (kN)
25
30 3 ×10
Fig. 5. Effect of clearance on static load-carrying capability for different radial loads.
12 10 8
effect in the other load cases analysed. The effects of clearance on the static load-carrying capacity for different radial loads are shown in Fig. 5, in which the radial load was in kN. Fig. 5 shows that the static load-carrying curves had similar trends under different radial loads. With an increasing radial load, the effect of clearance on the load-carrying capacity of bearing decreased.
6
load ic a l Rad 03 kN) (×1
4 2 0 2 4 6 8 10
0
25 15 5 20 10 5 (×10 kN) Axial load
Fig. 3. Precise static load-carrying capacity surface.
4.2.1. Effect of clearance on the load-carrying capacity of pitch bearings When the radial clearances were: 0 mm, 0.08 mm, 0.1 mm, and 0.2 mm, respectively, the static load-capacity curves are as shown in Fig. 4. It can be seen from Fig. 4 that the tilting moment load increased first and then decreased with increasing axial load at a clearance of 0, 0.08, and 0.1 mm, while the tilting moment load decreased with increasing axial load at a clearance of 0.2 mm. The maximum axial load which the bearing can carry decreased, while the maximum tilting moment which the bearing can carry increased with an increasing absolute value of the clearance. The clearance had a significant effect on the static load-carrying capacity of the bearing only in load cases with a rather small axial force and a large tilting moment, and it had little
4.2.2. Effect of coefficient of raceway groove curvature radius on load-carrying capacity of the bearing The coefficient of raceway groove curvature radius is an important parameter in bearing design. It affects the osculation and deformation of the ball–raceway contact, as well as the load capacity of the bearing. When the coefficients of raceway groove curvature radius were 0.515, 0.52, 0.525, 0.53, and 0.535, respectively, the static load-carrying capacity curves are as shown in Fig. 6. The tilting moment load decreased with increasing axial load when the coefficient of raceway groove curvature radius was less than 0.52. The tilting moment load first increased, and then decreased, with increasing axial load when the coefficient of raceway groove curvature radius was greater than 0.52. When the coefficient of raceway groove curvature radius increased, the maximum axial and tilting moment loads decreased, and the static load-carrying capacity decreased. This was mainly because the coefficient of raceway groove curvature radius affected the osculation of the ball–raceway contact: as the coefficient of the raceway groove curvature radius decreased, the osculation of the ball– raceway contact increased and the contact area increased resulting
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Y. Wang, J. Cao / International Journal of Mechanical Sciences 100 (2015) 209–215
3
16
16
fi=0.52
14
14
fi=0.525
12
fi=0.53
10
fi=0.535
8 6 4
Tilting moment (kN•m)
Tilting moment (kN•m)
× 10
fi=0.515
18
2 0
5
10
15 20 25 Axial load (kN)
30
35
α0=40 ° α0=45 ° α0=50 °
10
α0=55 °
8
α0=60 °
6 4 2
40 3 × 10
Fig. 6. Static load-carrying capacity curves for different coefficients of raceway groove curvature radius.
3
12
0
0
× 10
0
5
10
15
20
25
Axial load (kN)
303
× 10
Fig. 8. Static load-carrying capacity curves for different initial contact angles.
3
14
3
×10
fi=0.525
Fr=0
14
fi=0.52
Fr =200
12
Tilting moment (kN•m)
Tilting moment (kN•m)
16
10 Fr =400 8
Fr =600
6 4 2 0
Fr =800 Fr=1000 0
5
10 15 20 Axial load (kN)
25
12 10
in an increased load-carrying capacity. Fig. 7 shows the effect of the coefficient of raceway groove curvature radius on the static load-carrying capacity for different radial loads: the static loadcarrying capacity curves had similar trends at different radial loads. The smaller the radial load, the more significant the effect the coefficient of raceway groove curvature radius had on the static load-carrying capacity of the bearing.
4.2.3. Effect of initial contact angle on load-carrying capacity of bearing The initial contact angle of the bearing can also affect its loadcarrying capacity. The effect of the initial contact angle on the load-carrying capacity differed for different radial loads. When the radial force was zero, the static load-capacity curves for different initial contact angles are shown in Fig. 8. This showed that the tilting moment load first increased, and then decreased, with increasing axial load for initial contact angles of 401 and 451. The tilting moment load decreased with increasing axial load when the initial contact angles were 501, 551, and 601. As the initial contact angle increased, the static load-carrying capacity of the bearing increased. The effect of the initial contact angle on the static loadcarrying capacity curves for different radial loads are shown in Fig. 9. This showed that the effect of the initial contact angle on the static load-carrying differed for different radial loads. When the radial load varied between 0 to 200 kN, the load-carrying capacity increased with increasing initial contact angle. When the radial load varied from 400 to 600 kN, the initial contact angle had little
α0=45°
Fr=0
α0=50°
Fr=200
8 F =400 r 6 F =600 r 4 Fr=800 2 F =1000 0 r 0 5
30 3 ×10
Fig. 7. Effect of coefficient of raceway groove curvature radius on static loadcarrying capacity for different radial loads.
×10
10 15 20 Axial load (kN)
25
30 3
×10
Fig. 9. Effect of initial contact angle on static load-carrying capacity curves for different radial loads.
effect on the load-carrying capacity. When the radial loads exceeded 800 kN, the load-carrying capacity decreased with increasing initial contact angle.
5. Conclusion Static models taking account of clearance were proposed for a single-row, and a double-row, four-point contact pitch ball bearing. A precise computation method based, on the static model considering clearance, was presented for the static load-carrying capacity curves, or surface, of pitch bearings. With increasing radial load, the effect of clearance on the load-carrying capacity of the bearing decreased. The clearance had a significant effect on the static load-carrying capacity of the bearing only in load cases with a rather small axial force and a large tilting moment, it had little effect in the other load cases evaluated. The smaller the radial load, the more significant the effect of the coefficient of raceway groove curvature radius had on the static load-carrying capacity of the bearing. When the coefficient of raceway groove curvature radius increased, the maximum axial and tilting moment loads decreased as did the static load-carrying capacity decreased. When the radial load ranged from 0 to 200 kN, the load-carrying capacity increased with increasing initial contact angle. When the radial load ranged from 400 to 600 kN, the initial contact angle had little effect on the load-carrying capacity. When the radial loads exceeded 800 kN, the load-carrying capacity decreased as the initial contact angle increased.
Y. Wang, J. Cao / International Journal of Mechanical Sciences 100 (2015) 209–215
Acknowledgement This research was supported by the NSFC (Grant nos. 51475143 and 51105131), a Henan Province Key Scientific and Technological Project (Grant no. 142102210110), and the Tianjin Science and Technology Support Programme (Grant no. 14ZCDZGX00803).
References [1] Göncz P, Potočnik R, Glodež S. Load capacity of a three-row roller slewing bearing raceway. Procedia Eng 2011;10:1196–201. [2] Aguirrebeitia J, Abasolo Mikel, Avilés Rafael, et al. Theoretical calculation of general static load-carrying capacity for the design and selection of three row roller slewing bearings. Mech Mach Theory 2012;48:52–61. [3] Aguirrebeitia J, Avilés R, Bustos IF De, et al. General static load capacity in four contact point slewing bearings. Tribol Des 2010;66:17–23. [4] Aguirrebeitia J, Abasolo M, Avilés R, et al. General static load-carrying capacity for the design and selection of four contact point slewing bearings: finite element calculations and theoretical model validation. Finite Elem Anal Des 2012;55:23–30. [5] Olave Mireia, Sagartzazu Xabier, Damian Jorge, et al. Design of four contactpoint slewing bearing with a new load distribution procedure to account for structural stiffness. J Mech Des 2010;132:021006-1–10. [6] Robert Kunc, Prebil Ivan. Numerical determination of carrying capacity of large rolling bearings. J Mater Process Technol 2004;155–156:1696–703.
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[7] Kunc Robert, Žerovnik Andrej, Prebil Ivan. Verification of numerical determination of carrying capacity of large rolling bearing with hardened raceway. Int J Fatigue 2007;29:1913–9. [8] Amasorrain JI, Sagartzazu X, Damián J. Load distribution in a four contactpoint slewing bearing. Mech Mach Theory 2003;38:479–96. [9] Zupan S, Prebil I. Carrying angle and carrying capacity of a large single row ball bearing as a function of geometry parameters of the rolling contact and the supporting structure stiffness. Mech Mach Theory 2001;36:1087–103. [10] Kania Ludwik, Krynke Marek, Mazanek Eugeniusz. A catalogue capacity of slewing bearings. Mech Mach Theory 2012;58:29–45. [11] Kania Ludwik, Krynke Marek. Computation of the general carrying capacity of slewing bearings. Eng Comput: Int J Comput Aided Eng Softw 2013;30 (7):1011–28. [12] Göncz P, Potočni R, Glodež S. Computational model for determination of static load capacity of three-row roller slewing bearings with arbitrary clearances and predefined raceway deformations. Int J Mech Sci 2013;73:82–92. [13] Glodež S, Potočnik R, Flašker J. Computational model for calculation of static capacity and lifetime of large slewing bearing's raceway. Mech Mach Theory 2012;47:16–30. [14] Potočnik Rok, Göncz Peter, Glodež Srečko. Static capacity of a large double row slewing ball bearing with predefined irregular geometry. Mech Mach Theory 2013;64:67–79. [15] Wang Yanshuang, Yuan Qianqian. Contact force distribution and static loadcarrying capacity of large size double row four-point contact ball bearing. Def Technol 2013;4:229–36. [16] Harris TA, Kotzalas MN. Rolling bearing analysis. 5th ed. Boca Raton: Taylor & Francis Group; 2006. [17] Harris T, Rumbarger JH, Butterfield. CP. Wind turbine design guideline – DG03: Yaw and pitch rolling bearing life. Technical report. National Renewable Energy Laboratory, Golden; 2009.
Contents lists available at ScienceDirect
International Journal of Mechanical Sciences journal homepage: www.elsevier.com/locate/ijmecsci
Determination of the precise static load-carrying capacity of pitch bearings based on static models considering clearance Yanshuang Wang n, Jiawei Cao Tianjin University of Technology and Education, Tianjin 300222, China
art ic l e i nf o
a b s t r a c t
Article history: Received 6 December 2013 Received in revised form 12 June 2015 Accepted 28 June 2015 Available online 11 July 2015
Static models for a single row and a double row four-point contact pitch bearing, taking into account the clearances therein, were presented. A precise computation method was proposed for the static loadcarrying capacity curves of such pitch bearings, which can be used to pre-select pitch bearings and slewing bearings supposing rigid bearing rings. The effects on the static load-carrying capacity induced by changing the clearance, the raceway groove radius of curvature, and initial contact angle were analysed. The clearance has a significant effect on the static load-carrying capacity of the bearing only in load cases with a rather small axial force and a large tilting moment. When the coefficient of raceway groove curvature radius increases, the static load-carrying capacity decreases. The smaller the radial load, the more significant the effect of the coefficient of raceway groove curvature radius on the static load-carrying capacity of the bearing. When radial loads range from 0 to 800 kN, the load-carrying capacity increases with increasing initial contact angle. When the radial loads are above 800 kN, the load-carrying capacity decreases with increasing initial contact angle. & 2015 Elsevier Ltd. All rights reserved.
Keywords: Pitch bearing Negative clearance Static model Static load-carrying capacity
1. Introduction The pitch bearing of a wind-power generator is basically a single row four-point contact ball bearing or a double row fourpoint contact ball bearing. The running speed of a pitch bearing is usually very small, therefore, the static model and static loadcarrying capacity, which can be described by the static loadcarrying curves, are mainly considered in pitch bearing design. The impact load acting on the pitch bearing is very large, so a zero, or negative, clearance is considered to reduce fretting wear. Negative clearance not only affects the start-up torque, rotation precision, and stiffness of the bearing, but also affects the load distribution, load capability, and the life of the bearing. It is worth establishing a static model considering the effect of clearance and using it to study the static load-carrying capacity of pitch bearings. Göncz [1] analysed the contact force distribution for a threerow roller slewing bearing considering no clearances and no ring deformations. Aguirrebeitia [2–4] presented the derivation of the general static load-capacity of three-row roller slewing bearings and four-contact point slewing bearings, based on Rumbarger's method. Their calculations assume zero clearance in the contact and rigid rings. A procedure for obtaining the load distribution in a four-contact point slewing bearing considering the effect of the
n
Corresponding author. Tel.: þ 86 37964238583. E-mail address: [email protected] (Y. Wang).
http://dx.doi.org/10.1016/j.ijmecsci.2015.06.023 0020-7403/& 2015 Elsevier Ltd. All rights reserved.
structural elasticity was provided elsewhere [5]. Kunc [6,7] studied the actual carrying capacity of the rolling contact in single-row ball bearings using a computation model considering the material properties, which includes isotropic and kinematic hardening, and cumulative damage. Amasorrain [8] provided a static model taking into account the clearance for a single-row slewing ball bearing and analysed the load distribution; however, the effects induced by the supporting surface of the bearing were not considered. Zupan [9] discussed the carrying angle and carrying capacity of a large single-row ball bearing taking into account the osculation, actual carrying angle, and clearance. Kania [10] presented an FE method for computing the catalogue capacity of a slewing bearing taking into account the flexibility of its rings and clamping bolts. Kania also discussed the problem of calculating the real carrying capacity of a single-row roller slewing bearing by FEM, but did not give the analytical expressions for the proposed static model [11]. Göncz [12] provided a new computational model which considered ring deformations and clearances for determination of the internal contact force distribution and the determination of static load curves for a three-row roller slewing bearing. Glodež [13] presented a similar model for a large slewing ball bearing. However, the vector approach was used in the two reports [12,13] to give a mathematical description of the bearing geometry and the static force and moment equilibrium calculation. Potočnik [14] developed a design method for assessing the static capacity of a large double-row slewing ball bearing, but only gave the vector expression of the static model for a single-row slewing ball
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bearing because it is thought that the contact deformation and contact force of the upper row are same as those of the lower row for the double-row slewing ball bearing, which does not suit actual loading conditions. In the aforementioned papers discussing a double-row slewing ball bearing, the static model in the case of a single-row ball bearing has been described and the differences in contact formation and contact force of each row have not been pointed out [14]. In this paper, the proposed static model for a double-row fourpoint contact ball bearing was felt to be more reasonable than existing models as it could differ from the contact deformation and contact force between the upper row and the lower row. The determination of the contact deformation in existing static models is based on the changes in the initial and final coordinates of the centres of curvature of the raceways, which are complicated, especially for double-row ball bearings. In this research, new static models for single- and double-row ball bearings are presented, in which the determination of the contact deformation was based on the changes in the initial and final distances between the diagonally opposed centres of curvature of the inner and outer raceways. The new static models are as precise as the existing models taking the clearances into account, but they are easier to describe and realise than existing static models. In the aforementioned literatures and elsewhere [15], the static load-carrying capacity curves were presented in the form of a moment-axial-force diagram and the radial force was not taken into consideration; the method of establishing a static loadcarrying curve considering the radial force was not provided. In this research, the method used to establish the precise static loadcarrying curve, considering the effect of radial load, was presented on the basis of the static force model considering the effect of clearance. The effects of structure parameters on the load-carrying capacity of pitch bearings were also analysed. Here, the new static models are based on the supposition that the bearing rings were ideally stiff and elastic contact deformations only existed on the contact area between balls and their raceways. The study of the static models and static load-carrying capacity curves has significance with regards the pre-selection of slewing bearings.
2. Static model considering clearance Published research results [15] show that clearance greatly affects the magnitude of the contact force and its distribution. Accurate calculation of the contact force is essential when establishing the precise static load-carrying curve. So the clearance must to be taken into account when the static model used to compute the contact force is established. In this article, the static models considering clearance for a single-row four-point contact bearing, or a double-row four-point contact bearing, are provided, and a procedure for establishing the static model is not given as it may be found elsewhere [15]. The zero or negative clearance is always used in pitch bearings. Negative clearance not only affects the load distribution and load capability, but also affects the friction torque of the bearing. Taking a double-row four-point ball bearing with the geometry shown in Table 1 as an example, the start-up friction torques measured by the spring balance scale were 2.8 kN m and 3.5 kN m when the clearances were 0 and 0.01 mm respectively, with no external loads. So it was necessary to take the clearance into consideration in the static model of such a pitch bearing. The flexibility of rings and the flexibility of their fastening also affect the distribution of the contact force on the balls in a fourpoint contact ball bearing. Kania [10] has already discussed this
problem. In this research, the deformed states of the rings and support structures of the bearing are not taken into account. The procedure described here supposes that the rings are ideally stiff and only assumes elastic Hertzian contact deformations on the contact area between the balls and their raceways. The outer ring was supposed to be fixed and the external loads, which are an axial load Fa, a radial load Fr, and a tilting moment load M, were applied on the inner ring. Before loading, the distance between the diagonally opposed centres of curvature of the inner and outer raceways, with clearance, can be written as: 1 A ¼ f i þ f e −1 DW − ua cos α0 2
ð1Þ
The initial distance between diagonally opposed centres of curvature with zero clearance is: A0 ¼ ðf i þf e 1ÞDw
ð2Þ
where fi ¼ri/Dw is the coefficient of the inner raceway groove curvature radius, fe ¼re/Dw is the coefficient of the outer raceway groove curvature radius, Dw is the ball nominal diameter, ri and re are the inner and outer raceway groove curvature radii, respectively, ua is the bearing axial clearance, and α0 is the initial contact angle. After loading, the distances between diagonally opposed centres of curvature along the direction of contact pairs k (k ¼1, 2) for a single row four-point contact bearing were defined as Skφj , and can be computed thus: 8 1 < S1φj ¼ ½ðA sin α0 þ δa þ Ri θ cos φj Þ2 þ ðA cos α0 þ δr cos φj Þ2 2 : S2φ ¼ ½ðA sin α0 δa Ri θ cos φj Þ2 þ ðA cos α0 þ δr cos φj Þ2 12 j ð3Þ where the angular position of each ball inside the bearing is φj ¼ 2π ðj 1Þ=Z (j¼1, 2, 3…Z ), Z is the number of balls in a singlerow of the bearing, δa , δr and θ are the axial, radial, and angular displacements, respectively generated in the inner ring relative to the outer ring, and Ri is the radius of the track of the raceway groove curvature centre of the inner ring. Ri is given by: 1 1 Ri ¼ dm þ f i −0:5 Dw cos α0 − ua ð cos α0 Þ2 2 4
ð4Þ
where dm is the pitch diameter of the bearing. The distances between diagonally opposed centres of curvature along the direction of contact pairs k (k ¼1, 2, 3, 4) for a doublerow four-point contact bearing are defined as Akφj , they are: A1φj ¼
A sin α0 þ δa þRi θ cos φj
2
þ ðA cos α0 þ δr cos φj þ 0:5dc θ cos φj
2 12
ð5Þ A2φj ¼
A sin α0 δa Ri θ cos φj
2
þ ðA cos α0 þ δr cos φj þ 0:5dc θ cos φj
2 12
ð6Þ A3φj ¼
A sin α0 þ δa þRi θ cos φj
2
þðA cos α0 þ δr cos φj 0:5dc θ cos φj Þ2
12
ð7Þ A4φj ¼
A sin α0 δa Ri θ cos φj
2
þðA cos α0 þ δr cos φj 0:5dc θ cos φj Þ2
12
ð8Þ where dc is the distance between the centres of the two balls in the upper and lower rows. Under load, the contact angles at angular position φj along the directions of contact pairs k (k ¼1, 2) for a single-row four-point
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211
Table 1 Pitch bearing parameters. Parameter
Value
Pitch diameter of the bearing dm/mm Ball nominal diameter DW/mm Initial contact angle α0/1 Distance of the centres of the two balls between the upper row and the lower row dc/mm Coefficient of inner raceway groove curvature radius fi Coefficient of outer raceway groove curvature radius fe Number of balls Z Poisson's ratio of balls and rings v Elastic modulus of balls and rings E=GPa
2215 44.45 45 69 0.525 0.525 128 2 0.3 207
contact bearing are given by:
α1φj ¼ arcsin
A sin α0 þ δa þRi θ cos φj S1φj
α2φj ¼ arcsin
A sin α0 δa Ri θ cos φj S2φj
! ð9Þ ! ð10Þ
The contact angles at angular position φj along the directions of contact pairs k (k¼ 1, 2, 3, 4) for a double-row four-point contact bearing are given by: ! A sin α0 þ δa þRi θ cos φj α1φj ¼ arcsin ð11Þ A1φj
α2φj ¼ arcsin
A sin α0 δa Ri θ cos φj A2φj
α3φj ¼ arcsin
A sin α0 þ δa þRi θ cos φj A3φj
α4φj ¼ arcsin
A sin α0 δa Ri θ cos φj A4φj
! ð12Þ ! ð13Þ ! ð14Þ
The inner ring equilibrium equations for a single-row fourpoint contact bearing are: 8 2π X > > > ðQ 1φj sin α1φj Q 2φj sin α2φj Þ F a ¼ 0 > > > > > φj ¼ 0 > > > > 2π < X ðQ 1φj cos α1φj þ Q 2φj cos α2φj Þ cos φj F r ¼ 0 ð15Þ > > φj ¼ 0 > > > > 2π > X > > 1 > d ðQ 1φj sin α1φj Q 2φj sin α2φj Þ cos φj M ¼ 0 m > 2 > : φj ¼ 0
where Q kφj is the normal contact force at angular position φj along the directions of contact pairs k (k ¼1, 2). The inner ring equilibrium equations for a double-row fourpoint contact bearing are: 8 2π X > > > ðQ 1φj sin α1φj Q 2φj sin α2φj þ Q 3φj sin α3φj Q 4φj sin α4φj Þ F a ¼ 0 > > > > φj ¼ 0 > > > > > 2π > X > > > ðQ 1φj cos α1φj þ Q 2φj cos α2φj þ Q 3φj cos α3φj þ Q 4φj cos α4φj Þ cos φj F r ¼ 0 > > > < φj ¼ 0 2π X > > 1 > > d ðQ 1φj sin α1φj Q 2φj sin > >2 m > > φj ¼ 0 > > > > 2π > X > > > 1d c ðQ 1φj cos α1φj þ Q 2φj cos > 2 > > : φj ¼ 0
α2φj þ Q 3φj sin α3φj Q 4φj sin α4φj Þ cos φj þ α2φj Q 3φj cos α3φj Q 4φj cos α4φj Þ cos φj M ¼ 0
ð16Þ
where Q kφj is the normal contact force at angular position φj along the directions of contact pairs k (k ¼1, 2, 3, 4). At angular position φj along the direction of contact pair k, contact deformation δkφj due to normal contact force acting on the balls and their raceways can be calculated by using the following formulae: For a single-row four-point contact bearing:
δkφj ¼ Skφj A0 ðk ¼ 1; 2Þ
ð17Þ
For a double-row four-point contact bearing:
δkφj ¼ Akφj A0 ðk ¼ 1; 2; 3; 4Þ
ð18Þ
According to Hertz theory, at angular position φj along the direction of contact pair k, the relationship between the deformation of each ball, and the normal contact force which produces it, is given by [16]: 8 < K n δ1:5 kφj ; 8 δkφj Z 0 ð19Þ Q kφj ¼ : 0; 8 δkφ o 0 j where Kn is the load-induced deflection: for a steel ball–steel raceway contact, it is given by: h i 3=2 X X K n ¼ 2:15 105 nδi ð ρ Þ1=3 þ nδe ð ρ Þ1=3 ð20Þ i
e
P P where ρi and ρe are the total curvatures of the ball-inner and ball-outer raceway contacts respectively, nδi and nδe are dimensionless quantities relating to the functions of the curvature differences for the ball-inner and ball-outer raceway contacts respectively.
3. Determination of precise static load-carrying capacity curve 3.1. Coefficient of safety of the pitch bearing For a point contact, the maximum contact stress at the centre of the most heavily loaded ball–raceway contact is given by:
σ max ¼
3 Q max 2 πab
ð21Þ
where Qmax is the maximum contact force, a is the semi-major axis of the projected contact ellipse, b is the semi-minor axis of the projected contact ellipse: a and b can be calculated by [16]: 1 3Q max 3 η a ¼ na P ð22Þ 2 ρ b ¼ nb
1 3Q max 3 P η 2 ρ
ð23Þ
where na is the dimensionless semi-major axis of the projected contact ellipse, nb is the dimensionless semi-minor axis of the P projected contact ellipse, ρ is the total curvature of the ball–
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raceway contact, η ¼
ð1 ν21 Þ E1
þ
ð1 ν22 Þ E2
is the equivalent modulus of
elasticity, E1 and E2 are the moduli of elasticity of the contact pair, and ν1 and ν2 are Poisson's ratios of the contact pair. Substituting Eqs. (22) and (23) into Eq. (21) gives: " P #1=3 1 3 ρ 2 σ max ¼ Q max ð24Þ π na nb 2 η The allowable stress ½σ max is defined as the contact stress applied to a non-rotating bearing that will result in a total permanent deformation of 0.0001 of the ball diameter DW at the centre of the most heavily loaded element–raceway contact. If the ring is made of 42CrMo steel, its hardness is 55HRC and its hardened depth is larger than 0.1 DW, the allowable stress is suggested to be 4200 MPa [17]. The coefficient of safety of the pitch bearing according to the industry standard [17] can be expressed by: 3 f s ¼ ½σ max =σ max ð25Þ 3 1=3 when f s 4 ½σ max =σ max or σ max o ½σ max =f s , the bearing is in a safe operating condition with regard to the static load-carrying capacity. 3.2. The method used to establish the precise static load-carrying capacity surface or curves The points making up the static load-carrying capacity curve indicated the load combinations that result in static failure of the most loaded element. The method used to establish the precise static load-carrying capacity surface, or curves, is as follows: (1) In Eq. (15), for a single-row pitch ball bearing, or Eq. (16) for a double-row pitch ball bearing, let the radial load be zero and let axial force and tilting moment continuously change within a certain range, the maximum contact forces acting at the centre of the most heavily loaded ball–raceway contact corresponding to different axial forces and tilting moments can then be obtained. The maximum contact stresses corresponding to the maximum contact forces can be obtained by using Eq. (21). (2) If the value of the maximum contact stress approaches ½σ max =f 1=3 s , the corresponding axial forces, tilting moments, and radial forces are chosen as the points used to draw the static load-capacity curves or surface. (3) Change the value of radial load and repeat steps (1) and (2), the static load-carrying capacity surface or curves can then be obtained.
3.3. The method used to establish the simplified static load-carrying capacity curve as used widely in engineering The external axial load on a bearing can be calculated by: F a ¼ ½Q max z sin α0 J a ðε1 Þ for a single row ball bearing
ð26Þ
F a ¼ ½Q max z sin α0 J a ðε1 ; ε2 Þ for a double row ball bearing
ð27Þ
The external tilting moment on a bearing can be calculated by: M ¼ ½Q max zdm sin α0 J m ðε1 Þ for a single row ball bearing
ð28Þ
M ¼ ½Q max zdm sin α0 J m ðε1 ; ε2 Þ for a double row ball bearing ð29Þ where z is the number of bearing balls, J a ðε1 Þ and J a ðε1 ; ε2 Þ are the integral coefficients of the axial load, J m ðε1 Þ and J m ðε1 ; ε2 Þ are the integral coefficients of the tilting moment, and ε1 and ε2 are load
distribution factors: they satisfy the following relationship: ( ε1 þ ε2 ¼ 1 ε1 r1
ε2 ¼ 0
ε1 41
ð30Þ
½Q max is the allowable maximum contact force, which can be derived from Eq. (24) and given by: 2 η 2 P ½Q max ¼ ðπ na nb ½σ max Þ3 ð31Þ 3 ρ The method used to establish the simple static load-carrying capacity curve is as follows: (1) Let ε1 be any value, the corresponding value of ε2 may be obtained from Eq. (30). (2) Compute the allowable maximum contact force ½Q max . (3) The integral coefficients J a ðε1 Þ, J a ðε1 ; ε2 Þ, J m ðε1 Þ, and J m ðε1 ; ε2 Þ can be obtained by consulting the literature [16] or other work according to the values of ε1 and ε2 . (4) Substitute the values of ½Q max , J a ðε1 Þ, and J m ðε1 Þ into Eqs. (26) and (28) to obtain a series of values of axial load F a and tilting moment M for a single-row ball bearing. Substitute the values of ½Q max , J a ðε1 ; ε2 Þ, and J m ðε1 ; ε2 Þ into Eqs. (27) and (29) to obtain a series of values of axial load F a and tilting moment M for a double-row ball bearing. (5) The above axial and tilting moment loads are chosen as the points used to draw the simple static load-capacity curve. 4. Results and discussion 4.1. Static load-carrying capacity surface and curves of a pitch bearing A practical example was made of a double-row four-point contact ball bearing with the geometry shown in Table 1. The rings and balls in this bearing were made of 42CrMo steel. The raceways were induction quenched, their hardness was between 55 and 60 HRC and their depths were greater than 4 mm. The allowable contact stress was 4200 MPa, and the value of f s was taken as 1.5. The precise static load-carrying capacity curve at zero radial load is shown in Fig. 1. The corresponding simplified static loadcarrying capacity curve is shown in Fig. 2. Figs. 1 and 2 show that the scopes of the axial load and tilting moment that the bearing can carry, as obtained by the computing method used to find the precise static load-carrying capacity, are bigger than that found by the simplified static load-carrying capacity method as used widely in engineering practice. If the load-carrying capacity of the bearing is judged using the simplified static load-carrying capacity curve, an acceptable product may be considered unacceptable. The precise static load-carrying capacity surface of a double-row four-point ball bearing is shown in Fig. 3. 4.2. Effects of geometry parameters of bearing on the static loadcarrying capacity Some geometric pitch bearing parameters such as: the bearing basic bore diameter, bearing basic outside diameter, ball nominal diameter, and bearing pitch diameter have been standardised; however, there are some geometric parameters such as: clearance, coefficient of raceway groove curvature radius, and contact angle, which are designed by experience. The values of clearance, coefficient of raceway groove curvature radius, and contact angle affect the load-carrying capacity of such bearings. So it is necessary to know the effects of these parameters on the bearing performance.
Y. Wang, J. Cao / International Journal of Mechanical Sciences 100 (2015) 209–215
× 10
3
3
12 10 8 6 4 2 0
×10
14
Tilting moment ( kN • m )
Tilting moment ( kN • m )
14
0
213
5
10 15 20 Axial load(kN)
25
ur=-0.08mm
10
ur=-0.1mm
8
ur=-0.2mm
6 4 2 0
30 × 10
ur=0mm
12
3
0
5
10
15
20
25
30 ×10
Axial load (kN)
3
Fig. 1. Precise static load-carrying capacity curve. Fig. 4. Static load-carrying capacity curves for different clearances.
× 10
3
10
14
8 6 4 2 0
0
5
10 15 Axial load(kN)
20
25 × 10
3
Fig. 2. Simplified static load-carrying capacity curve.
3 kN m) Tilting moment(×10
14
Tilting moment ( kN • m )
Tilting moment (kN • m)
12
3
12 10
×10
Fr=0 Fr =200
ur=0mm ur=-0.08mm
8 Fr =400 6 F =600 r 4
Fr =800
2 0 Fr=1000 0 5
10 15 20 Axial load (kN)
25
30 3 ×10
Fig. 5. Effect of clearance on static load-carrying capability for different radial loads.
12 10 8
effect in the other load cases analysed. The effects of clearance on the static load-carrying capacity for different radial loads are shown in Fig. 5, in which the radial load was in kN. Fig. 5 shows that the static load-carrying curves had similar trends under different radial loads. With an increasing radial load, the effect of clearance on the load-carrying capacity of bearing decreased.
6
load ic a l Rad 03 kN) (×1
4 2 0 2 4 6 8 10
0
25 15 5 20 10 5 (×10 kN) Axial load
Fig. 3. Precise static load-carrying capacity surface.
4.2.1. Effect of clearance on the load-carrying capacity of pitch bearings When the radial clearances were: 0 mm, 0.08 mm, 0.1 mm, and 0.2 mm, respectively, the static load-capacity curves are as shown in Fig. 4. It can be seen from Fig. 4 that the tilting moment load increased first and then decreased with increasing axial load at a clearance of 0, 0.08, and 0.1 mm, while the tilting moment load decreased with increasing axial load at a clearance of 0.2 mm. The maximum axial load which the bearing can carry decreased, while the maximum tilting moment which the bearing can carry increased with an increasing absolute value of the clearance. The clearance had a significant effect on the static load-carrying capacity of the bearing only in load cases with a rather small axial force and a large tilting moment, and it had little
4.2.2. Effect of coefficient of raceway groove curvature radius on load-carrying capacity of the bearing The coefficient of raceway groove curvature radius is an important parameter in bearing design. It affects the osculation and deformation of the ball–raceway contact, as well as the load capacity of the bearing. When the coefficients of raceway groove curvature radius were 0.515, 0.52, 0.525, 0.53, and 0.535, respectively, the static load-carrying capacity curves are as shown in Fig. 6. The tilting moment load decreased with increasing axial load when the coefficient of raceway groove curvature radius was less than 0.52. The tilting moment load first increased, and then decreased, with increasing axial load when the coefficient of raceway groove curvature radius was greater than 0.52. When the coefficient of raceway groove curvature radius increased, the maximum axial and tilting moment loads decreased, and the static load-carrying capacity decreased. This was mainly because the coefficient of raceway groove curvature radius affected the osculation of the ball–raceway contact: as the coefficient of the raceway groove curvature radius decreased, the osculation of the ball– raceway contact increased and the contact area increased resulting
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Y. Wang, J. Cao / International Journal of Mechanical Sciences 100 (2015) 209–215
3
16
16
fi=0.52
14
14
fi=0.525
12
fi=0.53
10
fi=0.535
8 6 4
Tilting moment (kN•m)
Tilting moment (kN•m)
× 10
fi=0.515
18
2 0
5
10
15 20 25 Axial load (kN)
30
35
α0=40 ° α0=45 ° α0=50 °
10
α0=55 °
8
α0=60 °
6 4 2
40 3 × 10
Fig. 6. Static load-carrying capacity curves for different coefficients of raceway groove curvature radius.
3
12
0
0
× 10
0
5
10
15
20
25
Axial load (kN)
303
× 10
Fig. 8. Static load-carrying capacity curves for different initial contact angles.
3
14
3
×10
fi=0.525
Fr=0
14
fi=0.52
Fr =200
12
Tilting moment (kN•m)
Tilting moment (kN•m)
16
10 Fr =400 8
Fr =600
6 4 2 0
Fr =800 Fr=1000 0
5
10 15 20 Axial load (kN)
25
12 10
in an increased load-carrying capacity. Fig. 7 shows the effect of the coefficient of raceway groove curvature radius on the static load-carrying capacity for different radial loads: the static loadcarrying capacity curves had similar trends at different radial loads. The smaller the radial load, the more significant the effect the coefficient of raceway groove curvature radius had on the static load-carrying capacity of the bearing.
4.2.3. Effect of initial contact angle on load-carrying capacity of bearing The initial contact angle of the bearing can also affect its loadcarrying capacity. The effect of the initial contact angle on the load-carrying capacity differed for different radial loads. When the radial force was zero, the static load-capacity curves for different initial contact angles are shown in Fig. 8. This showed that the tilting moment load first increased, and then decreased, with increasing axial load for initial contact angles of 401 and 451. The tilting moment load decreased with increasing axial load when the initial contact angles were 501, 551, and 601. As the initial contact angle increased, the static load-carrying capacity of the bearing increased. The effect of the initial contact angle on the static loadcarrying capacity curves for different radial loads are shown in Fig. 9. This showed that the effect of the initial contact angle on the static load-carrying differed for different radial loads. When the radial load varied between 0 to 200 kN, the load-carrying capacity increased with increasing initial contact angle. When the radial load varied from 400 to 600 kN, the initial contact angle had little
α0=45°
Fr=0
α0=50°
Fr=200
8 F =400 r 6 F =600 r 4 Fr=800 2 F =1000 0 r 0 5
30 3 ×10
Fig. 7. Effect of coefficient of raceway groove curvature radius on static loadcarrying capacity for different radial loads.
×10
10 15 20 Axial load (kN)
25
30 3
×10
Fig. 9. Effect of initial contact angle on static load-carrying capacity curves for different radial loads.
effect on the load-carrying capacity. When the radial loads exceeded 800 kN, the load-carrying capacity decreased with increasing initial contact angle.
5. Conclusion Static models taking account of clearance were proposed for a single-row, and a double-row, four-point contact pitch ball bearing. A precise computation method based, on the static model considering clearance, was presented for the static load-carrying capacity curves, or surface, of pitch bearings. With increasing radial load, the effect of clearance on the load-carrying capacity of the bearing decreased. The clearance had a significant effect on the static load-carrying capacity of the bearing only in load cases with a rather small axial force and a large tilting moment, it had little effect in the other load cases evaluated. The smaller the radial load, the more significant the effect of the coefficient of raceway groove curvature radius had on the static load-carrying capacity of the bearing. When the coefficient of raceway groove curvature radius increased, the maximum axial and tilting moment loads decreased as did the static load-carrying capacity decreased. When the radial load ranged from 0 to 200 kN, the load-carrying capacity increased with increasing initial contact angle. When the radial load ranged from 400 to 600 kN, the initial contact angle had little effect on the load-carrying capacity. When the radial loads exceeded 800 kN, the load-carrying capacity decreased as the initial contact angle increased.
Y. Wang, J. Cao / International Journal of Mechanical Sciences 100 (2015) 209–215
Acknowledgement This research was supported by the NSFC (Grant nos. 51475143 and 51105131), a Henan Province Key Scientific and Technological Project (Grant no. 142102210110), and the Tianjin Science and Technology Support Programme (Grant no. 14ZCDZGX00803).
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