Simulation of torque caused by the

lubrication fluid in a ball bearing

David Westsson

March 11, 2015

Abstract

The purpose of the study has been to get a first estimation of howwell adapted Computational Fluid Dynamics (CFD) based on the Fi-nite Element Method (FEM) is for studying the large scale flow oflubrication fluid within a ball bearing and the torque arised thereby.Focus has been on calculating the velocities and forces of the lubri-cation fluid in all three directions in order to estimate the unwantedtorque.The code used throughout the study is a fully parallel code developedat Umea University and the results show that it is well suited forthe task. Moreover the simulations show that accurate results can beachieved with a reasonable number (16) of CPUs within a few hoursof simulation.

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Sammanfattning

Ma let med studien har varit ha fa en fo rsta uppfattning om hur va lanpassad finita elementmetoden a r fo r att studera det storskaligaflo det hos smo rjva tskan i ett kullager samt det moment som dettager upphov till. Tyngdpunkten i arbetet har varit att bera knahastigheterna och krafterna orsakade av smo rjva tskan i alla tredimensioner och da rigenom uppskatta det oo nskade momentet.Under hela studiens ga ng har en fullsta ndigt parallell kod -utvecklad pa Umea universitet - anva nts och resultaten visar att dena r va l anpassad fo r att klara av uppgiften. Da ruto ver visarsimuleringarna att noggranna resultat kan fa s med ett rimligt antalprocessorer (16 st) efter ett par timmars simulering.

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Contents

1 Introduction 41.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2 Finite Element Method . . . . . . . . . . . . . . . . . . . . . 51.3 Ball Bearings . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Simulation set-up 62.1 Included . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Excluded . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3 Solver Method . . . . . . . . . . . . . . . . . . . . . . . . . . 72.4 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.5 Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.6 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . 112.7 Model Parameters . . . . . . . . . . . . . . . . . . . . . . . . 12

3 Results 133.1 Runtime and Computational Performance . . . . . . . . . . . 133.2 Forces and Torque Inside the Ball Bearing . . . . . . . . . . . 13

3.2.1 Torque . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.2.2 Forces in the Different Directions . . . . . . . . . . . . 153.2.3 The velocity field . . . . . . . . . . . . . . . . . . . . . 18

4 Conclusions and Discussion 19

5 Future and On Going Work 20

6 References 21

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1 Introduction

1.1 Background

This is the final report of a Bachelor’s Thesis done at the Department ofMathematics and Mathematical Statistics Umea University. The purpose ofthe study has been to get a first estimation of how well adapted Computa-5

tional Fluid Dynamics (CFD) based on the Finite Element Method (FEM)is for studying the large scale flow of lubrication fluid within a ball bearingand the torque arised thereby. Focus has been upon finding a good testmodel and then calculating the velocities and forces of the lubrication fluidin all three directions in order to estimate the unwanted torque.10

My part of the project has mainly been to design a ball bearing model, andbased thereupon supply suitable meshes. The design work has been done inSolidWorks (see Figure 1 and 2 below) and the meshes in ANSYS CFX. Allthe calculations have been performed in the Akka-cluster of the HPC2N withhelp of a fully parallel code, developed at the Department of Mathematics15

and Mathematical Statistics.

Figure 1: Intermediate drawing from which the full geometry is obtained byrotation and mirroring.

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Figure 2: Showing the numbers of the balls in relation to the coordinatesystem.

1.2 Finite Element Method

The main idea of all numerical methods is to consider a continous problem,to descretize it into a finite problem and solve it with the help of a computer.The Finite Element Method (FEM) is a numerical method for solving partialdifferential equations (PDE’s). Particularly for the viscous incompressibleNavier-Stokes equation the problem to be solved can be formulated as:

Find the velocity (u) and the pressure (p) such that:

u− ν∆u+ (u · ∇)u+∇p = f ∈ Ω× I (1.1)

∇ · u = 0 ∈ Ω× I (1.2)

u = g ∈ ΓD × I (1.3)

νn · ∇u− p · n = 0 ∈ ΓN × I (1.4)

u(·, 0) = 0 ∈ Ω (1.5)

on the domain Ω, which boundary δΩ is divided into a Dirichlet part (ΓD)with a ”non-slip” condition and a Neumann part (ΓN ) with a ”do-nothing”condition. g is a given function, n is the unitary vector outward normal to5

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the boundary, ν is the kinematic viscosity and f a known force. I = (0, T ∗)is the applied time interval.

1.3 Ball Bearings

The main purpose of a ball bearing is to decrease the rotational frictionand to support radial and axial loads. Though depending upon a particular5

application the design of the ball bearing can be adapted in enumerousdifferent ways, in order to optimize the behaviour and life length of thebearing. Another important aspect for deminishing the friction forces is thelubrication fluid used. Depending upon the rotational speed of the bearingin combination with the viscosity of said fluid, the frictional forces will vary10

and in order to minimize them it is important to choose a lubrication fluidwith e.g. a vicosity suitable for the rotational speed of interest.Yet another important feature is the number of balls within the bearing andtheir arrangement. Like in the self-aligned classical ball bearing (SABB) bySven Wingquist, the balls can also be lined up in more than one plane and15

thus allow a certain misalignment between the outer and the inner races.Further, the balls are normally held in place by a so called cage, whichpresences of course also effects the ease of lubrication and thus the suitableviscosity of the lubrication fluid.

2 Simulation set-up20

The project is a first trial in preparation for a more detailled study of thefrictional forces in ball bearings, thus the geometry and conditions definedbelow have been kept simple enough for not requiring a too complicatedset-up and adaptation of the code, but yet thourough enough for testing themethodology and evaluating the quality of the results.25

2.1 Included

The trial was set to include:

1. a simple six balls geometry

2. ridgid balls

3. perfect rotation of the balls30

4. a low Reynold’s number

5. the inner of the ball bearing perfectly filled with lubrication oil

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2.2 Excluded

No particular regards have been taken to:

1. true geometrical models of existing ball bearings

2. non-perfect rotation of the balls

3. Reynold’s numbers with highly turbulent flow5

4. radial force from work load

5. linear deformation of the balls

6. a two phase flow - with two lubrication oils or lubrication oil mixedwith air

2.3 Solver Method10

There are different possible approches for solving the Navier-Stokes equa-tion, however all of them have advantages and disadvantages emerging fromthe complexity of a time dependent, non-linear equation.The method applied in the solver is based on Chorin’s Projection Methodwhich is based on the introduction of the addition and subtraction of anintermediate velocity, u, in each time step, n of Equation 1.1:

un+1 + u− u− un

dt− ν∆un + (un · ∇)un +∇p = fn (2.1)

decouples the problem into one Crank-Nicolson part for the velocities andone pressure correction part. This splitting introduces extra calculation inform of intermediate velocities, but since the code is completely parallel itis a problem easily overcome by using some extra computational power.

2.4 Geometry15

The geometrical model was first made in SolidWorks and then exported asan IGS-file to the ANSYS CFX mesh generator, but any common computeraided design program could have been used. The model was constructed ina right-handed coordinate system with the z-axis along the rotational axisof the inner axis.20

Considering the geometry of the ball bearing (see Figure 3 below), two lay-outs were tested for avoiding mesh problems at the area between the balland the inner and outer ring respectively. As seen in Figure 4 the balls allhave a small collar at the contact zone to make the transition smootherand avoid poorly shaped mesh elements. Figure 5 shows the second layout,25

which has a small gap between the balls and the respective rings.During the preparational test runs the collar layout showed a lot of uneven

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flow close to the collars and considering the purpose of the project, it wasdecided that free hanging balls should be used for further simulations.Another simplification made in order to allow for simple boundary condi-tions on the side walls were the elongated sides of the bearing. In Figure 6the longitudinal dimensions are shown.5

Figure 3: An isometric, wireframe view of the ball bearing showing the outershell, the balls and the inner axis.

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Figure 4: A partial cut view from the side of the ball bearing with collarsshowing the outer shell, the balls and the inner axis.

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Figure 5: A wireframe front view of the ball bearing showing the dimensionsin millimeters.

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Figure 6: A wireframe side view of the ball bearing showing the dimensionsin millimeters.

2.5 Mesh

After importation of the IGS-file into the ANSYS CFX module, a fluid body,consisting of the hollow space between the inner wall of the outer shell, theballs and the inner axis, was created and meshed. Throughout the wholeproject unstructured tetrahedral elements were used, due to there ability to5

adapt to different types of geometries. All hanging nodes were deleted andthe mesh was saved in the TetGen-format in order to be compatible withthe solver.Although most of the mesh sizes created were around 100k-800k nodes, testruns were made with as many as 2.2M nodes.10

2.6 Boundary Conditions

Since the mesh generator and solver are not automatically connected, astatic mesh had to be used. Thus to impose a rotation of the inner axison the surface nodes of the balls, the point of gravity of the ball bearingwas set to coicide with origo and the surface nodes of the balls were given15

corresponding velocity.Throughout all meshes, a right-handed coordinate system has been used

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with the z-axis along the rotational axis (ω) of the inner axis of the ballbearing.

2.7 Model Parameters

During the simulations the only physical parameters varied were the (kine-matic) viscosity (ν) and the number of rotations per minute (RPM). Since5

the rotation of both the outer and inner ring had to be considered, RPMwill equal the sum of their number of rotations per minute.

The values used for the viscosity and the rotation per minute were:

1. ν = 4.0 · 10−3 until 4.0 · 10−3m2s−1 and

2. RPM = 500 to 5000 RPM .10

Other parameters of interest - that either have been kept constant or havebeen calculated from set parameters - are:

1. The dimensionless Reynolds number (Re) defined as:

Re =L · Uν

, (2.2)

where L is the characteristic length scale of the bearing - i.e. thedistance from the center of the bearing to the centre of the balls; U is15

the characteristic velocity of the bearing - i.e. the velocity of the innerring at the contact surface with the fluid; ν is the kinematic vicosityof the fluid.

2. Number of nodes (nno) of the mesh.

3. Total simulated time (T ∗) - set to T ∗= 10 s20

4. Mesh size (h) defined as the shortest side of the smallest element inthe mesh.

5. Constrained Delaunay tetrahedralisation (CDT ) is a constant thatis selected to assure the quality of the mesh through fulfilling theDelaunay criteria. Throughout the simulations the vale of CDT = 0.0525

has been used.

6. Time step dt is calculated for each iteration through:

dt =CDT · h2

2 · ν + U · h(2.3)

7. Total number of iterations (niter) is defined as:

niter =T ∗

dt(2.4)

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8. Finally the ramp parameter (ramp) - for gradually accelerating thesystem during the first 5th of the simulation time - was defined as:

ramp =5 · iterniter

, (2.5)

where iter is the number of the current iteration.

The ramping allows a smoother, more realistic acceleration and since it isonly applied for a relatively short period in the beginning of the simulation,any eventual effects due to the change from this linear acceleration to a5

constant speed of the system will be small or even neglectible compared to asystem where the full speed is applied directly. Moreover, during this phaseit is also possible to save computational power without losing accuracy, byonly updating the velocity vectors every 5th iteration.

3 Results10

The results for the two separate purposes of this study - evaluating theperformance of the method and code in the aspects of runtime and accuracy- are presented in the this section below.

3.1 Runtime and Computational Performance

In order to get an understanding for the speed of which the simulation is15

performed with different numbers of CPUs for the different mesh sizes, theruntimes for the runs where measured (see table 1).

3.2 Forces and Torque Inside the Ball Bearing

For all the different mesh sizes, the torque on the inner axis and the forceson the balls were calculated using a drag function in the code. Below, a20

part of these results is plotted in order to more comprehensively see theeffect the mesh sizes have on the result and later on use that to balancethe total runtime and the accuracy. Where the different runs have hadsimilar characteristiques, the results from the 797k run have for the sake ofconsistency and comparision been taken as example.25

3.2.1 Torque

The torque along the axis of the ball bearing as seen in Figure 8 is as expectedmuch larger (at least 10 times during the ramp up phase and then even more)than the torque in the x- and y-direction (see Figure 7 below).

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#nodes ν [Pa· s] #cpu Re #time steps runtime [h:m:s]

32 439 4e-4 16 50 103 890 00:15:44

32 439 4e-5 16 500 65 790 00:10:28

32 439 4e-6 16 5000 61 980 00:09:23

81 896 4e-4 16 50 176 565 01:18:26

81 896 4e-5 16 500 97 477 00:43:27

81 896 4e-6 16 5000 89 568 00:41:18

462 987 4e-4 64 50 763 291 09:26:01

462 987 4e-5 64 500 287 662 03:35:59

462 987 4e-6 64 5000 240 110 03:06:11

797 567 4.e-4 128 50 998 038 12:04:33

797 567 4.e-5 128 500 307 058 04:19:52

797 567 4.e-6 128 5000 281 960 03:33:58

797 567 4.e-6 64 5000 281 960 06:43:19

1 532 432 4e-5 256 500 983 604 12:53:18

1 532 432 4e-6 256 5000 654 390 08:42:20

Table 1: Various performance data for the simulation runs.

Figure 7: The torque [Nm] in the x- and y-direction on the inner axis dueto the lubrication fluid for the 797k mesh.

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Among the different z-torques of the different mesh sizes there is a minordifference and also a certain periodic fluctuation over time. All graphs showinstability during the ramp-up phase.

Figure 8: The torque [Nm] in the z-direction on the inner axis due to thelubrication fluid for the different mesh sizes.

The means and the standard deviation (taken after the ramp-up phase)of the different torques shown in Figure 8 are:5

1. mean32k= 1.07e-6 Nm and std32k = 3.36e-8 Nm

2. mean81k= 1.16e-6 Nm and std81k = 2.70e-8 Nm

3. mean462k= 1.18e-6 Nm and std462k = 3.46e-8 Nm

4. mean797k= 1.23e-6 Nm and std797k = 3.93e-8 Nm

3.2.2 Forces in the Different Directions10

Since the torques are calcualated from the net forces in the different direc-tions, a part of the information is lost and it is of value to consider the

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forces acting on the balls in the respective directions in order to evaluate toconsistency of the results. In Figure 9, Figure 10 and Figure 11 this is givenfor the 797k run.

Figure 9: The forces [N] in the x-direction on the balls in the bearing dueto the lubrication fluid for the 797k mesh.

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Figure 10: The forces [N] in the y-direction on the balls in the bearing dueto the lubrication fluid for the 797k mesh.

Figure 11: The forces [N] in the z-direction on the balls in the bearing dueto the lubrication fluid for the 797k mesh.

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3.2.3 The velocity field

In the search for unexpected behavior it is also informative to depict thevelocity field and to some extent visualize the velocity field and the majormodes of the solution of the posed problem.

Figure 12: The velocity field in multiple slices in the xy-plane (taken fromthe center and outwards) for the 797k mesh, clearly showing a nonzero z-component, since the velocity arrows are pointing out of the disc.

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Figure 13: The velocity field for a 232k run with ν = 8e−3 of the lubricationfluid between two of the balls in the bearing.

4 Conclusions and Discussion

Studying the result presented in Figure 8, there are a couple of conclusionsthat can be directly drawn; the torque of the 32k run differs significantlyfrom the other mesh sizes and the higher number of nodes, the higher valuefor the top peak of the oscillation. It also seems as the 797k results in two5

resonance peaks with one of them slowly increasing while the amplitud ofthe other three rather seem to flaten out. The 462k mesh appears to bethe most periodic one. The torque in x- and y-direction respectively seen inFigure 7 is most likely an effect of a not perfectly symmetric mesh, but theeffect of it - if needed - is expected to decrease by even a finer meshes.10

Shifting to figures 9 and 10, one notices a particularity while comparingthe forces in the x- and y-direction respectively. For the x-direction theforces seem to pair up with (see Figure ?? in the appendix) balls 1 and 6,balls 2 and 5, and balls 3 and 4 having comparable values. However in they-direction this symmetry seems broken, since there the pairs with compa-15

rable forces are ball 6 and 4, and ball 2 and 3. The force on ball 1 is larglynegative and the force on ball 5 is the positively largest one.

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As shown in Figure 11 the force in the z-direction also appears to have oneanomali consisting of the force on ball 1 which is not zero as bluntly ex-pected. The reason therefore is assumed also to be due to a non-symmetricmesh, possibly due to an asymmetry introduced while creating the bearingin SolidWorks, even if whenever possible mirroring functions in SolidWorks5

were used.Figure 13 shows an appearently smooth velocity field between two of the ballsand intuitively opposite balls should have opposite forces acting on them,but since the solution has different eigenmodes (with different wavelengthsand directions), what appears to be asymmetries in one or two dimensions10

in fact is symmetric on the whole in three dimensions. Part of the resultof these eigenmodes are pictured in Figure 12, wherefrom it also (especiallyin slice number 2 and 3 from the top) is clear that the z-component of thevelocity field is nonzero with a certain periodicity.

Probably the amplitud of this z-component would be smaller when using15

a true geometrical model of a ball bearing with a cage, since there is lessfree space to build up on. The cage might also even out the force betweenthe balls making the x- and y-forces more symmetrical.Combining the run times with the qualitative results of the toruqes and theforces, one realizes that the 81k run is almost 5 times faster than the 462k20

run with a fourth of the number of CPUs. Yet the result is comparable tothe much finer mesh, what definitely makes it suitable as a test mesh duringa forthcoming study, where short runtimes are preferred.

5 Future and On Going Work25

There is a lot of loose ends in this project, but a natural continuation wouldbe to run the code with SKF’s reference geometry (see Figure 14 below) andsettings. After confirmed results yet another step would be to adapt the codeand boundary conditions in order to be able to run the simulation with atrue geometrical model and conditions. With these preparations everything30

is set up for simulations that can really deepen the understanding of allkinds of ball bearing performances and the important interaction betweenthe geometry of the bearing, the external forces and the lubrication fluid.

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Figure 14: Drawing of SKF’s benchmark problem with a possibility to adda two phase flow through not filling the whole space with lubrication fluid.

6 References

1. Concepts And Applications Of Finite Element Analysis - Robert DavisCook and Michael E. Plesha (2001)

2. Finite Element Mesh Generation - B.H.V. Topping and J. Muylle andR. Putanowicz (2002)5

3. Applied Finite Element Analysis - Larry J. Segerlind (1985)

4. An Introduction to Computational Fluid Dynamics - H. K. Versteeg(2007)

5. Applied Computational Fluid Dynamics Techniques: An IntroductionBased on Finite Element Methods - Rainald Lohner (2008)10

6. An Introduction to Parallel Computational Fluid Dynamics - FrancescoPapetti and Sauro Succi

7. Simulation of Lubricant Flow in Ball Bearings - Mats G. Larson andPer Vesterlund (2010)

8. Multiphase Flow Dynamics 1: Fundamentals - Nikolay I. Kolev (2004)15

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9. Beta Mathematics Handbook: Concepts, Theorems, Methods, Algo-rithms, Formulas, Graphs, Tables - Lennart Rade and Bertil Wester-gren (1997)

10. Schaum’s Mathematical Handbook of Formulas and Tables - MurrayR. Spiegel (1998)5

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