*Corr. Author’s Address: University of Ljubljana, Faculty of Mechanical Engineering, Aškerčeva 6, 1000 Ljubljana, Slovenia, miha.boltezar@fs.uni-lj.si 289
Strojniški vestnik - Journal of Mechanical Engineering 69(2023)7-8, 289-298 Received for review: 2023-03-30
© 2023 The Authors. CC BY 4.0 Int. Licensee: SV-JME Received revised form: 2023-05-03
DOI:10.5545/sv-jme.2023.592 Original Scientific Paper Accepted for publication: 2023-05-31
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© 2017 Journal of Mechanical Engineering. All rights reserved.
Review Paper — DOI: 10.5545/sv-jme.2017.4027
Received for review: 2016-11-04
Received revised form: 2017-01-14
Accepted for publication: 2017-02-12
TheDynamicsofTapered-rollerBearings–ABottom-up
ValidationStudy
MatejRazpotnik
1
-ThomasBischof
2
-MihaBoltežar
1,*
1
UniversityofLjubljana,FacultyofMechanicalEngineering,Slovenia
2
ZFFriedrichshafenAG,Germany
Rolling-element bearings are one of the most important elements when predicting the noise of rotating machinery. As a major connecting point between the
rotating and non-rotating parts, their dynamic properties have to be accurately known. In this investigation we present a bottom-up approach to characterising
the dynamics of the rolling-element bearing. A special test device was designed and built to assess the quality of the well-established analytical modelling
approach of Lim and Singh. Two types of bearings were tested, i.e., the ball and tapered-roller types. The dynamic properties were observed by investigating
the frequency-response functions. In addition, non-rotating as well as rotating test scenarios were checked. It was shown that the ball bearing model
adequately predicts the system’s response, whereas the tapered-roller bearing model requires modifications. These results were further confirmed with a
quasi-static load-displacement numerical evaluation, where a full finite-element model serves as the reference.
Keywords: dynamic bearing model, tapered-roller bearing, bearing stiffness matrix, vibration transmission
Highlights
• Test device for bottom-up investigation of bearing’s dynamics is built.
• Ball and tapered-roller bearings are tested at different speeds and axial preloads.
• The system is evaluated numerically and experimentally.
• Ball bearing model is validated whereas tapered-roller one needs improvements.
0 INTRODUCTION
Every rotating machinery contains bearings. They
represent the connecting points between the rotating
and non-rotating parts and so are very important
elements in the chain of vibration transmission. The
dynamic properties of rolling-element bearings have
been studied for many decades; however, due to their
complex contact-related phenomena the topic remains
importantinongoingresearch.
A first general theory for elastically constrained
ball and roller bearings was developed by Jones [1].
Later this theory was further extended by Harris [2].
The theory was very general and focused more on
staticandfatigue-lifecalculationsthanonthevibration
transmission through the bearings. Simplified bearing
models were introduced by other researchers, with
the bearings being modelled as an ideal boundary
condition for the shaft, as presented by Rao [3].
Meanwhile,theideaofinterpretingthebearingswitha
simpleone-ortwo-degrees-of-freedom(DOFs)model
with linear springs was introduced by While [4]
and Garigiulo [5]. Later, more accurate dynamic
bearing models were derived. A major improvement
in predicting the vibration transmission through
rolling-element bearings was made by Lim and
Singh [6] and [7] and in parallel by de Mul [8].
They derived a model that provides a comprehensive
bearing-stiffness matrix. The model is capable of
describing the nonlinear relation between the load and
the deflection, taking into account all six DOFs and
their interplay. These authors also presented system
studies [9] and [10] for model-validation purposes.
A good agreement between the measurements and
the analytical model was shown for the ball and the
cylindrical roller bearings as the two most distinct
examples of different contact types. The six-DOFs
model is the basis for the widely used industrial
standard ISO/TS 16281 [11] as well as for many
subsequent studies. Recently, a thorough review
of mechanical model development of rolling-element
bearing was presented by Cao et al. [12]. The authors
classify modelling approaches into five different
techniques and comprehensively discuss the current
progress of development as well as identify future
trends for research. Despite great computational
power available these days, modelling of the bearings
primary remains on the analytical level. Contact
related phenomena and non-linearities lead to huge
and often unstable finite-element method (FEM)
models. However, connecting analytical models
with numerical ones is crucial in predicting the
proper behaviour of a modern system. Guo and
Parker [13] presented a stiffness-matrix calculation
*Corr. Author’s Address: Name of institution, Address, City, Country, xxx.yyy@xxxxxx.yyy 1

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using a finite-element/contact-mechanics model. On
theotherhandanalyticalapproachesdemandadvanced
methods to solve the system of nonlinear equations.
Fang et al. [14] recently presented a comprehensive
study of the speed-varying stiffness of ball bearings
under different load conditions. They proposed a
novel mathematical method for solving an implicit set
of nonlinear equations based on a new assumption
of the initial conditions. To mitigate the numerical
difficulties of time integration procedure, induced
by rolling-elements coming and leaving the contact,
Razpotnik et al. [15] extended the model from
Lim and Singh [6]. They implemented modular
smoothing of the load-displacement characteristics in
the region of contact-state transition. To improve the
calculation accuracy of non-hertzian contact pressure
the high-precision half space theory was adopted by
Kabusetal.[16]and[17].
Tapered-roller bearings (TRBs) are widely used
in rotor dynamics. They are usually treated as a
special case of a cylindrical roller bearing with a
non-zero contact angle. The difference, in fact,
is much more significant because TRBs have two
different contact angles (the inner ring-roller and
outer ring-roller contacts) and also because of their
additional roller-flange contact. Cretu et al. [18]
and [19] analysed the dynamics of TRBs under
fully flooded conditions. The assumption of an
elastohdydrodynamic (EHD) lubrication regime is
common to the majority of TRB studies. In this
way the friction forces can be either calculated
or, even more commonly, neglected. Tong and
Hong [20] analytically studied the characteristics of
TRBs subjected to combined radial and moment
loads. Thesameauthors[21]investigatedtheinfluence
of the roller profile and the speed on the stiffness
of a TRB. Zhao et al. [22] studied the effect of
gyroscopic moment on the damage of a tapered-roller
bearing, which are found to occur under high-speed
and high-load conditions, such as high-speed trains.
Roda-Casanova and Sanchez-Marin [23] presented an
illustrative study of the contribution of the deflection
of the TRB to the misalignment of the pinion
in a pinion-rack transmission. They stressed the
importance of having accurate knowledge of the
elasticity of the bearings. Houpert [24] studied the
torque generated by the friction forces in a TRB,
where he emphasised an important fact, i.e., TRBs
are subjected to a high roller-flange torque. The
roller-flange contact, which largely affects the power
loss,wasinvestigatedalsobyAietal.[25]. Inaddition,
Tongetal.[26]madenumericalevaluationoftheeffect
of misalignment on the generated friction forces and
consequentlyevaluatedthepowerlossofaTRB.Itwas
shown that already a small misalignment can have a
significantinfluenceonthegeneratedtorque.
Experimental investigations of a TRB’s dynamics
are rare in the literature. Zhou and Hoeprich [27]
measured the torque generated at different contacts
in a TRB; however, they focused on the losses and
not on the dynamics. Gradu [28] also analysed the
TRB losses and compared them with equivalent ball
bearing. Wrzochal et al. [29] presented a new device
for measuring the friction torque in rolling-element
bearings of different types, where the main goal
was to establish a reliable device for quality control
measurement. Discrepancy between theoretical and
measured friction torque was presented and discussed.
A comparative study, as presented by Zhang et
al. [30] for angular-contact ball bearings, would also
be beneficial for TRBs. Further, since TRBs are often
usedinapplicationsthatdonotrequirehighspeed,the
influence of friction on the dynamic properties would
begenerallywelcomed.
In this paper a numerical and experimental
characterisation of a TRB’s dynamics is presented.
First, a general bearing modelling technique is
introduced, where the analytical model of Lim
and Singh [6, 7] is embedded into a FEM model.
Afterwards, a special test device is presented. There
follows a description of a workflow for a bottom-up
validation study. A TRB is mainly investigated,
whereas ball bearing is also tested. The results in
the form of frequency-response functions (FRFs) are
compared for the measurements and the simulations.
The non-rotating as well as low-speed-rotating
scenarios are presented. Finally, a quasi-static
load-displacement numerical analysis was performed
toadditionallyverifytheresults.
1 BEARING MODELLING TECHNIQUE
Rolling-element bearings can be modelled as a part
of a wider system in several different ways. Most
often the system is studied by utilising a FEM model.
The bearings are, due to their complex contact-related
phenomena, represented by a special element that
embeds the analytically calculated bearing-stiffness
matrix K
b
. This technique introduces the so-called
spider elements (commercially known as RBE3
element), as shown in Fig. 1. A spider element
connects a raceway of a ring to one, central node.
The motion of that central node depends on the
weighted average of the motions at a set of connected
grid points [31]. Two spider elements are needed,
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(a)
K
b
(b)
K
b
Fig. 1. A cross-section view of a bearing in a FEM model with zoomed-in
areas; a) ball bearing, and b) TRB
each for one bearing ring. Central nodes, connected
by K
b
, are located at exactly the same position.
Fig. 1 separates them just for illustrative purposes.
Bearing rings can be coarsely meshed since the mesh
is only used to represent the shape of the ring
and to serve as a connecting body for the spider
element. Rolling elements are not modelled, since all
contact-relatedphenomenaarecoveredbythestiffness
matrix. The source of the accuracy of the presented
technique is not in the spider element itself, but in the
bearing-stiffness matrix. In this study we implement
the bearing-stiffness matrix from Lim and Singh [6]
and [7]; however, any relevant bearing theory yielding
the stiffness matrix can be implemented.
There are two types of FEM analyses used in
the presented study, i.e., frequency-response modal
analysis and quasi-static load-displacement analysis.
Both assume that rolling elements are not rotating.
However, the former is used to obtain dynamic
response of the system when excited and subjected
under different axial loads, and the latter is used to
obtain load-displacement characteristics of a bearing.
2 TEST DEVICE
A simple test device was designed and built for
validation purposes. It consists of the housing, a
long shaft and a special nut, as shown in Fig. 2. It
is important to note the shape of the nut, which is
only in contact at both ends. This design ensures
that the load dependency of the thread contact is
negligible. A bottom-up validation approach was
housing shaft
nut
Fig. 2. A simplified technical drawing of the test device, where all three
possible configurations are shown
utilised. Therefore, solid rings are inserted at first in
order to prove the linearity of the system. Afterwards,
two types of rolling-element bearings were tested,
i.e., ball bearing and TRB, with the properties given
in Table 1. In order to eliminate the influence of
the surroundings, the device was tested with free-free
boundary conditions (BCs). These conditions were
achieved by hanging the test device via housing by
thinropes. TheFRFsweremeasuredbetweendifferent
Table 1. Bearings used in the test device.
type designation d [mm] D [mm] B [mm]
ball 6006 30 55 13
tapered-roller 32006-X 30 55 17
parts of the system. An excitation was applied
with a modal hammer, whereas the acceleration was
measuredbytheaccelerometer. Thetransferpathfrom
the shaft to the housing is of special interest, since
thebearing’sdynamicsarethemostclearlyseenthere.
The test device makes it possible to apply different
axial preloads to the bearings and consequently to the
entire system by turning the nut with respect to the
shaft. The applied axial force is measured with the
strain gauges located at the housing ribs. Additionally,
the system can be investigated while the shaft is either
stationary or rotating (up to 6000 rpm). For this
purposeaspecialmotorcanbemountedtothesystem.
Indoingso,thefree-freeBCsaremaintained,asshown
in Fig. 3.
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Fig. 3. Experimental setup with attached motor
3 WORKFLOW
The main goal was to evaluate the quality of the
analytically calculated dynamic bearing model. The
assessment was made for a non-rotating scenario by
comparing the dynamic properties of the system in
the form of FRFs. We focused on a representative
FRF, namely Accelerance, where the excitation was
performed on the shaft and acceleration was obtained
at the housing, as shown in Fig. 4. All of the
possible setups (solid rings, ball bearings, TRBs)
undergo the same testing procedure. Four different
axial preloads were inserted into the system. Thus,
four different FRFs were obtained. Fig. 5 shows
Fig. 4. Excitation point and accelerometer position
the corresponding workflow. The same workflow is
followed for measurements and simulations. Finally,
the FRFs are compared.
The rotating version of the test device was
investigated only experimentally. A run-up
investigation was utilised. By that we can see
the potential change of the system’s dynamics due
to rotor-dynamic effects appearing in the rolling
bearings, e.g., centrifugal forces and gyroscopic
effects. Some researchers have pointed out these
effects during high-speed applications [1], [14], [21],
[22], and [32], where the effects start to become
noticeable in the region between 5000 rpm and 10000
rpm, depending on the bearing type and load case.
However,itisimportanttonotethatourtestingdidnot
exceed6000rpm. Whenthesystemrotates,thereisno
additional external excitation. The system is mainly
excited by the white noise coming from the bearings.
The resulting acceleration is measured and shown in a
Campbell diagram.
Preload
100 N 300 N 600 N 1000 N
FRFs
Comparison between measured and calcluated FRFs
measurements/
simulations
Fig.5. Workflowofthevalidationprocedureforthenon-rotatingtestdevice
4 RESULTS
4.1 Solid Rings
With solid rings inserted it is possible to verify
whether the system without bearings is linear or not.
The linearity implies the load-independent dynamic
properties of the system. Ideally, the presented
system should be load-independent; however, due to
contact issues, especially the thread contact, the load
independence has to be experimentally proven. Fig. 6
shows the results. The measured FRFs, given in the
form of Accelerance (A), are shown with a gray and
blackcolour. Theredcurvecorrespondstosimulation.
Thepreloadwasconsideredinsimulationsaswell,but
itseffectiscompletelyunnoticeable,thusonlyoneline
sufficientlyrepresentsthesimulationresults. Itisclear
that all the measured curves correlate well with each
otherregardingtheeigenfrequencyposition. Damping,
however, decreases with an increased preload. Also,
the calculated FRF predicts the measured behaviour
correctly. However, the peak around 1.45 kHz is more
damped in the measured results.
Proving the linearity of the system with solid
rings is an important step. All the non-linearities in
the succeeding investigations (when real bearings are
inserted) can now be associated with the bearing’s
behaviour.
4.2 Ball Bearings
Theballbearings,asgiveninTable1,areinsertedinto
the system. Fig. 7 shows the amplitude comparison
between the measured and calculated results. It is
clear that some peaks move their position with the
increased preload, while the others do not. Those
involvingthemodesoftheshaftareaffected,whilethe
others are not. Fig. 8 shows the modes of the marked
regions from Fig. 7. The stiffness of the bearing plays
a crucial role there. All of them are pure modes of
the shaft, only the mode at 2490 Hz is a combination
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10
−3
10
−2
10
−1
10
0
10
1
10
2
10
3
10
4
A [1/kg]
meas, 100N meas, 300N meas, 600N meas, 1000N sim
−π
0
π
φ [rad]
0 500 1000 1500 2000 2500 3000 3500 4000
Frequency [Hz]
0
1
γ
2
[/]
Fig. 6. Load-dependency of the FRF (from top to bottom: magnitude, phase, coherence) for the test device with the inserted solid rings
of shaft and the local movement of the housing ribs.
Additionally, it is clear that the higher the preload, the
higher is the bearing stiffness. This also causes the
eigenfrequencies to increase. Comparing the spectra it
can be seen that the frequency span of each affected
eigenfrequency is around 100 Hz for the measured as
well as the calculated FRFs. Some peak positions,
however, differ slightly, but the general behaviour is
well predicted. It is important to note that no FEM
model updating was performed. Doing so would
obviously align the calculated results completely with
themeasuredones.
So far, we have presented the non-rotating
version of the test device. Since the bearings are
meant to rotate it is crucial to determine whether
the analytical bearing-stiffness matrix is an adequate
representation of the bearing’s dynamics also under
operating conditions. A run-up test was performed.
An extension to the motor was additionally mounted
to the test device while maintaining its free-free BCs
(see Fig. 3). The run-up test sequentially increases the
motor speed from 500 rpm to 6000 rpm with a step of
100 rpm. At each step the acceleration on the housing
(the same position as for the FRF investigation) was
measured. The resulting Campbell diagram is shown
in Fig. 9 for the preloads of 300N and 1000N. The
preload of 100N was found to be too loose for the
run-up investigation. Already a slight torque induced
by the motor caused a disturbance that changed the
axialpreload. At300Nthiseffectisnotnoticeableany
more. Itisclearthattheeigenfrequenciesgovernedby
the bearing stiffness do change with a higher preload
in a similar manner to the non-rotating version. They
are marked with red arrows. On the other hand, their
position does not change while increasing the RPM
in the investigated RPM region. Another important
conclusionisthatthelocationsoftheeigenfrequencies
remain at practically the same positions as in the
non-rotatinginvestigation.
The distinct change of eigenfrequencies
dominated by the bearing stiffness is evident.
Comparingtheresultsforthenon-rotatingandrotating
versionswenoticethatonedominatingpeakismissing
in the rotating version, i.e., the one at 2490Hz. The
eigenmode of this peak is a combination of housing
andshaftmovementsandisapparentlychangeddueto
theextensionmountedtothehousing.
The comparison between the measurements and
the simulations shows good agreement for the
non-rotating as well as for the rotating setup. As such
itcanbeconcludedthattheanalyticalbearing-stiffness
matrix seems to be an adequate representation of the
actual bearing’s dynamics for the ball type in the
observedspeedrange.
4.3 Tapered-roller Bearings
The TRBs, as given in Table 1, are inserted
into the system. Fig. 10 shows the amplitude
comparison between the measured and calculated
results. Both spectra have marked regions where
the eigenfrequencies shift with respect to the inserted
preload. Comparing the spectra it is clear that
the positions of the regions differ tremendously.
Investigating the eigenmodes gives us an insight
into the problem. Fig. 11 shows all the calculated
eigenmodes of the test device within the marked
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10
−3
10
−2
10
−1
10
0
10
1
10
2
10
3
10
4
A [1/kg]
meas, 100N meas, 300N meas, 600N meas, 1000N
0 500 1000 1500 2000 2500 3000 3500 4000
Frequency [Hz]
10
−3
10
−2
10
−1
10
0
10
1
10
2
10
3
10
4
A [1/kg]
sim, 100N sim, 300N sim, 600N sim, 1000N
Fig. 7. Comparison between the measured (upper part) and calculated (lower part) load-dependency of the chosen FRF for the test device with inserted ball
bearings
(a) (b)
(c) (d)
Fig. 8. Selected calculated eigenmodes of the test device with ball
bearings; a) Eigenmode at 590Hz, b) Eigenmode at 1650Hz, c)
Eigenmode at 2490Hz, and d) Eigenmode at 3085Hz
frequency band from Fig. 10. All the presented modes
are governed by the bearing stiffness and do actually
correspond to the first, second and third modes of the
shaft. To show the measured modes we need to do
a complete experimental modal analysis (EMA). For
this purpose we used an approach with a high-speed
camera. The measured eigenmodes are shown in
Fig. 12. The first eigenmode obviously represents
the first mode of the shaft, whereas the second mode
represents the combination of the dominant local
housing movement and the second mode of the shaft.
The arrows in Fig. 12 indicate the direction of motion,
where a different colour stands for a different phase.
From the presented results it can be concluded that the
analytically calculated bearing-stiffness matrix for the
TRBs exhibits behaviour that is much too weak. As
suchthecurrentmodellingapproachisnotanadequate
representation of the TRB’s dynamics.
Since the non-rotating scenario has a huge gap
betweenthemeasurementsandthecalculationswedid
not continue to the rotating scenario. Instead we tried
to shed some light on possible causes for the observed
differences with the help of a detailed FEM bearing
model, as discussed in detail in the next section.
5 NUMERICAL INSIGHT
The analytical bearing-stiffness model seems to be a
good representation of reality for the ball bearing, but
considerably worse for the TRB. To find the origin of
the problem we built a complete, detailed, full FEM
bearing model for both bearing types (see Table 1)
as depicted in Fig. 13. The goal is to compare
the load-displacement characteristics, where the load
is exerted incrementally in the axial direction. A
quasi-staticload-displacementanalysiswasperformed
on the full FEM bearing model. Further, the slope of
the load-displcement curve is extracted, representing
the total axial stiffness in the loaded direction, which
is also compared. Due to its completeness, the FEM
model represents the reference. Besides the full FEM
modelandtheanalyticalmodelfromLimandSingh[6]
and [7], the results from the widely used standard
ISO/TS 16281 [11] are also included.
The results in the form of load-displacement
characteristics and the corresponding stiffness for the
ballbearingareshowninFig.14. Allthreeapproaches
result in similar characteristics. There is a minor gap
betweenbothanalytical approaches, whereastheFEM
yields a slightly higher stiffness at a high preload.
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Fig. 9. Campbell diagrams for the test device with inserted ball bearings loaded under 300N and 1000N
10
−3
10
−2
10
−1
10
0
10
1
10
2
10
3
10
4
A [1/kg]
meas, 100N meas, 300N meas, 600N meas, 1000N
0 500 1000 1500 2000 2500 3000 3500 4000
Frequency [Hz]
10
−3
10
−2
10
−1
10
0
10
1
10
2
10
3
10
4
A [1/kg]
sim, 100N sim, 300N sim, 600N sim, 1000N
Fig. 10. Comparison between the measured (upper part) and calculated (lower part) load dependency of the chosen FRF for the test device with inserted
TRBs
(a) (b)
(c)
Fig. 11. Selected calculated eigenmodes of the test device with TRB;
a) Eigenmode at 820Hz, b) Eigenmode at 1960Hz, c) Eigenmode at
3300Hz
However, it can be concluded that all the models are
sufficiently well correlated.
In the same manner we incrementally load the
TRB in the axial direction. The results are shown in
Fig. 15. The curves quantitatively differ significantly;
however, they reflect the same tendency. Concerning
the load-displacement characteristics, it is interesting
that already both analytical approaches differ to a
great extent. The same two approaches exhibit a
similar level in the stiffness characteristic. The FEM
model, on the other hand, yields a factor of two higher
stiffness.
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(a)
(b)
Fig. 12. Measured eigenmodes of the test device with inserted TRB (using
high-speed camera); a) Eigenmode at 1250Hz, b) Eigenmode at 2490Hz
(a)
(b)
Fig. 13. Full FEM model of a bearing; a) ball bearing, b) TRB
6 DISCUSSION
Modelling the dynamics of ball bearings with
the presented approach seems appropriate. The
discrepancy between the measurements and the
simulations is negligible. On the other hand, the TRBs
haveasignificantmismatchbetweenthemeasuredand
calculated results. This is true for the dynamic testing
(comparing FRFs) as well as for a simple quasi-static
load-displacement investigation. The reason for such
a disagreement is the inadequate bearing-stiffness
model. The shortcomings can be outlined as:
1. The TRB has different contact angles in between
the inner ring-roller and the outer ring-roller.
The theory assumes that all the contacts are
0 100 200 300 400 500 600 700 800 900 1000
Fz [N]
(a)
0.00
0.02
0.04
0.06
0.08
0.10
0.12
δz [mm]
FEM
Analytical
Standard
0 100 200 300 400 500 600 700 800 900 1000
Fz [N]
(b)
0
1
2
3
4
5
kz [N/mm]
×10
4
FEM
Analytical
Standard
Fig. 14. Ball bearing being incrementally loaded in the axial direction;
a) load-displacement characteristic, b) corresponding total stiffness
happening at the nominal contact angle, which is
defined for the axis going through the centre of a
roller. Consequently, no axial force is generated
that pushes the rollers out of the initial contact
and no flange is needed to prevent the rollers
from escaping the contact. The roller-flange
contact is thus neglected. This contact carries
only minimal load; however, it should not be
neglected, especially if the TRB is loaded in
the axial direction only. It is important to note
that the literature provides some theories that
take into account different contact angles and
flange contacts [8] and [21]. However, those
theories provide a negligibly different stiffness
matrix compared to the theory, which does not
include the mentioned contacts. Having looked
at the results, the stiffness should be of factorial
difference.
2. Friction effects are neglected in all models, i.e.,
analytical, standard and FEM model. When the
bearing operates in the hydrodynamic regime,
the friction is expected to be very small and as
such it is justified to neglect it in the stiffness
calculation. However, when the bearing operates
in the boundary or mixed-lubrication regime,
the friction coefficients are expected to have a
significant influence on the bearing’s stiffness.
Further, the dynamic load from the eigenmodes
is expected to be small enough to not cause the
transition from the stick to the slip state in the
contact along the roller’s line of action. That
8 Razpotnik, M, – Bischof, T. – Boltežar, M.

Strojniški vestnik - Journal of Mechanical Engineering 69(2023)7-8, 289-298
297 The Dynamics of Tapered-roller Bearings – A Bottom-up Validation Study
9  REFERENCES
[1] Jones, A.B. (1960). A general theory for elastically constrained 
ball and radial roller bearings under arbitrary load and speed 
conditions. Journal of Basic Engineering, vol. 82, no. 2, p. 
309-320, DOI:10.1115/1.3662587.
[2] Harris, T.A. (1984). Rolling Bearing Analysis, John Wiley, New 
York.
[3] Rao, J.S. (1983). Rotor Dynamics, John Wiley, New York.
[4] While, M.F. (1979). Rolling element bearing vibration 
transfer characteristics: Effect of stiffness. Journal 
of Applied Mechanics, vol. 46, no. 3, p. 677-684, 
DOI:10.1115/1.3424626.
[5] Gargiulo, E.P. (1980). A simple way to estimate bearing 
stiffness. Machine Design, vol. 52, p. 107.
[6] Lim, T.C., Singh, R. (1990). Vibration transmission through 
rolling element bearings, part 1: Bearing stiffness formulation. 
Journal of Sound and Vibration, vol. 139, no. 2, p. 179-199, 
DOI:10.1016/0022-460X(90)90882-Z.
[7] Lim, T. C., Singh, R. (1994). Vibration transmission through 
rolling element bearings, part 5: Effect of distributed contact 
load on roller bearing stiffness matrix. Journal of Sound 
and Vibration, vol. 169, no. 4, p. 547-553, DOI:10.1006/
jsvi.1994.1033.
[8] Mul, J. M., Vree, J.M., Maas, D.A. (1989) Equilibrium and 
Associated Load Distribution in Ball and Roller Bearings 
Loaded in Five Degrees of Freedom While Neglecting 
Friction-Part I: General Theory and Application to Ball 
Bearings. Journal of Tribology, vol. 111, no. 1, p. 142-148, 
DOI:10.1115/1.3261864.
[9] Lim, T.C., Singh, R. (1990). Vibration transmission through 
rolling element bearings, part 2: System studies. Journal 
of Sound and Vibration, vol. 139, no. 2, p. 201-225, 
DOI:10.1016/0022-460X(90)90883-2.
[10] Mul, J.M., Vree, J.M., Maas, D.A. (1989). Equilibrium and 
associated load distribution in ball and roller bearings 
loaded in five degrees of freedom while neglecting friction-
Part II: Application to roller bearings and experimental 
verification. Journal of Tribology, vol. 111, no. 1, p. 149-155, 
DOI:10.1115/1.3261865.
[11] ISO/TS 16281:2008. Rolling Bearings - Methods for 
Calculating the Modified Reference Rating Life for 
Universally Loaded Bearings. International Organization for 
Standardization, Geneva.
[12] Cao, H., Niu, L., Xi, S., Chen, X. (2018). Mechanical model 
development of rolling bearing-rotor systems: A review. 
Mechanical Systems and Signal Processing, vol. 102, p. 37-
58, DOI:10.1016/j.ymssp.2017.09.023.
[13] Guo, Y., Parker, R. G., (2012). Stiffness matrix calculation 
of rolling element bearings using a finite element/contact 
mechanics model. Mechanism and Machine Theory, vol. 51, 
p. 32-45, DOI:10.1016/j.mechmachtheory.2011.12.006.
[14] Fang, B., Zhang, J., Yan, K., Hong, J., Yu Wang, M. (2019). 
A comprehensive study on the speed-varying stiffness of 
ball bearing under different load conditions. Mechanism 
and Machine Theory, vol. 136, p. 1-13, DOI:10.1016/j.
mechmachtheory.2019.02.012.
“output” — 2023/6/22 — 8:23 — page 9 — #9
Strojniški vestnik - Journal of Mechanical Engineering 63(2017)3, XXX-4
0 100 200 300 400 500 600 700 800 900 1000
Fz [N]
(a)
0.000
0.002
0.004
0.006
0.008
δz [mm]
FEM
Analytical
Standard
0 100 200 300 400 500 600 700 800 900 1000
Fz [N]
(b)
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
kz [N/mm]
×10
5
FEM Analytical Standard
Fig. 15. TRB being incrementally loaded in the axial direction;
a) load-displacement characteristic, b) corresponding total stiffness
being said, the stiffness of the TRB would have
been significantly increased when the friction
phenomena were also taken into account.
7 CONCLUSION
A bottom-up approach to characterise a bearing’s
dynamics is presented. A special test device was
designed and built to assess the quality of the
well-established modelling approach. The dynamic
properties of the system were measured in the form of
FRFs, where load-dependent nonlinearities, resulting
from the bearings were observed. It was shown
that the ball bearing model yields appropriate results,
whereas the TRB model requires modifications.
These outcomes were confirmed with a quasi-static,
load-displacement numerical insight, where a full
FEM model serves as a reference.
In future work it will be of great interest to see
how a TRB reacts when loaded in other than the pure
axial direction. In addition, the influence of a different
lubricationregimewouldshedsomelightonquestions
relating to the friction. The presented study represents
a good starting point for a possible new TRB model
derivation.
8 ACKNOWLEDGEMENT
TheauthorswouldliketothanktheZFFriedrichshafen
AG for supporting this research.
9 REFERENCES
[1] Jones, A. B., A general theory for elastically
constrained ball and radial roller bearings under
arbitrary load and speed conditions, Journal of Basic
Engineering, ASME 82 (1960) 309.
[2] Harris,T.A., Rolling bearing analysis, JohnWiley,New
york, 1984.
[3] Rao, J. S., Rotor dynamics, John Wiley, New york,
1983.
[4] While, M. F., Rolling element bearing vibration transfer
characteristics: Effect of stiffness, Journal of Applied
Mechanics, ASME46 (1979) 677.
[5] Gargiulo, E. P., A simple way to estimate bearing
stiffness, Machine design 52 (1980) 107.
[6] Lim,T.C.andSingh,R., Vibrationtransmissionthrough
rolling element bearings, part 1: bearing stiffness
formulation, JournalofSoundandVibration139(1990)
179.
[7] Lim,T.C.andSingh,R., Vibrationtransmissionthrough
rolling element bearings, part 5: Effect of distributed
contact load on roller bearing stiffness matrix, Journal
of Sound and Vibration169 (1994) 547 .
[8] Mul, J. M., Equilibrium and Associated Load
Distribution in Ball and Roller Bearings Loaded in Five
Degrees of Freedom While Neglecting Friction—Part
I: General Theory and Application to Ball Bearings,
Journal of Tribology 111 (1989) 142.
[9] Lim,T.C.andSingh,R., Vibrationtransmissionthrough
rolling element bearings, part 2: system studies,
Journal of Sound and Vibration 139 (1990) 201.
[10] Mul, J. M., Vree, J. M., and Maas, D. A., Equilibrium
and Associated Load Distribution in Ball and Roller
Bearings Loaded in Five Degrees of Freedom While
Neglecting Friction—Part II: Application to Roller
Bearings and Experimental Verification, Journal of
Tribology 111 (1989) 149.
[11] ISO/TS 16281:2008(E), Rolling bearings – methods
for calculating the modified reference rating life for
universally loaded bearings, Standard, International
Organization for Standardization, Geneva, CH, 2008.
[12] Cao, H., Niu, L., Xi, S., and Chen, X., Mechanical
model development of rolling bearing-rotor systems: A
review, Mechanical Systems and Signal Processing
102 (2018) 37.
[13] Guo,Y.andParker,R.G.,Stiffnessmatrixcalculationof
rolling element bearings using a finite element/contact
mechanicsmodel, MechanismandMachineTheory51
(2012) 32.
[14] Fang, B., Zhang, J., Yan, K., Hong, J., and Yu Wang,
M., A comprehensive study on the speed-varying
stiffness of ball bearing under different load conditions,
Mechanism and Machine Theory136 (2019) 1.
[15] Razpotnik, M.,
ˇ
Cepon, G., and Boltežar, M., A
smooth contact-state transition in a dynamic model
of rolling-element bearings, Journal of Sound and
Vibration 430 (2018) 196.
[16] Kabus, S., Hansen, M. R., and Mouritsen, O., A
New Quasi-Static Cylindrical Roller Bearing Model to
Accurately Consider Non-Hertzian Contact Pressure in
The Dynamics of Tapered-roller Bearings – A Bottom-up Validation Study 9

Strojniški vestnik - Journal of Mechanical Engineering 69(2023)7-8, 289-298
298 Razpotnik, M. – Bischof, T. – Boltežar, M.
[15] Razpotnik, M., Čepon, G., Boltežar, M. (2018). A smooth 
contact-state transition in a dynamic model of rolling-element 
bearings. Journal of Sound and Vibration, vol. 430, p. 196-
213, DOI:10.1016/j.jsv.2018.05.041.
[16] Kabus, S., Hansen, M.R., Mouritsen, O. (2012). A new quasi-
static cylindrical roller bearing model to accurately consider 
non-hertzian contact pressure in time-domain simulations. 
Journal of Tribology, vol. 134, no. 4, art. ID 041401, 
DOI:10.1115/1.4007219.
[17] Kabus, S., Hansen, M., Mouritsen, O. (2014). A new quasi-
static multi-degree of freedom tapered roller bearing model 
to accurately consider non-Hertzian contact pressures in 
time-domain simulations. Proceedings of the Institution 
of ol. Mechanical Engineers, Part K: Journal of Multi-body 
Dynamics, vol. 228, no. 2, DOI:10.1177/1464419313513446.
[18] Creju, S., Bercea, I., Mitu, N. (1995). A dynamic analysis of 
tapered roller bearing under fully flooded conditions. Part 
I: Theoretical formulation. Wear, vol. 188, no. 1-2, p. 1-10, 
DOI:10.1016/0043-1648(94)06551-9.
[19] Cretu, S., Mitu, N., Bercea, I. (1995). A dynamic analysis of 
tapered roller bearings under fully flooded conditions part 2: 
results. Wear, vol. 188, no. 1-2, p. 11-18, DOI:10.1016/0043-
1648(94)06552-7.
[20] Tong, V.C., Hong, S.W. (2014). Characteristics of tapered 
roller bearing subjected to combined radial and moment 
loads. International Journal of Precision Engineering and 
Manufacturing-Green Technology, vol. 1, no. 4, p. 323-328, 
DOI:10.1007/s40684-014-0040-1.
[21] Tong, V.C., Hong, S.W. (2015). Effects of roller profile on the 
stiffness of tapered roller bearings. Journal of Automation and 
Control Engineering, vol. 3, no. 2, p. 151-156, DOI:10.12720/
joace.3.2.151-156.
[22] Zhao, J.-S. Liu, W., Zhang, Y., Feng, Z.-J., Ye, J., Niu, Q.-B. 
(2013). Effects of gyroscopic moment on the damage of a 
tapered roller bearing. Mechanism and Machine Theory, vol. 
69, p. 185-199, DOI:10.1016/j.mechmachtheory.2013.06.002.
[23] Roda-Casanova, V., Sanchez-Marin, F. (2017). Contribution of 
the deflection of tapered roller bearings to the misalignment 
of the pinion in a pinion-rack transmission. Mechanism 
and Machine Theory, vol. 109, p. 78-94, DOI:10.1016/j.
mechmachtheory.2016.11.013.
[24] Houpert, L. (2002). Ball bearing and tapered roller 
bearing torque: analytical, numerical and experimental 
results. Tribology Transactions, vol. 45, no. 3, p. 345-353, 
DOI:10.1080/10402000208982559.
[25] Ai, S., Wang, W., Wang, Y., Zhao, Z. (2015). Temperature rise of 
double-row tapered roller bearings analyzed with the thermal 
network method. Tribology International, vol. 87, p. 11-22, 
DOI:10.1016/j.triboint.2015.02.011.
[26] Tong, V.-C., Hong, S.-W. (2016). The effect of angular 
misalignment on the running torques of tapered roller 
bearings. Tribology International, vol. 95, p. 76-85, 
DOI:10.1016/j.triboint.2015.11.005.
[27] Zhou, R.S., Hoeprich, M.R. (1991). Torque of tapered roller 
bearings. Journal of Tribology, vol. 113, no. 3, p. 590-597, 
DOI:10.1115/1.2920664.
[28] Gradu, M. (2000). Tapered roller bearings with improved 
efficiency and high power density for automotive 
transmissions. SAE 2020 World Congress, DOI:10.4271/2000-
01-1154.
[29] Wrzochal, M., Adamczak, S., Domagalski, R., Piotrowicz, 
G., Wnuk, S. (2022). New device proposed for industrial 
measurement of rolling bearing friction torque. Strojniški 
vestnik - Journal of Mechanical Engineering, vol. 68, no. 10, p. 
610-622, DOI:10.5545/sv-jme.2022.275.
[30] Zhang, J., Fang, B., Zhu, Y., Hong, J. (2017). A comparative 
study and stiffness analysis of angular contact ball 
bearings under different preload mechanisms, Mechanism 
and Machine Theory, vol. 115, p. 1-17, DOI:10.1016/j.
mechmachtheory.2017.03.012.
[31] Altair Optistruct Eng. (2017). Altair Optistruct Reference 
Guide, Troy.
[32] Sheng, X., Li, B., Wu, Z., Li, H. (2014). Calculation of ball bearing 
speed-varying stiffness. Mechanism and Machine Theory, vol. 
81, p. 166-180, DOI:10.1016/j.mechmachtheory.2014.07.003.