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Rolling Element Bearing Stiffness Matrix 

Determination

Yi Guo

National Wind Technology Center

National Renewable Energy 

Laboratory

Robert G. Parker

Department of Mechanical 

Engineering

Virginia Tech

NREL/PR-5000-60639
2

Motivation

• Limited work on stiffness matrix in the literature

o Diagonal matrix approximation typically used

• Elastic deformation of race causes nondiagonal terms

0

0

0

0

0

0

x y

x y

x y

xx xy

y y

xx xy xz x x

yy yz y y

zz z z

Symmetric

k k kk k

k kk k

kk k

k k

k

θ θ

θ θ

θ θ

θθ θθ

θ θ

 

 

 

Coupling between the radial

and rotational displacement

Coupling between the axial

and rotational displacement

Coupling between the axial

and radial displacement
3

Finite Element/Contact Mechanics Model

• Three-dimensional finite element model includes 

microgeometry

• Analyze contact between rolling elements and races

o Contact searched at every time instant as bearing rotates

Radial Ball Bearing Cylindrical Bearing Contact Grid
4

Accuracy Order of Finite Element Analysis

• Numerical Jacobian used to compute K

• Order of Jacobian approximation formula should be comparable 

to the accuracy order of finite element analysis (FEA)

• Method to obtain the accuracy order of FEA

Accuracy Order FEA FEA/Contact

p1 1.11 1.94

: finite element solution

: exact solution

: finite element size

: order of accuracy

h e

1 2

1 2 ... p p V V ch c h h =+ + +

1 1

32 3 2

1 1

21 2 1

2 3

1 2

p p

hh h h

p p

hh h h

ee VV h h

ee VV h h

− − − = ≈ −− −

3 1

3 2

2

3

1

1

2

{( ) }1

3

( )

h h

h h

h log V V h log V V h log h p h log h

−

−

=

1

Vh

V

h

p
5

Comparison Against Published Experiments

• Calculated stiffnesses by FEA agree with experiments

Experiment

FEA Experiment

Experiment

FEA

FEA FEA

Experiment

(Royston et al. 1998) (Kraus 1987)

Axial Load (N)

Stiffness (kN/mm)

Radial Stiffness

Axial Stiffness

Radial Stiffness

Axial Stiffness

Radial and Axial Load (N)

Stiffness (kN/mm)
6

Comparison Against Theoretical Models 

• FEA stiffness agrees with the theoretical model

o Only with unrealistic races that match Harris’s assumptions

• Theoretical models predict higher stiffness with design 

dimensions

FE, thick race

FE, actual race

FE, thick race

FE, actual race

Harris, Theoretical

Harris, Theoretical
7

Bearing Stiffness Is Time-Varying

• Number of rollers in contact changes periodically

• Can excite gearbox vibration

Four Rollers in Contact

Five Rollers in Contact

Ball pass period
8

Off-Diagonal Stiffnesses Affect Gear Vibration

• Gear dynamics with off-diagonal stiffnesses differ 

from gear dynamics with a diagonal stiffness matrix

o Need to include off-diagonal stiffnesses

Full matrix

Diagonal matrix

Photo by Yi Guo, NREL 
9

Conclusions

• A method developed to determine bearing stiffness matrices

• Method validated by experiments

• Comparison against theoretical models expose their 

limitations

• Bearing contact is nonlinear and time-varying

• Bearing microgeometry affects stiffness

• Off-diagonal bearing stiffnesses affect gear dynamics
10

Thank You!

This study was supported by the National Aeronautics and Space 

Administration Glenn Research Center. The authors would also like to 

thank the National Renewable Energy Laboratory for follow-on support. 

NREL’s contribution to this work was funded by the U.S. Department of 

Energy under Contract No. DE-AC36-08GO28308.

yi.guo@nrel.gov

303-384-7187