CHAPTER 7 FINITE ELEMENT ANALYSIS OF DEEP GROOVE BALL BEARING
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1 113 CHAPTER 7 FINITE ELEMENT ANALYSIS OF DEEP GROOVE BALL BEARING 7. 1 INTRODUCTION Finite element computational methodology for rolling contact analysis of the bearing was proposed and it has several advantages compared with analytical and numerical approaches. First, it is a realistic 3D finite element model and it can accurately calculate the 3D stress response in the contact region. Second, it includes both material and geometric non-linearity, i.e. elasto-plastic material behaviour and contact stress analysis and it can be used to simulate the total contact stress with different negative internal clearance. In this chapter, the effect of internal clearance on fatigue life variation has been evaluated by using fatigue analysis software (fe-safe). The radial operating clearance was calculated theoretically and the same has been used for finite element modelling of the bearing. The contact stress analysis was carried out with the calculated radial clearance, stress plots were extracted and the same has been used to evaluate the fatigue life of the bearing. The effect of internal clearance on radially loaded deep groove ball bearing load distribution and fatigue life was determined for four-clearance group. Life declines gradually with positive clearance and rapidly with
2 114 negative clearance (Oswald et al., 2012). In this chapter, the finite element method and analytical calculations have been performed to determine the contact stress and to estimate the fatigue life variation of bearing with different radial internal clearances. 7.2 STEPS INVOLVED IN THE FINITE ELEMENT ANALYSIS The main steps involved in the finite element analysis are: (1) Construction of 3D geometric model.(2) Development of finite element modelling.(3) Definition of material properties.(4) Defining contact elements and its properties. (5) Application of boundary conditions and loading. (6) Extraction of results Construction of 3D geometric model A 3D geometrical model was created parametrically by using the APDL (ANSYS Parametric Design Language) in the ANSYS software (Version 13.0) to vary the internal clearance between the ball and raceways. The APDL code is given in Appendix 7. The geometric dimensions of the inner ring, outer ring and balls were accurately modelled as per the dimensions given in the bearing catalogue. The plane sketch and 3D model of the bearing is shown in Figure 7.1. Bearing dimensions (6211) as follows: Outer ring diameter : 100 mm Inner ring bore diameter : 55 mm Width : 21 mm Ball diameter (d) : mm
3 115 Number of balls : 10 Pitch circle diameter : mm Raceway groove conformity for inner raceway (R/d) : Raceway groove conformity for outer raceway (R/d) : Raceway groove curvature radius of outer ring : mm Raceway groove curvature radius of Inner ring : mm Most of the commercial bearing raceway groove conformity varies from0.51 to 0.53, which is indicated by Harris & Kotzalas (2007). Radial clearance Figure 7.1 Plane sketch and 3D geometric model Development of finite element model The inner ring and outer ring were meshed with solid 185 elements and it is defined by eight nodes having three degrees of freedom at each node
4 116 in translational direction X, Y and Z. In order to have more number of nodes in the contact surfaces, the mapped meshing was done on inner and outer ring as shown in Figure 7.2. The cage was not modelled and its effects were neglected and the steel balls were meshed with solid 187 elements. Since the geometry is symmetric, only half of the bearing was meshed and symmetric boundary conditions have been applied to take care of the material continuity. Figure 7.2 Meshed model Definition of material properties The homogeneous and isotropic material was assumed and from the chemical analysis, it was confirmed that bearing material is EN31 (Equivalent to AISI 52100). The Young s modulus (E) of MPa and Poisson s ratio (ν) of 0.3were considered in this analysis Defining contact elements and its properties The contact pair consists of target and contact surface to establish the contact between the balls and raceways. The inner and outer ring raceway surface nodes were selected and assigned as target area and ball surface nodes were selected as a contact area. Generally, the surface that has a larger curvature will be chosen as the contact surface, thus ball surface was assigned
5 117 with CONTA174 (3-D 8-Node Surface-to-Surface Contact). The raceway surfaces of outer race and inner race were assigned with TARGE170 and it is used to represent various 3-D target surfaces for the associated contact elements. The contact normal direction is shown in Figure 7.3 and contact region was established only on the contact area between the ball and raceway. Contact normal Contact region Figure 7.3 Direction of contact normal and contact region The number of rolling elements (balls) sharing the total load depends on the interference/clearance between the ball and raceway. If interference is high, the more number of balls are used to share the total radial load. If the clearance is high, less number of balls are used to share the total radial load in each load cycle. It is considered that, the five numbers of balls in case of negative clearances shares the total radial load. The following contact properties have been considered in the analysis: Coefficient of friction between the ball and raceway surface : Behaviour of contact surface : Standard
6 118 Contact algorithm : Augmented Lagrange Initial penetration : Include everything Automatic contact adjustment : close gap Contact surface offset : 0 Contact detection : on Gauss points Mesh independence study The bearing elements were meshed with hexahedral and tetrahedral elements with different element size to achieve the accurate results. The meshing was done to achieve better results which are independent of element size and number of nodes. The mesh intensity has been increased from coarse to fine to attain the mesh independent results. In order to reduce the solving time, the mesh independent study was carried out with one particular radial operating clearance and the same mesh quality was maintained for all other analysis. The mesh independence study was performed for 0.04 mm negative clearance and contact stress values are given in Table 7.1 with an increased number of nodes and elements. The stress plots are given in Appendix 8. Table 7.1 Mesh independent study for 0.04 mm negative clearance Iteration Clearance value (mm) Nodes Elements Contact stress (MPa)
7 Application of boundary conditions and loading The connecting rod big end bearing is subjected to radial load as shown in Figure 7.4. The bearing has been isolated from the assembly, and the following boundary conditions were applied. All the nodes on the outer surface of the outer ring were constrained in all directions. The symmetric boundary condition was applied on the transverse section of the inner and outer ring to take care of the material continuity. The radial load of 4057was applied to the inner ring and its calculation details are shown in TableA1.1. The interference (negative radial operating clearance) between the ball and raceway was calculated analytically and the same was considered for FEA modelling to estimate the contact stress. The radial operating clearance and its details are given in Table 3.4. Symmetric boundary condition Total radial load 4057 N Figure 7.4 Direction of radial load
8 Extraction of results The analyses have been carried out after establishing the contact elements and applying the boundary conditions. The contact stress plots have extracted for various operating clearances and its values are compared against the analytical values as shown in Table 7.3. The stress contour at the contact region is shown in Figure 7.5 for the radial clearance of -0.01mm. Contact stress contour at contact region Figure 7.5 Contact stress contour in the contact region for the radial clearance of -0.01mm
9 121 The stress contour at the contact region is shown in Figure 7.6 and Figure 7.7 for the radial clearances of mm and mm. The remaining plots are given in Appendix 7. Figure 7.6 Contact stress contour in the contact region for the radial clearance of mm Figure 7.7 Contact stress contour in the contact region for the radial clearance of mm
10 122 Oswald et al., (2012), explained the total load shared by the number of balls with respect to internal clearance and based on the above literature, the number of balls was considered for this FE analysis Analytical calculation of contact stress Radial load on the bearing was calculated from the mean indicated pressure in the high pressure cylinder and its details are given in Table A1.1. The maximum surface contact pressure was calculated for the selected bearing (6211) by using the formula given in the bearing manufacturer s catalogue. P max = K 1 F r 1/3 Where K 1 = 133 (NSK Technical report, given in Appendix 3) F r =4057 N (Radial load acting on the bearing) P max = MPa. The above formula is used to calculate the approximate contact stress for deep groove ball bearing. The analytical calculation was carried out to find the maximum ball load and contact stress for different radial clearances are shown in Table 7.2. The formula used in this calculation is referred from bearing manufacturer catalogue (NSK Technical report 2009).
11 Table 7.2Analytical calculation of contact stress for different radial clearance Description Unit Values Total radial load (F r ) N Radial operating clearance(δ r ) mm Load factor (ε) is estimated from Figure A Radial integral (J r ) is calculated from Figure A Load factor function f(ε) =Δr.(D w ) 1/3 / (F r /Z) 2/3, Where ball diameter (D w ) = mm Maximum ball load (Q max ) = F r /(J r Z osα), where contact angle α=0 and number of balls(z) = N γ =D w. cosα/d pw, Ball pitch diameter (D pw ) = mm Cosτ=(1/f±2γ/(1 γ))/(4-1/f±2γ/(1 γ)) Total major curvature ( ρ) = Where ratio of groove radius to ball diameter (f) = Maximum contact pressure P max = A 1 [( ρ) 2/3 Q 1/3 ]/µν,where µ = 3.1 and ν = 0.42 and A 1 = 858 (Refer Appendix 3) MPa
12 Description Unit Values Total radial load (F r ) N Radial operating clearance(δ r ) mm Load factor (ε) is estimated from Figure A Radial integral (J r ) is calculated from Figure A γ =D w. cosα/d pw, Ball pitch diameter (D pw ) =77.67 mm Cosτ =(1/f±2γ/(1 γ))/(4-1/f±2γ/(1 γ)) Table 7.2(Continued) Load factor function f(ε) =Δr.(D w ) 1/3 / (F r /Z) 2/3, Where ball diameter (D w ) = mm Maximum ball load (Q max ) =F r /(J r Z osα), where contact angle α=0 and number of balls (Z) =10 N Total major curvature ( ρ) = Where ratio of groove radius to ball diameter( f) =0.525 Maximum contact pressure P max = A 1 [( ρ) 2/3 Q 1/3 ]/µν, where µ =3.1 and ν=0.42 and A 1 = 858 (Refer Appendix 3) MPa
13 125 Table 7.3 Comparison of FEA results with analytical results Radial operating clearance (Δr) in mm Maximum contact stress by analytical calculation (MPa) Maximum contact stress by FEA (MPa) Deviation of FEA results from analytical results % % % % % % % % % % % The percentages of error between the analytical and FEA results are falling within 9% in the case of negative clearances as shown in Table 7.3. The percentage of error is more in case of positive clearances. Since the clearance is on positive side, the only three balls were considered for sharing the total radial load. Instead of applying the radial load, the theoretical deflection was calculated and given as a boundary condition. The deflection was estimated by using the formula δ max = (ε/1-ε)δ r, where ε is the load factor. In order to find the effect of the internal operating clearance on the fatigue life of the bearing elements, the stress values have been extracted from the ANSYS software and the same was given as an input to the fatigue analysis software. Most of the existing rolling contact fatigue models use a simplified stress calculation technique, such as Hertz analytical solution or simplified finite element analysis with applied Hertz contact pressure. Due to the
14 126 complex geometry of the ball/raceway contact area, it is more appropriate to use a 3D finite element method to calculate stress response. The Hertzian theory assumes that the contact area is small compared to the body dimension and surface curvature. In this section, proposed finite element computational methodology to calculate the complex 3D stress histories of ball/raceway contact. The stress histories are used for fatigue life prediction using fatigue analysis software. 7.3 CALCULATION OF FATIGUE LIFE In this work, specialized fatigue analysis software was used to study the fatigue life of the bearing. The stress and strain datasets were obtained from the FEA for the maximum applied load was used to carry out the fatigue life analysis. The each stress and strain tensor datasets were scaled by a ratio (N l /N a ) of the loading history of the constant amplitude load and it was used to represent the variation of applied load with time (T). The ratio of the loading history of the constant amplitude load is shown in Figure Basic steps involved in the fatigue analysis software a) Define the loading case: The cyclic load histories are defined in the load settings. The stress value has been calculated for the maximum load by using FEA and a load history represents each load as a proportion of the maximum load. Figure 7.8 Loading history
15 127 b) Define the sub group options as a whole group. Since the all the elemental results are used for fatigue analysis, the whole group option was considered. c) Define the surface finish, it was assumed that the whole group has the mirror- polished surface finish (<0.25µm). d) Define the material properties and then store it in the database and retrieved for fatigue analysis. e) Define the material data of the bearing steel AISI as follows Fatigue strength exponent (b) : Fatigue strength coefficient (σ f ) : 2642 MPa Fatigue ductility exponent (c) : Fatigue ductility coefficient (ε f ) : Young s modulus (E) : GPa f) Define the in-plane residual stress, in this case the residual stress was assumed to be zero. g) Define the analysis algorithm Brown Miller combined strain criterion was used to evaluate the fatigue life of rolling bearing, because it proposes that the maximum fatigue damage occurs on the plane, which experiences the maximum shear strain amplitude and that the damage is a function of both the shear strain and the strain normal to this plane. h) Configuring the factor of strength analysis: Infinite design life 10 7 (use materials endurance limit) i) Define the output file
16 128 j) Run the analysis k) Extraction of fatigue life contour Fatigue life contour In order to understand the fatigue life variation with respect to clearance, the following fatigue life contour plots were extracted from the output of the fatigue life software. The fatigue life varies from to 10 7 cycles for the internal operating clearance of mm as shown in Figure 7.9. Figure 7.9 Fatigue life contour for mm interference
17 129 The fatigue life varies from to 10 7 cycles for the internal operating clearance of mm as shown Figure Figure 7.10 Fatigue life contour for mm interference From the above plots, it is observed that, the negative clearance decreases the fatigue life of the bearing in the contact region where the stress levels are higher. 7.4 DISCUSSION The FEA results have shown that the contact stress has varied drastically when the negative clearance increases beyond a certain limit. Oswald et al., (2012), explained the total load shared by the number of balls with respect to internal clearance and based on the above literature, the number of balls was considered for this FE analysis. The maximum load on each ball was calculated by using the analytical equations given Table 7.2 for different radial internal clearances. The maximum ball load variation with respect to internal clearance is shown in Figure It is observed that the ball load increases rapidly with increased negative internal clearance.
18 Maximum ball load with internal clearance Maximum ball load (N) Internal clearance (Interference) in mm Figure 7.11 Maximum ball load variation with internal clearance It is observed from Figure 7.12, rapid increase in contact stress occurred with a slight increase in negative clearance beyond a certain limit. The more contact pressure between the ball and raceways and it will deteriorate the lubricating property. In the modified compressor with C3 model bearings, the maximum internal clearance is mm and the maximum ball load is 48% lesser than the normal clearance bearing. In order to understand the variation of life with respect to internal operating clearance, a graph has been plotted between load factor(ε) and life ratio (L ε /L), where L ε is the life of the bearing with clearance Δ r and L is the life of the bearing with zero internal clearance. It is observed from Figure 7.13, the life ratio falls off rapidly reaching 0.11 at mm clearance, because the maximum ball load has increased considerably.
19 131 It is observed from fatigue life contour, the life of the bearing decreases with increased negative operating clearances. Contact stress with internal clearance 3400 Contact stress (MPa) Internal clearance (Interference) in mm Figure 7.12 Contact stress variations with internal clearance Life ratio Lε/L Life ratio variations with internal clearance Internal clearance (Interference) in mm Figure 7.13 Life ratio variations with internal clearance
20 CONCLUSION From the above results, it was concluded that the fatigue life of the bearing decreases with increased negative internal clearances. The life ratio variation plot clearly indicates that, life of the bearing has drastically reduced beyond mm of negative clearance. Therefore, it was proven that C3 class of bearing with modified fit would have more fatigue life than normal clearance bearing for this application.
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