dynamic analysis model of spherical roller bearings with

Transcription

Behnam Ghalamchi
DYNAMIC ANALYSIS MODEL OF SPHERICAL
ROLLER BEARINGS WITH DEFECTS
Thesis for the degree of Doctor of Science (Technology) to be presented with
due permission for public examination and criticism in the Auditorium 1382
at Lappeenranta University of Technology, Lappeenranta, Finland on the 14th
of November, 2014 at noon.
Acta Universitatis
Lappeenrantaensis 596
Supervisors
Professor Aki Mikkola
Department of Mechanical Engineering
Lappeenranta University of Technology
Finland
Professor Jussi Sopanen
Finland
Reviewers
Professor Viktor Berbyuk
Chalmers University of Technology
Sweden
Associate Professor Morten Kjeld Ebbesen
University of Agder
Norway
Opponents
Professor Viktor Berbyuk
Chalmers University of Technology
Sweden
Associate Professor Morten Kjeld Ebbesen
University of Agder
Norway
ISBN 978-952-265-669-8
ISBN 978-952-265-670-4 (PDF)
ISSN 1456-4491
Yliopistopaino 2014
Preface
The research work of this thesis was carried out during the years 2011 - 2014 in
laboratories of Machine Design and Machine Dynamics at Lappeenranta University
of Technology.
Acknowledgements
First of all, I would like to express my sincere gratitude to my supervisors Professor Aki
Mikkola and Professor Jussi Sopanen for all the supports they provided during this entire
research. Your encouragement and enthusiasm have been a source of inspiration which
kept me going forward. I am grateful for the opportunity I was given to work in this
interesting research project.
I am grateful to Professor, Viktor Berbyuk and Associate Professor, Morten Kjeld Ebbesen
for reviewing this thesis and giving valuable comments and suggestions.
It is a pleasure to Thanks to all the members of Laboratories of Machine Design and
Machine Dynamic over years. Janne Heikkinen, Antti Valkeapää, Emil Kurvinen, Oskari
Halminen, Adam Klodowski, Marko Matikainen, Scott Semken, Ezral Bin Baharudin,
John Bruzzo, and Elias Altarriba, I feel honored to be part of this particular research
groups.
Special thanks to my friend and my former college Tuomas Rantalainen for his helps to
make me more familiar with these groups atmosphere. The financial support of Tekniikan
edistämissäätiö is highly acknowledged.
I will be eternally grateful for my parents for their encouragements and supports.
Zahra, without you I wouldn’t be where I am. You made it all possible and I will be
always grateful for your unconditional love and support.
Lappeenranta, November 2014
Behnam Ghalamchi
Abstract
Behnam Ghalamchi
Dynamic Analysis Model of Spherical Roller Bearings with Defects
Lappeenranta, 2014
90 pages
Acta Universitatis Lappeenrantaensis 596
Dissertation. Lappeenranta University of Technology
ISBN 978-952-265-669-8
ISBN 978-952-265-670-4 (PDF)
ISSN 1456-4491
Rolling element bearings are essential components of rotating machinery. The spherical
roller bearing (SRB) is one variant seeing increasing use, because it is self-aligning and
can support high loads. It is becoming increasingly important to understand how the SRB
responds dynamically under a variety of conditions. This doctoral dissertation introduces
a computationally efficient, three-degree-of-freedom, SRB model that was developed to
predict the transient dynamic behaviors of a rotor-SRB system. In the model, bearing
forces and deflections were calculated as a function of contact deformation and bearing
geometry parameters according to nonlinear Hertzian contact theory. The results reveal
how some of the more important parameters; such as diametral clearance, the number
of rollers, and osculation number; influence ultimate bearing performance. Distributed
defects, such as the waviness of the inner and outer ring, and localized defects, such
as inner and outer ring defects, are taken into consideration in the proposed model.
Simulation results were verified with results obtained by applying the formula for the
spherical roller bearing radial deflection and the commercial bearing analysis software.
Following model verification, a numerical simulation was carried out successfully for a
full rotor-bearing system to demonstrate the application of this newly developed SRB
model in a typical real world analysis. Accuracy of the model was verified by comparing
measured to predicted behaviors for equivalent systems.
Keywords: Spherical roller bearing, dynamic analysis, defect
UDC 624.078.54:621.822:621.822.6:004.94
C ONTENTS
1
2
3
4
Introduction
1.1 Rolling Element Bearings . . . . . . . . . . . .
1.2 Spherical Roller Bearings (SRB) . . . . . . . .
1.3 Objective and Scope of the Dissertation . . . .
1.4 Outline of the Dissertation . . . . . . . . . . .
1.5 Scientific Contributions and Published Articles
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Rotor-Bearing System Analysis
2.1 Rotor Dynamic Analysis . . . . . . . . . . . . . . . . . . . . . . .
2.1.1 Rigid Rotor . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.2 Flexible Rotor . . . . . . . . . . . . . . . . . . . . . . . .
2.1.3 Solving the Governing Equations of Motion . . . . . . . . .
2.2 Modeling the Spherical Roller Bearing . . . . . . . . . . . . . . . .
2.2.1 Dynamic Model of the Spherical Roller Bearing . . . . . .
2.2.2 Geometry of Contacting Elastic Solids . . . . . . . . . . . .
2.2.3 Geometry of Spherical Roller Bearing . . . . . . . . . . . .
2.2.4 Contact Deformation in Spherical Roller Bearing . . . . . .
2.2.5 Elastic Deformation in Spherical Roller Bearing . . . . . .
2.2.6 Calculating the Stiffness Matrix of Spherical Roller Bearing
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Modeling of Non-idealities in the Spherical Roller Bearing
3.1 Localized Defects . . . . . . . . . . . . . . . . . . . . .
3.1.1 Outer Race Defects . . . . . . . . . . . . . . . .
3.1.2 Inner Race Defects . . . . . . . . . . . . . . . .
3.2 Distributed Defects . . . . . . . . . . . . . . . . . . . .
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Numerical Examples and Experimental Verification
4.1 Case 1: Single Bearing Numerical Simulations . . . . . . . . . .
4.1.1 Single Bearing Load Analysis - Contact Forces . . . . . .
4.1.2 Elastic Deformation, Displacement, and Load . . . . . . .
4.1.3 Clearance, Displacement, and Load . . . . . . . . . . . .
4.1.4 Osculation Number, Displacement, and Load . . . . . . .
4.1.5 Number of Rollers, Displacement, and Load . . . . . . . .
4.1.6 Angular Alignment of Side-by-Side Roller Arrays . . . .
4.2 Case 2: Rigid Rotor Supported by Ideal Spherical Roller Bearings
4.2.1 Studied Structure . . . . . . . . . . . . . . . . . . . . . .
4.2.2 Simulation Results . . . . . . . . . . . . . . . . . . . . .
4.3 Case 3: Single Spherical Roller Bearings with Localized Defects .
4.3.1 Studied Structure . . . . . . . . . . . . . . . . . . . . . .
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4.4
4.5
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4.3.2 Single Bearing Load Analysis - Outer Ring Defects . . . . . . .
4.3.3 Single Bearing Load Analysis - Inner Ring Defects . . . . . . .
Case 4: Rotor Supported with Spherical Roller Bearings with Localized
Defects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4.1 Studied Structure . . . . . . . . . . . . . . . . . . . . . . . . .
4.4.2 Measurement Setup . . . . . . . . . . . . . . . . . . . . . . . .
4.4.3 Simulation Results - Rotor with Outer Ring Defects . . . . . .
4.4.4 Simulation Results - Rotor with Inner Ring Defects . . . . . . .
Case 5: Rotor Supported with Spherical Roller Bearings with Distributed
Defect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5.1 Studied Structure . . . . . . . . . . . . . . . . . . . . . . . . .
4.5.2 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . .
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Conclusions
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Bibliography
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Appendices
87
S YMBOLS AND ABBREVIATIONS
S YMBOLS
a
Aelem
A
A
b
B
cd
Cos
C
C
d
D
eub
e
E
F
F
G
G
I
I
J
kelem
ke
ks
K
K
L
m
me
melem
Mt
M
ne
N
N
P
major axis
area
amplitude
steady state matrix
minor axis
bearing widht
clearance
osculation
damping coefficent
damping matrix
diameter
depth
unbalance mass eccentricity
displacement of a bearing
elastic modulus
force
force vector
shear modulus
gyroscopic matrix
moment of inertia (axis in plane)
identity matrix
moment of inertia (axis perpendicular to plane)
single element stiffness matrix
elliptic parameter
shear correction factor
stiffness coefficent
stiffness matrix
lengt of an element
mass of an object
semi major axis of an ellipsoid
consistent mass matrix
total mass
mass matrix of the system
semi minor axis of an ellipsoid
number of rolling elements
shape function matrix
order of waviness
q
q̇
q̈
r
R
t
T
vector of generalized coordinates
first time derivative of generalized coordinates
second time derivative of generalized coordinates
radius
effective radius
time
transformation matrix
G REEK LETTERS
ψ
β
δ
ζ
θ
Θ
ν
<
ξ
ρ
τ
φ0
φ
ω
Ω
I , II
tilting motion of a rotor
roller angular position
displacement / compression
elliptic integral parameter
angular position
momentum
Poisson’s ratio
ratio between inner and outer radius of rotor element
elliptic integral parameter
density
parameter given by φ from the ellipse center
auxiliary angle
phase angle
rotational frequency
rotational velocity
1, 2
S UBSCRIPTS
0
1, 2, 3, 4
A
B
b
c
cg
con
d
D
e
elem
ex
g
gy
i
in
j
p
P
out
r
rpof
rpif
R
S
t
T
ub
W
x, y, z
initial state
1st , 2nd , 3rd and 4th variable
solid A
solid B
bearing
center
cage
contact
difference
defect
ellipse
element
external
gravitational
gyration
index of roller in a row
inner
index of a row
pitch
order of waviness
outer
roller
roller pass outer ring frequency
roller pass outer ring frequency
rotor
support
total
tangential
unbalance
waviness
about / in direction of x, y or z -axis
S UPERSCRIPTS
b
i
in
g
n
bearing
index of body i
inner
global coordinates
number of step in the Newton-Raphson method
out
tot
T
outer
total
transpose of vector or matrix
A BBREVIATIONS
DOF
FEM
FFT
ICP
ODE
RIF
SF
RMS
SRB
VC
Degrees Of Freedom
Finite Element Method
Fast Fourier Transform
Integrated Circuit Piezoelectric Sensor
Ordinary Differential Equation
Roller Pass Inner Ring Frequency
Shaft Rotating Frequency
Root Mean Square
Spherical Roller Bearing
Varying Compliance
C HAPTER 1
Introduction
Rotating machines are important in industrial machinery that contains several components.
Figure 1.1 shows some of those parts that have the most effect on the behavior of the
systems, such as the main rotor and bearings. Since there is a wide range of industrial
applications for rotor-bearing systems, it is becoming increasingly important to study the
dynamic behavior of rotating systems. The initial approach in rotor dynamic analysis
was to study different components individually. However, in order to have accurate
enough results to compare with real rotor-bearing systems, it is important to consider the
interaction between various elements using a computer simulation approach.
Figure 1.1: Rotating system with bearings
The topic of rotor dynamic analysis has been interesting and useful for researchers for
many decades. A large number of books have been published, and research in this field
is ongoing. Calculating the critical speed of the rotor as a major concern in designing
process of rotating machineries, finding a stable and safe operation region for the system,
and reducing the unexpected vibrations due to geometric imbalance in the rotor were
15
16
1 Introduction
studied at the early stages [46, 49, 64] of interest.
One of the simplest rotor-bearing systems is called the Jeffcott rotor, which contains a
bulk mass disk carried by a massless flexible shaft and supported by two linear bearings,
one at each end. However, in complex rotating systems, these simplifications are no
longer valid. For this purpose, a transfer matrix method and a finite element method
were used to predict the natural frequencies, mode shapes, and unbalance response of
systems. The transfer matrix method is an analytical approach that is used by Prohl [62].
Ruhl and Booker [67] applied a finite element method to a rotor dynamic analysis for
the first time. In this process, the rotor was defined by a divided rigid mass segment.
Childs [13] developed the finite element method for complicated models. In this approach,
the flexibility of the rotor and also bearing force effects could be described directly.
Recently, researchers have tried to develop the finite element methods to improve the
computational efficiency of simulation [76, 11]. They used a new modal analysis solution
which is effective in determining the dynamic characteristics of a rotor dynamic system
while in rotating condition.
The recent studies on rotor dynamic systems are mostly focused on considering the effect
of nonlinearity effects in the dynamic analysis of rotating equipment. Nonlinearities
in rotor dynamic analysis can be caused by different elements, for instance, geometric
nonlinearities [52, 56], rotor-base excitations [16, 17, 65], oil film in sliding bearings
[2, 79, 50], and magnetic bearings [39]. Another type of non-linearity in rotor systems
comes from the non-linear terms of bearing stiffness and damping.
This doctoral thesis introduces nonlinear dynamic model of spherical roller bearing to
predict the nonlinear dynamic behaviors of a rotor-bearing system. In general, rotor
dynamic analysis could be done in both the frequency and the time domain. Working in
the frequency domain is computationally efficient compared with a time domain analysis.
It can predict the stability and natural frequencies of the rotor system. However, for
analyzing the nonlinearity of the system which is come from the bearings, the steady
state responses are not adequately capable. It usually needs linearization of the bearing
data. Thus, the time transient analysis becomes necessary. For this purpose, by applying
modal synthesis method [42, 49, 82], we can decrease the total number of system’s
degrees of freedom, and by applying numerical integrators, such as different orders of the
Runge-Kutta method, the governing equations of motion for a rotor system can be solved
in a straightforward manner.
1.1
Rolling Element Bearings
Bearings are one of the more important components in mechanical systems, and their
reliable operation is necessary to ensure a safe and efficient operation of rotating machinery [28]. For this reason, a multipurpose dynamic rolling element bearing model capable
of predicting the dynamic vibration responses of rotor-bearing systems is important.
17
1.2 Spherical Roller Bearings (SRB)
However, bearings introduce nonlinearities, often leading to unexpected behaviors, and
these behaviors are sensitive to initial conditions. For rolling element bearings, the
significant sources of nonlinearity are the radial clearance between the rolling elements
and raceways and the nonlinear restoring forces between the various curved surfaces in
contact. A special type of non-linearity is introduced to the system if the contact surfaces
have distributed defects, such as waviness, or localized defects, such as inner or outer ring
defects. In order to study the rotating machinery vibration caused by the nonlinearity of a
bearing, transient analysis on detailed simulation should be used. As an example, El Saidy
[18] studied the shaft supported by a nonlinear rolling element bearing. A finite element
equation was applied and solved in this model. An analytical model of a high speed
rotor with a nonlinear rolling element bearing is introduced by Harsha [29]. Implicit type
numerical integration was applied to solve the differential equation of motion.
1.2
Spherical Roller Bearings (SRB)
As Figure 1.2 shows, a spherical roller bearing by itself can contain two rows of rollers,
an outer ring with a spherical raceway, and an inner ring with two spherical raceways
perpendicular to bearing axes, and a cage. The locking feature makes the inner ring
captive within the outer ring in the axial direction only. With their two rows of large
rollers, they have a high load-carrying capacity, axial forces in both directions, and high
radial forces, which make them irreplaceable in many heavy industrial applications.
Rollers
Cage
Exploded
view
Outer ring
Figure 1.2: Components of SRB
Inner ring
18
1 Introduction
Because of symmetrical rollers and spherical raceways, a spherical roller bearing compensates shaft tilting and misalignments without increasing friction or reducing bearing
service life (Figure 1.3).
Force
Figure 1.3: Shaft tilting and misalignments in SRB
Spherical roller bearings are used in many applications wherever a rotation shaft might
need to change the alignment of its axis or to carry high loads. Some examples of these
applications include different types of stone crushers (jaw or impact crushers), vibrating
screens, bulk mass handling, marine equipment, geared transmissions and modern high
power wind turbines.
By investigating the available literature on the dynamic analysis of rolling element
bearings, we could find that extensive research has been conducted on the dynamics of
ball bearings while studies related to spherical roller bearings have received short shrift.
For example, Goenka and Booker [24] extended the general applicability of the finite
element method to include spherical roller bearings (SRBs). In their research, triangular
finite elements with linear interpolation functions were used to model the lubricant
film. The loading conditions for spherical roller bearings with elastohydrodynamic and
hydrodynamic lubrication effects were analyzed by Kleckner and Pirvics [45]. They
simulated the mechanical behavior of spherical roller bearings in isothermal conditions.
1.3 Objective and Scope of the Dissertation
1.3
19
Objective and Scope of the Dissertation
This doctoral thesis introduces a new general-purpose spherical roller bearing model
developed to act as an interface element between a spinning rotor and its supporting
structure. Spherical roller bearings experience point contact between the inner race,
rolling element and outer race in the no-load condition and elliptical contact when
loaded. The modeling approach presented in this dissertation accounts for the loaded
condition and has three degrees of freedom. Its simplifying assumptions make the model
computationally efficient. It is accurate enough for engineering analysis, since it can
capture the most important dynamic properties of the bearing. It is also studied in [21, 22].
Model performance is demonstrated by comparing the results of two basic numerical
simulations to the results obtained using both commercial bearing analysis software
and the bearing radial deflection formula proposed by Gargiulo [19]. The simulations
focused on the more important design parameters: diametral clearance, number of rollers,
and osculation. A third numerical simulation of a full bearing system was performed
to demonstrate the application of this new SRB model in a typical real world analysis.
In the model, bearing forces and deflections were calculated as a function of contact
deformation and bearing geometry parameters according to nonlinear Hertzian contact
theory. The bearing force calculation routine can be used as a stand-alone program or
as part of a bearing stiffness matrix calculation routine in a multibody or rotor dynamic
analysis code.
For design and analysis purposes, a versatile dynamic model capable of modeling bearing
defects is important. Bearing defects are classified as distributed defects, such as surface
roughness, waviness, misaligned races, and off-size rolling elements, and local defects
such as cracks, pits, and spalls on the rolling surfaces [71, 73]. The defects are usually
the result of manufacturing error, improper installation, or abrasive wear.
These two categories of bearing imperfections are studied separately in the international
journals and conference proceedings, and make up the main core of this doctoral dissertation. Effect of defect shape in dynamic behavior of the bearing is studied in [23].
Two kinds of defects were studied: elliptical and simple defects in both inner and outer
rings. For the elliptical shape, the roller path the defect area without losing connection
with rings. This leads to varying contact stiffness in the defect area due to variation in
the contact geometry between the defect surfaces and the bearing components. In simple
defect, it is assumed that the roller immediately lose its connection when it is in the defect
area which does not impact contact stiffness. Finally, the proposed model is compared to
and verified by measured data taken from a rotor bearing system with a predefined local
defect. Bearing waviness as geometric inaccuracies might excite the rotating system to
vibrate. This distributed defect is studied in [34].
20
1.4
1 Introduction
Outline of the Dissertation
The thesis is divided into seven chapters with the following outline:
Chapter 1: Introduction
It provides the motivation, background, and objectives of the thesis. The operation
principle, characteristics, and applications of spherical roller bearings are introduced.
Scientific contributions and publications according to this research are also presented.
Chapter 2: Rotor-bearing systems analysis
Two general types of rotor modeling are presented in this chapter. Governing equation
of motion for both rigid and flexible rotors is introduced. In the following, nonlinear
model of spherical roller bearing model is proposed in this chapter. It contains the whole
process of calculation of bearing for in general coordinate system as well as introducing
the bearing geometry, nonlinear contact Hertzian contact theory and elastic deformation
in spherical roller bearing under external loads.
Chapter 3: Modeling of non-idealities in spherical roller bearing
Different kinds of non-idealities in spherical roller bearing are introduced in this section.
It provides information about applying defect parameters in bearing modeling for both
local and distributed defects.
Chapter 4: Numerical examples and experimental verification
This chapter is introduced five different case studies. To verify the new bearing model, a
series of verifying numerical calculations were carried out for a single SRB subjected to
a simple radial load. Some of the simulation results were compared to results obtained by
applying the formula for the spherical roller bearing radial deflection and the commercial
bearing analysis software. Also, numerical simulation was carried out of a full rotorbearing system comprising a rigid rotor supported by SRBs on either end of the rotor
axle. Finally, rotor-bearing system includes bearing non-idealities are presented.
Chapter 5: Conclusions
Conclusions and discussion of this doctoral dissertation is provides in this chapter
1.5
Scientific Contributions and Published Articles
The doctoral thesis provides the following scientific contributions:
• a computationally efficient three-degree-of-freedom model of a spherical roller
bearing is proposed. In the model, bearing forces and deflections were calculated
as a function of contact deformation and bearing geometry parameters according
to nonlinear Hertzian contact theory. Transient dynamic behaviors of a rotor-SRB
system and vibration cause by nonlinearity of bearing are studied;
1.5 Scientific Contributions and Published Articles
21
• distributed defects, such as the waviness of the inner and outer ring, and localized
defects, such as inner and outer ring defects are taken into consideration in the
proposed model. Using this model to derive a catalog of behaviors as a function of
defect type and location, it is possible to identify an actual defect by examining its
measured behavior and comparing it to the catalog of derived behaviors, and
• the effects of important SRB parameters, such as diametral clearance, the number
of rollers, and osculation number on ultimate bearing performance are presented.
Some of the results presented in this thesis have also been published in the following
papers:
• Ghalamchi Behnam, Sopanen Jussi, Mikkola Aki, ‘Simple and Versatile Dynamic
Model of Spherical Roller Bearing.’ International Journal of Rotating Machinery.
2013.
• Heikkinen Janne, Ghalamchi Behnam, Sopanen Jussi, Mikkola Aki, 2014, ‘TwiceRunning-Speed Resonances of a Paper Machine Tube Roll Supported by Spherical
Roller Bearings – Analysis and Comparison with Experiments’, Proceedings of
the ASME International Design Engineering Technical Conferences IDETC2014,
August 17–20, 2014, Buffalo, NY, USA
• Ghalamchi Behnam, Sopanen Jussi, Mikkola Aki, ‘Nonlinear Model of Spherical Roller Bearing Including Localized Defects’ , 9th IFToMM International
Conference on Rotor Dynamics, 2014, Milan, Italy
• Ghalamchi Behnam, Sopanen Jussi, Mikkola Aki, 2014, ‘Dynamic model of spherical roller bearing’, Proceedings of the ASME International Design Engineering
Technical Conferences IDETC2013, 2013, Portland,OR, USA
• Ghalamchi Behnam, Sopanen Jussi, Mikkola Aki, ‘Nonlinear Model of Spherical
Roller Bearing Including Localized Defects’. (To be submitted).
22
1 Introduction
C HAPTER 2
Rotor-Bearing System Analysis
2.1
Rotor Dynamic Analysis
In scientific research, rotors are described as two general types: rigid and flexible. In the
flexible rotor, all flexible eigenfrequencies of a system at low frequencies that are crossed
during the run-up and run-down are considered. In the rigid rotor, the effects of these
flexible eigenfrequencies are neglected [35]. In fact, a pure rigid rotor is just a simplified
model of a real rotor, which could decrease the total number of degrees of freedom in
systems and prevent complex mathematical modeling. The governing equation of motion
in both rigid and flexible rotors, based on Newton’s II law of motion, can be expressed as
follows:
Mq̈(t) + (C + ΩG)q̇(t) + Kq(t) = F (t)
(2.1)
where M is the mass matrix, q is the displacement vector, q̇ and q̈ are the first and second
time derivatives of generalized coordinates C is the damping matrix, Ω is rotation speed,
G is the gyroscopic matrix, K is the stiffness matrix, F is a force vector, and t is a time
variable which will not be shown in the following equations.
2.1.1
Rigid Rotor
For a rigid rotor, the effect of internal damping can be neglected [20], so it should be
equal to zero ( i.e., C = 0). According to rigid rotor assumptions, the stiffness matrix
also should be zero (K = 0). Therefore, the equation of motion of a rigid rotor in the
center of mass coordinates can be written as:
MRc q̈ Rc + ΩGRc q̇ Rc = F Rc
23
(2.2)
24
2 Rotor-Bearing System Analysis
T
In this equation, q Rc = x y βx βy , showing the transversal and tilting motions
of the rotor in the x and y directions (Figure 2.1), and subscripts R and c respectively,
refer to the rotor and center of mass of the rotor. Equation (2.3), which follows, presents
the mass matrix MRc , the gyroscopic matrix GRc , and the force vector F Rc in center of
gravity coordinates.
MRc
GRc
F Rc

mR 0
 0 mR
=
 0
0
0
0

0 0 0
0 0 0
=
0 0 0
0 0 −1
 
Fx
 Fy 

=
Θx 
Θy

0 0
0 0

Ix 0 
0 Iy

0
0
I
1 z
0
(2.3a)
(2.3b)
(2.3c)
In Equation (2.3), mR is the rotor mass, and Ix and Iy are the transversal moments of
inertia about the x- and y axes, respectively, and Iz is the polar moment of inertia about
the z axis. F and Θ denote force and moments on their axes. The rotor is assumed to be
axisymmetric, so Ix = Iy .
y
Bearing II
ψy
Bearing I
z
ψx
CS
KS
dI
dII
CS
KS
Figure 2.1: Rigid rotor modeling using four degrees of freedom
According to Equation (2.2) The equation of motion of rigid rotor in bearing coordinates
is as follows:
25
2.1 Rotor Dynamic Analysis
(2.4)
T
Subscript b refers to the bearings. In bearing coordinates, q Rb = xI yI xII yII
shows the rotor displacements at the SRBs in positions I and II in the x and y directions.
MRb and GRb can be calculated as below:
MRb q̈ Rb + ΩGRb q̇ Rb = F Rb
MRb = TT
2 MRc T2
(2.5a)
GRb = TT
2 GRc T2
(2.5b)
T
where T2 = T−
1 . Transformation matrix T1 can be defined as follows:

1
 0
T1 = 
 0
−dI
0
1
−dI
0
1
0
0
dII

0
1 

dII 
0
(2.6)
In Equation (2.4), F Rb includes the external forces (F ex ), the bearing forces (F b ), is calculated in Equation (2.67), the gravity forces for the rigid rotor (F g ), and the unbalance
forces as shown by Equation (2.7).
F Rb = F ex + F b + F g + F ub
(2.7)
For the bearing housing, the equation of motion also can be written as:
MS q̈ S + CS q̇ S + KS q S = 0
(2.8)
T
where subscript S refers to the supports and q S = xSA ySA xSB ySB , shows
the displacements of the bearings housing in the x and y directions. The mass matrix
MS , the damping matrix CS , and the stiffness matrix KS are presented as follows:
MS = mS I4
(2.9a)
CS = CS I4
(2.9b)
KS = KS I4
(2.9c)
where I4 is a 4×4 identity matrix. Finally, for the whole rotor-SRB system, the assembly
matrix according to Equations, (2.4) and (2.8) can be written as follows:
MRb 0
q̈ Rb
ΩGRb 0
q̇ Rb
0 0
q Rb
F Rb
+
+
=
0
MS
q̈ S
0
CS
q̇ S
0 KS
qS
0
(2.10)
26
yi
ψy,i
xi
ψz,i
zi
ψx,i
z
yi+1
ψy,i+1
xi+1
ψz,i+1
zi+1
ψx,i+1
Figure 2.2: Three dimensional beam element
2.1.2
Flexible Rotor
In rotor dynamics the finite element method (FEM) is the most convenient method to
describe complex real-world machinery. In this study, the basic element is a 12-degreeof-freedom three-dimensional two-node beam element (Figure 2.2), whose stiffness,
damping, and gyroscopic matrices can be found in [1]. Each node of the element has
six-degrees-of-freedom.
The stiffness matrix kelem for the used 3D beam element can be written as:
 Aelem E

L
kelem


0


0



0


0


0

=
 Aelem E
− L


0


0



0



0
0
ay
0
az
0
0
GJ
L
0
cz
−cy
0
0
0
ey
0
ez
0
0
0
0
0
Aelem E
L
0
0
0
cz
−cy
0
0
0
ay
0
az
− GJ
L
0
0
0
0
0
GL
L
fy
0
0
fz
0
0
0
−cy
cz
0
0
0
−ay
0
0
−az
0
0
0
cz
−cy
0
Symmetric
0
0

























ey
0 ez
(2.11)
27
where Aelem is element area, E is elastic modulus, L is element length, J is perpendicular
moment of inertia, and G is shear modulus. Note that the cross-section of the rotor is
assumed to be symmetrical. This means that coefficients ay = az , ey = ez , cy = cz ,
fy = fz . These coefficients are identical as follows:
12EI
L3 (1 + φ)
(2.12)
6EI
+ φ)
(2.13)
ey = ez =
(4 + φ)EI
L(1 + φ)
(2.14)
fy = fz =
(2 − φ)EI
L(1 + φ)
(2.15)
ay = az =
cy = cz =
L2 (1
where
φ=
12EI
ks GAelem L2
(2.16)
where I is moment of inertia of the beam cross-section and ks is the shear correction factor
that takes into account the non-uniform shear stress distribution over the beam crosssection. The shear correction factor can be defined for a hollow circular cross-section as
follows [12]:
ks =
6(1 + ν)(1 + <2 )2
(7 + 6ν)(1 + <2 )2 + (20 + 12ν)<2
(2.17)
where ν is posson’s ration and < is the ratio between inner and outer radius of rotor
element.. The consistent mass matrix melem for the used 3D beam element can be written
as:
28
1

3
melem

 0 Ay

0
0
Az


0
0
0


0
−Cy
0

0
 0 Cz
= Mt 
1
0
0
6

0 B
0
y

0
0
Bz


0
0
0


0
0
Dy
0 −Dz
0
Symmetric
Jx
3A
0
0
Ey
0
Ez
0
0
0
0
0
Jx
6A
0
Dy
−Dz 0
0
0
1
3
0
0
Ay
0
Az
0
0
0
0
0
Fy
0
0
Fz
0
0
Cy
0 −Cz 0
Jx
3A
0
0

























Ey
0 Ez
(2.18)
where
Mt = ρAelem L
Ay = Az =
By = Bz =
Cy = Cz =
11
( 210
+
Dy = Dz =
Ey = Ez =
Fy = Fz =
13
( 420
+
1
+
( 105
1
−( 140
+
1
60 φ
1
60 φ
+
13
35
+
7
10 φ
+ 13 φ2 + 65 ( Lr )2
(1 + φ)2
9
70
+
3
10 φ
+ 16 φ2 − 65 ( Lr )2
(1 + φ)2
11
120 φ
+
1 2
24 φ
3
40 φ
+
1 2
24 φ
+
1
+ ( 10
− 21 φ)( Lr )2 )L
(1 + φ)2
1 2
120 φ
1 2
120 φ
(2.19)
(2.20)
(2.21)
(2.22)
1
− ( 10
− 21 φ)( Lr )2 )L
(1 + φ)2
(2.23)
2
+ ( 15
+ 16 φ + 31 φ2 )( Lr )2 )L2
(1 + φ)2
(2.24)
1
+ ( 30
+ 61 φ − 16 φ2 )( Lr )2 )L2
(1 + φ)2
(2.25)
29
and where the radius of gyration is defined as:
r
rgy =
I
Aelem
(2.26)
Matrices are formed using a standard assembly procedure of FEM. A generalized
displacement, that is, the nodal displacement is obtained using a shape function matrix
N.
q g = Nq
(2.27)
The superscript g refers to global coordinates. The shape function matrix is constructed
using beam elasticity theory ([12, 47]). The complete system consists of the generalized
displacements of all the nodes in global coordinates. The equation of motion of the
complete system is
Mg q̈ g + (Cg + ΩGg )q̇ g + Kg q g = F g
(2.28)
where q g is the global displacement vector.
2.1.3
Solving the Governing Equations of Motion
Rotor dynamic analysis could be done in both frequency and time domain. In general,
working on frequency domain is computationally efficient, compare with time domain
analysis. It can provide the stability and natural frequencies of the rotor system. By
applying the state variable form of a dynamic system, second order differential equations
can be expressed as a set of simultaneous first order differential equations. To reduce
the differential equation which is shown in Equation (2.28) into a system of first order
differential equations, let
q = x1
(2.29a)
q̇ = x2
(2.29b)
q̈ = x3
(2.29c)
From Equation (2.29) we have
dq
= x2
dt
dq̇
ẋ2 =
= x3
dt
ẋ1 =
(2.30a)
(2.30b)
30
From Equations (2.29 ) and (2.30), Equation (2.28) can be re-written as:
ẋ2 = M−1 F − M−1 (C + ΩG)x2 − M−1 Kx1
(2.31)
Finally, the state space form of equation of motion is calculated as follows
ẋ1
x1
0
=A
+
ẋ2
x2
M−1 F
(2.32)
where A is a state matrix which defined as
0
I
A=
−M−1 K −M−1 (C + ΩG)
(2.33)
In this case, according to the eigenvalues solution from Equation ( 2.33 ), a Campbell
diagram, which relates the rotational speed and the natural frequencies of the system, and
also system mode shapes could be calculated.
Frequency Domain Solution: The term frequency response, refers to the steady state
response of a system. It is the response of the system to harmonic excitation. In this case,
steady state solution can be obtained by defining the displacement vector , by sinusoidal
function, as follows:
q(t) = asinωf t + bcosωf t
(2.34)
where the vectors a and b are coefficient vectors and t is time. By deriving the first and
second order of Equation ( 2.34 ) and applying those to Equation ( 2.28 ), steady state
solution can be obtained.
Transient Analysis: In the real world, in rotor bearing systems, it sometimes happens
that the frequency domain analysis does not provide adequate capability. Therefore time
transient analysis becomes necessary. In this case, the governing equation of motions is
integrated directly by applying numerical integrators such as different orders of RungeKutta method.
2.2
Modeling the Spherical Roller Bearing
A spherical roller bearing consists of a number of parts, including a series of rollers, a
cage, and the inner and outer raceways. Describing each component in detail can result
in a simulation model with a large number of degrees-of-freedom. Additionally, as with
all radial rolling bearings, spherical roller bearings are designed with clearance. This
clearance also increases the computational complexity of the system. However, bearing
2.2 Modeling the Spherical Roller Bearing
31
analysis computation should be efficient so it can be used to simulate the dynamics of
complete machine systems.
Creju et al. [14, 15] improved the dynamic analysis of tapered roller bearings by
improving integration of the differential equations describing the dynamics of the rollers
and bearing cage. Their study considered the effects of centrifugal forces and the
gyroscopic moments of the rollers. The effects of the correction parameters for roller
generators in spherical roller bearings were discussed by Krzemiński-Freda and Bogdan
[48]. In their study, they focused on determining a proper ratio of osculation coefficients
for both races to obtain the self-stabilization of the barrel shaped roller and to minimize
friction losses.
Olofsson and Björklund [58] performed 3D surface measurements and an analysis on
spherical roller thrust bearings that revealed the different wear mechanisms. A theoretical
model for estimating the stiffness coefficients of spherical roller bearings was developed
by Royston and Basdogan [66] showing that coefficient values are complicated functions,
dependent on radial and axial preloads. While this work is useful for qualitative analysis,
it cannot deliver the dynamic insights needed for understanding the high performance
machine systems.
Olofsson et al. [57] simulated the wear of boundary lubricated spherical roller thrust
bearings. A wear model was developed in which the normal load distribution, tangential
tractions, and sliding distances can be calculated to simulate the changes in surface
profile due to wear. Taking into account internal geometry and preload impacts, Bercea et
al. [6] applied a vector-and-matrix method to describe total elastic deflection between
double-row bearing races. However this study focused only on static analysis. Therefor,
it is not capable of delivering a detailed analysis of the complex dynamic behaviors of
spherical roller bearing systems involving nonlinear interactions between rollers and
inner/outer races.
Cao et al. [9, 10] established and applied a comprehensive spherical roller bearing
model to provide quantitative performance analyses of SRBs. In addition to the vertical
and horizontal displacements considered in previous investigations, the impacts of axial
displacement and load were addressed by introducing degrees-of-freedom in the axial
shaft direction. The point contacts between rollers and inner/outer races were considered.
These bearing models have a large number of degrees-of-freedom since there is one
degree-of-freedom (DOF) for each roller and an additional 3 to 5 DOFs for the inner
race. Its high complexity makes this bearing model unattractive for the analysis of
complete rotor-bearing systems. For example, a single gear-box can contain up to ten
roller bearings.
The effect of centrifugal forces on lubricant supply layer thickness in the roller bearings
was considered by Zoelen et al. [75]. In particular, this model is used to predict lubricant
layer thickness on the surface of the inner and outer raceways and each of the rollers. In
32
this extended model it is assumed that the lubricant layers for each of the roller raceway
contacts are divided equally between the diverging surfaces.
Although a large number of ball bearing models exist, there has been little study of
spherical roller bearing dynamics. For example, Harsha et al. [30, 32] studied the rolling
element dynamics for certain imperfect configurations of single row deep-grooved ball
bearings. The study revealed dynamic behaviors that are extremely sensitive to small
variations in system parameters, such as the number of balls and the number of waves.
A dynamic model of deep-groove ball bearings was proposed by Sopanen and Mikkola
[69, 70]. They considered the effects of distributed defects such as surface waviness and
inner and outer imperfections.
In this study, to improve the computational efficiency of the proposed spherical roller
bearing model the following simplifications have been introduced.
• cage movement is based on the geometric dimensions of the bearing; therefore, it is
assumed that no slipping or sliding occurs between the components of the bearing
and all rollers move around the raceways with equal velocity.
• the inner raceway is assumed to be fixed rigidly to the shaft.
• there is no bending deformation of the raceways. Only nonlinear Hertzian con-tact
deformations are considered in the area of contact between the rollers and raceways.
• the bearings are assumed to operate under isothermal conditions.
• rollers are equally distributed around the inner race, and there is no interaction
between them; and
• the centrifugal forces acting on the rollers are neglected.
2.2.1
Dynamic Model of the Spherical Roller Bearing
The bearing stiffness matrix and bearing force calculation routines are implemented
according to the block diagrams shown in Figure 2.3. The bearing geometries, material
properties and the displacements between the bearing rings are defined as inputs. For the
stiffness matrix calculation routine, the external force on the bearing is given as an input.
The bearing force calculation routine can be used as a stand-alone program or as part of
a bearing stiffness matrix calculation routine in a multi-body or rotor dynamic analysis
code.
33
Start
Read input
Read input
• SRB Geometry
• Displacements
External force
• SRB material
• Step size
Initial Displacements
Contact
stiffness
coefficents
Rows
Rollers
Updated Displacements
Contact elastic deformation
Bearing
force
calculation
Bearing
force
No
|External force −
Bearing force|
< tolerance?
Elastic compression > 0?
No
Yes
Roller contact force
Summation of
rollers contact forces
Yes
Saving KT as
bearing stiffness
matrix
Roller number
> total number
of the rollers?
No
Yes
Row number
> total number
of the rows?
No
Yes
Bearing force
End
Figure 2.3: Block diagram for the bearing stiffness matrix and bearing force calculation
In the following sections, the theory behind the bearing force and bearing stiffness matrix
34
calculation is explained in detail.
2.2.2
Geometry of Contacting Elastic Solids
Two solids that have different radii of curvature in two directions (x and y) are in point
contact when no load is applied to them. When the two solids are pressed together by a
force F , the contact area is elliptical. For moderately loaded spherical roller bearings,
the contact conjunction can be considered elliptical [45], as shown in Figure 2.4. The
following analysis will assume the curvature is positive for convex surfaces and negative
for concave surfaces [25].
F
rAx
rBy
rBx
y
rAy
x
Figure 2.4: Elliptical contact conjunctions
where rAx , rAy , rBx , and rBy are the radii of curvatures for two solids (A and B). The
geometry between two solids in contact can be expressed in terms of the curvature sum
R, and curvature difference Rd as follows [8, 26]
1
R
Rd
1
1
+
and
Rx Ry
1
1
= R
−
Rx Ry
=
(2.35)
(2.36)
The curvature sums in x and y are defined as per Equations (2.37) and (2.38).
1
Rx
1
Ry
=
=
1
+
1
rAx rBx
1
1
+
rAy
rBy
and
(2.37)
(2.38)
Variables Rx and Ry represent the effective radii of curvature in the principal x- and
y-planes. When the two solids have a normal load applied to them, the point expands to
35
an ellipse with a being a major axis and b being the minor axis. The elliptic parameter is
defined as [25]:
ke =
a
.
b
(2.39)
The elliptic parameter can be re-defined as a function of the curvature difference Rd and
the elliptic integrals of the first ξ and second ζ kinds as follows [26]:
2ξ − ζ (1 + Rd )
ke =
ζ (1 − Rd )
1/2
,
(2.40)
Equations (2.41) and (2.42) define the first and second kinds ξ and ζ .
−1/2
1
2
1 − 1 − 2 sin ϕ
ξ =
dϕ and
ke
0
1/2
Z π/2 1
1 − 1 − 2 sin2 ϕ
ζ =
dϕ
ke
0
Z
π/2 (2.41)
(2.42)
Brewe and Hamrock [8] used numerical iteration and curve fitting techniques to find
the following approximation formulas for the ellipticity parameter ke and the elliptical
integrals of the first ξ and second ζ kinds as shown below.
ke = 1.0339
Ry
Rx
0.6360
,
Rx
,
Ry
Ry
ζ = 1.5277 + 0.6023 ln
.
Rx
ξ = 1.0003 + 0.5968
2.2.3
(2.43)
(2.44)
(2.45)
Geometry of Spherical Roller Bearing
The most important geometric dimensions of the spherical roller bearing are shown in
Figure 2.5. Diametral clearance is the maximum diametral distance that one race can
move freely. Osculation is defined as the ratio between the roller contour radius and the
race contour radius as written in Equation (2.46).
Cos =
rr
rin , rout
(2.46)
36
Subscripts r, in, and out refer to roller, inner race, and outer race, respectively. Perfect
osculation is when Cos is equal to 1. In general, maximum contact pressure between the
race and the roller decreases as osculation increases. Decreasing contact pressure reduces
fatigue damage to the rolling surfaces; however, there is more frictional heating with
increasing conformity. A reasonable value for osculation and one that can be used in the
roller contour radius definition is 0.98 [28].
rin
φ0
cd
4
dr
rr
rout
dp
2
B
Figure 2.5: Dimension of spherical roller bearing
Figure 2.6 illustrates the radii of curvature between roller, outer race, and inner race of an
SRB.
out
O–1
in ,r out
rAx
Ax
out
rAy
in
rBy
in
out
rBx
in
rAy
in
rBx
out
rBy
O–2
Figure 2.6: Radii of curvature between roller, outer race, and inner race
37
Figure 2.6 suggests the radii of curvature for the roller-to-inner race contact area can be
written as follows:
dr
,
2
= rr ,
(2.47)
in
rAx
=
in
rAy
(2.48)
dp − dr cos φ0 −
2 cos φ0
= −rin .
in
rBx
=
in
rBy
cd
2
cos φ0
,
(2.49)
(2.50)
Similarly, the equations for the radii of curvature for roller-to-outer race contact can be
written:
dr
,
2
= rr ,
(2.51)
out
rAx
=
out
rAy
(2.52)
dp + dr cos φ0 +
2 cos φ0
= −rout .
out
rBx
= −
out
rBy
2.2.4
cd
2
cos φ0
,
(2.53)
(2.54)
Contact Deformation in Spherical Roller Bearing
From the relative displacements between the inner and outer race the resultant elastic
deformation of the ith rolling element of the j th row located at angle βji from the xaxis can be determined. The initial distance, A0 , between the inner and outer raceway
curvature centers (O–1, O–2) can be written, again based on Figure 2.6, as given by
Equation (2.55).
out in + rBy − dr − cd
A0 = rBy
2
(2.55)
The corresponding loaded distance for roller i in row j can be written as follows:
A
βji
=
r
i
δzj
2
2
i
+ δrj
(2.56)
i and δ i are the displacements for roller i in row j in axial and radial directions,
where δzj
rj
respectively, which can be determined using these equations.
38
(a)
ith
(b)
y
A-A plane
y
A
roller
ωout
βji
ωin
ez
ey
x
z
ey
A
ex
Figure 2.7: (a) Axial and (b) transverse cross-section in the A-A plane of spherical roller
bearing
i
δzj
i
δrj
= A0 sin(φ0 ) + ez ,
= A0 cos(φ0 ) +
(2.57)
ex cos(βji )
+
ey sin(βji ),
(2.58)
The variables ex , ey and ez represent displacements in the global xyz-coordinate system,
and βji is the attitude angle of roller i in row j (See Figure 2.7). The initial contact angle
is negative for the 1st row and positive for the 2nd row of the bearing.
The distance between race surfaces along the common normal is given by Equation (2.59).
out in + rBy − A βji .
d βji = rBy
(2.59)
Elastic compression becomes:
δβ i = dr − d βji .
(2.60)
j
And, the loaded contact angle in each roller element can be defined as follows:
φij
−1
= tan
i
δzj
i
δrj
!
.
(2.61)
39
2.2.5
Elastic Deformation in Spherical Roller Bearing
In a single rolling element, total deflection is the sum of the contact deflections between
the roller and the inner and outer races. The deflection between the roller and the race
can be approximated as given by Equation (2.62) [45].
δ0 =
Fcon
Kcon
2/3
,
(2.62)
the Fcon denotes normal load, and Kcon is the contact stiffness coefficient, which can be
calculated using the elliptic integrals and ellipticity parameter in this manner.
s
Kcon = πke E
0
Rξ
,
4.5ζ 3
(2.63)
The effective modulus of elasticity E 0 is defined as follows:
1
1
=
2
E0
2
2
1 − νA
1 − νB
+
EA
EB
,
(2.64)
E and ν are the modulus of elasticity and Poisson’s ratio of solids A and B. The total
stiffness coefficient for both inner and outer race contact areas can be expressed with the
next equation.
tot
Kcon
=
1
in )−2/3 + (K out )−2/3
(Kcon
con
3/2 .
(2.65)
According to Equations (2.60) and (2.65), the contact force for roller i in row j can be
calculated in this manner
3/2
tot
Fji = Kcon
δβ i
.
j
(2.66)
Finally, the total bearing force components acting upon the shaft in the x, y, and z
directions can be written according to following equations.
40
Fxb = −
Fyb = −
Fzb = −
2 X
N
X
Fji cos φij cos βji ,
(2.67)
Fji cos φij sin βji , and
(2.68)
Fji sin φij ,
(2.69)
j=1 i=1
2 X
N
X
j=1 i=1
2 X
N
X
j=1 i=1
The variable N is the number of rolling elements in each row. In Equations (2.67-2.69),
only positive values of the contact force Fji are taken into account and Fji = 0 for the
negative values.
2.2.6
Calculating the Stiffness Matrix of Spherical Roller Bearing
In general, an accurate estimation of the SRB stiffness matrix is needed which, since it
has a significant effect on the static and dynamic analyses of rotating mechanical systems.
According to Equation (2.67), for a known given load, the displacements of bearing are
calculated using the Newton-Raphson iteration procedure as follows:
(n)
q (n+1) = q (n) − (KT )−1 F (n)
(2.70)
(n)
The displacement values can be calculated at step n + 1. In Equation (2.70), KT is the
tangent stiffness matrix, and vector F (n) includes the bearing forces and external forces
at iteration step nth as follows:
(n)
F (n) = F b
− F (n)
ex
(2.71)
The tangent stiffness matrix can be written as:
(n)
KT =
∂F (n)
∂q (n)
(2.72)
It can be seen that the tangent stiffness matrix is essentially a matrix of sensitivities. In
particular, it is the sensitivities of the internal member forces to perturbations in the nodal
displacement DOFs of the system. In this calculation process, the convergence criterion
for the iteration is defined as follows:
|F | < 0.001 · |F ex |
(2.73)
C HAPTER 3
Modeling of Non-idealities in the Spherical Roller
Bearing
Thus far, a large number of bearing dynamic behavior studies have been published
that target distributed defects such as waviness. Wardle [78] derived a vibration force
analysis approach to determine the wavelength of surface imperfections in thrust loaded
ball bearings. In ball bearing defect analysis, most studies have targeted non-linear
dynamic bearing behaviors resulting from surface waviness [59, 36, 37, 38, 30, 33, 5, 31].
Typically, a sinusoidal function is defined as the model of waviness. In some cases,
waviness studies include the effects of centrifugal force and gyroscopic moment as well
as the internal clearances of the ball bearings. Also, there are some vibration studies for
angular contact ball bearings [3, 4] and for roller bearings [32, 77] with inner and outer
races and with rolling surface waviness.
For local defects in ball and roller bearings, several works offer mathematical models for
the detection of bearing defects. McFadden and Smith [53, 54] carried out one of the
first mechanical modeling studies of localized bearing defects. Initially, they presented a
vibration model with single point defects in the inner raceway, also considering the effects
of bearing geometry, shaft speed, and bearing load distribution. Finally, they extended
their model to include multiple point defects under radial load. Vibration was modeled as
the product of a series of impulses that occur at the frequency of the interaction between
rolling elements and defects, olso known as the rolling element passing frequency.
Tandon and Choudhury [73] presented a model for predicting the vibration frequencies of
rolling bearings resulting from a localized defect on the outer race, inner race, or on one
of the rolling elements under radial and axial loads. They showed, in the characteristic
defect frequencies, that a discrete spectrum has a peak. The “healthy and faulty” model
for a rolling element bearing based on a multi-body dynamic approach was proposed
by Nakhaeinejad and Bryant [55]. They studied the effects of a localized fault type on
vibration response in rolling element bearings.
41
42
3 Modeling of Non-idealities in the Spherical Roller Bearing
Sopanen and Mikkola [69, 70] proposed a model for deep-groove ball bearings include
localized and distributed defects. They defined the distributed defects as a Fourier cosine
series, and the shapes of local defects were described using their length and height. This
model could be used in general multi-body or rotor dynamics code as an interference
element between the rotor and the housing. Rolling element bearings analysis with
various defects on different components of the bearing structure was proposed by Kiral
and Karagulle [44]. They employed a finite element vibration analysis to detect defects in
rolling element bearings, and also studied the effects on the time and frequency domain
parameters of defect location and quantity.
Rafsanjani et al. [63] studied the topological structure and stability of rotor roller bearing
systems, including surface defects and internal clearance, by applying the classical
Floquet theory. They linearized the nonlinear equation of motion and solved the equations
numerically by using the modified Newmark time integration technique. For deep groove
ball bearings with single and multiple race defects, vibration analysis was presented by
Patel et al. [60], wherein the vibration results were presented in the frequency domain;
ignored in earlier studies.
Kankar et al. [40] studied fault diagnosis for high speed rolling element bearings with
local defects in inner and outer races and also rollers by using a response surface technique.
This model predicts peaks in a discrete spectrum at the characteristic frequencies. The
model was validated with measurement data. Tadina and Boltezar [72] improved the
vibration simulation of ball bearing with local defects during run-up. In the proposed
model, local defects are defined using an ellipsoidal depression in the inner and outer
races and a flattened sphere on the roller, which leads to different contact stiffness in the
defect area.
Liu et al. [51] studied vibration analysis methods for ball bearings with a local defect by
applying a piecewise response function. The amplitude of the impulse generated by a
ball bearing passing over a local defect was determined with respect to the shape and size
of the local defect. The effects of localized defects, such as micron-sized spalls, on the
vibration response of ball bearings were studied by Kankar et al. [41]. They proposed a
mathematical model based on Hertzian elastic contact deformation theory to predict the
characteristic defect frequencies and amplitudes under radial load.
There are also many studies on measurement techniques for the detection of bearing
defects and health monitoring. For one of the first studies, Tandon and Choudhury
[74] reviewed the vibration and acoustic measurement methods for the detection of
localized and distributed defects in rolling element bearings. They compared several
time domain parameters, such as overall Root Mean Square (RMS) level, crest factor,
probability density, and kurtosis, and found kurtosis to be the most effective. These
kinds of studies have continued, and especially in recent years, new methods have been
presented [7, 43, 50, 61, 68, 80, 81].
According to literature reviews, extensive research has been conducted to study the
3.1 Localized Defects
43
analysis of ball bearing defects. There are fewer studies related to spherical roller
bearings. Cao and Xiao [10] presented a dynamic model for a spherical roller bearing
that included the effect of surface defects, preloads, and radial clearance in the dynamic
behavior of the bearing. They assumed point and line contacts between rollers and the
inner and outer races according to different bearing forces. They also defined race and
roller surface waviness as sinusoidal functions. As a regular Hertz point contact problem
for the defect point, they were able to consider large point defects in their numerical
calculations, assuming two degrees of freedom for each roller, which led to a large number
of degrees of freedom for the system and a correspondingly computationally inefficient
model.
The objective of this study is to propose a simple and efficient dynamic model of the
spherical roller bearing with local defects in the inner and outer races. The contact
forces between rollers and the inner and outer races are described according to nonlinear
Hertzian contact deformation. The effect of radial clearance has been taken into account.
The defect geometry for the inner and outer race is an impressed ellipsoid. This leads
to varying contact stiffness in the defect area due to variation in the contact geometry
between the defect surfaces and the bearing components. In addition, a simple defect
shape, which does not impact contact stiffness, is modeled. The results are compared to
those of elliptical defects.
3.1
Localized Defects
In most cases, the non-idealities, such as the waviness of the rings, misalignments and
bearing defects, are the main reason for the vibration in the bearing. In this research, the
vibration response according to the local defects on the inner and outer race, and also on
rollers, is studied. In most of the previously published studies of rolling element bearings
with local defects, it was usually assumed that the rolling elements in the region of the
defect lose their connection to the inner or outer race, thus causing a large impulsive
force on rotor bearing systems [73]. Some recent studies [72] have proposed modeling
the bearing defect as an ellipsoidal shape while taking into account the contact stiffness
variation in the defect location. In this study, both these modeling approaches are studied
and compared in the case of a spherical roller bearing.
3.1.1
Outer Race Defects
It is assumed that the center of defect on the outer race is located at angle φ from the
horizontal axis. Correspondingly, the angle occupied by the defect is equal to φD and
the maximum depth of the defect is Dd as shown in Figure 3.1. When a roller with an
attitude angle of βji passes the defect, the contact elastic deformation can be calculated as
follows:
44
δ(βji )D = dr − d(βji ) − δ + (βji )
(3.1)
In comparing Equations (3.1) and (2.60), δ + (βji ) is the additional displacement in the
dent area, which is subtracted from total displacement (see Appendix A). Note that the
additional displacement is defined only when φ − φ2D < βji < φ + φ2D .
φ
βji
δ + (βji )
θcon
con
φD
Dd
Figure 3.1: Outer race defect
As mentioned above, the elliptical defect leads to different contact stiffness in the dent
area, since the radii of curvature in the dent area is different compared to that of the race.
According to Hertzian contact theory, a new curvature sum and difference as well as new
contact stiffness can be evaluated (Equations (2.35) to (2.44)). However, the evaluation of
new contact stiffness in the dent area, in the case of the elliptical defect, is straightforward;
out (Equation (2.53)) has be re-evaluated as follows [72]:
only the radius of curvature rBx
out
rBx
=
m e ne
3
2
(m2e sin τ + n2e cos2 τ ) 2
(3.2)
where me is the semi major axis, ne is the semi minor axis of the ellipsoid and τ is a
parameter given by the polar angle φ from the ellipse center as:
−1
τ = tan
me
tan φ
ne
(3.3)
From Equations (2.66) and (3.1) and by re-calculating contact stiffness, the contact force
for roller i in row j can be calculated as follows:
45
3.1 Localized Defects
32
i
tot
Fj,D
= kcon,D
δ βji ,D cos βji − θcon
(3.4)
tot
where subscript D refers to the defect area, kcon,D
is the contact stiffness, and the θcon
is the contact angle of the roller (see Figure 3.1) to get the exact force direction to the
center of bearing.
3.1.2
Inner Race Defects
Assuming that the elliptical defect on an inner race with a length equal to φD and with a
maximum depth of Dd (Figure 3.2), the contact elastic deformation when the roller with
attitude angle of βji passes the defect, will be calculated as follows:
δ βji
,D
(3.5)
= δ + βji − dr + d βji
and the contact force for roller i in row j can be calculated as follows:
23
i
tot
Fj,D
= kcon,D
δ βji ,D cos θcon − βji
(3.6)
tot
where kcon,D
is the contact stiffness which follows the same procedure of Equations (3.2)
and (3.3).
βji
Dd
θcon
δ + βji
φD
Figure 3.2: Inner race defect
46
3.2
Distributed Defects
The waviness in the bearing ring can be presented as a Fourier cosine series as follows
R(θ) =
n
X
AP cos (P θ + φP )
(3.7)
P =1
where AP is the amplitude and φP is the phase angle of P th order waviness, and θ is
the angular coordinate of the bearing ring which is shown in Figure 3.3. The amplitudes
and phase angles of the harmonic components can be obtained from measurements by
analyzing the results with the fast Fourier transform (FFT).
y
ith roller
x
Am
Figure 3.3: Race waviness model
The effect of the waviness of the bearing ring can be included in the elastic deformation
of roller i in row j, (Eq. 2.24) as follows:
δβ i
j
,W
= δβ i +
n
X
j
P =1
i
Ain
P cos P βj − θin + φP
where the summation describes the waviness of the inner ring.
(3.8)
C HAPTER 4
Numerical Examples and Experimental
Verification
4.1
Case 1: Single Bearing Numerical Simulations
This study introduces a computationally efficient, three-degree-of-freedom, SRB model
that was developed to predict the transient dynamic behaviors of a rotor-SRB system. To
verify the new bearing model, a series of verifying numerical calculations were carried out
for a single SRB subjected to a simple radial load. The SRB modeled was a double-row
spherical roller bearing (FAG 21322-E1-TVPB) with 16 roller elements in each row.
Table 4.1 gives the relevant dimensions and parameters of the roller bearing, which were
used to define the model. All numerical calculations were performed using Matlab-2011b.
Some of the Matlab results were compared to results obtained by applying the formula for
the spherical roller bearing radial deflection (Gargiulo [19]) and the commercial bearing
analysis software BearinX provided by the Schaeffler Group.
The model verification analysis series comprised six sets of Matlab numerical calculations,
each focused on a specific area of behavior. In the first set, roller contact forces were
calculated for four levels of radial load. The second set explored the relationship between
bearing displacement and load. A third set of calculations established how SRB elastic
deformation changes with load as a function of diametral clearance. The effect of
osculation number on bearing displacement for different levels of radial loading was
the area of focus of the fourth set of calculations. The fifth set looked at changes in
displacement as a function of radial load and the number of bearing rollers. Finally, the
last pair of verification calculations looked at bearing displacement with respect to time
for two separate arrangements of the caged side-by-side roller arrays; when they are
aligned and when they are staggered.
47
48
4 Numerical Examples and Experimental Verification
Table 4.1: Dimensions and parameters of the spherical roller bearing
free contact angle
roller diameter
inner raceway contour radius
outer raceway contour radius
roller contour radius
bearing width
clearance
pithc diameter
number of rows
number of rolling elements in one row
modulus of elasticity
Poisson’s ratio
4.1.1
φ0
dr
rin
rout
rr
B
cd
dp
nz
N
E
ν
7.92
29
106.61
106.61
103.95
50
41
175
2
16
206
0.3
degree
mm
mm
mm
mm
mm
µm
mm
GPa
-
Single Bearing Load Analysis - Contact Forces
The first set of calculations was performed to verify that the newly developed SRB model
would correctly simulate how roller contact forces change with increasing load. Figure
4.1 shows the calculated contact force distribution for radial loads in the y direction of 4,
6, 8, and 10 kN. The figure demonstrates, as theory predicts, that at the input diametral
clearance of cd = 41 µm, fewer rollers support the lowest applied radial load, and as the
load increases, the number of supporting rollers increases.
4.1.2
Elastic Deformation, Displacement, and Load
The second set of calculations explored the relationship between bearing displace-ment
and load. Figure 4.2 illustrates the predicted relationship between the applied radial force
and bearing displacement, and shows that displacement increases with the increasing
load. This eccentric displacement of the rotating bearing centers is a result of bearing
radial clearance (cd = 41 µm) and the elastic deformations occurring at the regions of
roller-to-race contact. In Figure 4.2, the red curve shows the behavior predicted by the
BearinX commercial bearing software and the black one shows the behavior predicted by
the Gargiulo [19] formula for the spherical roller bearing radial deflection.
49
4.1 Case 1: Single Bearing Numerical Simulations
F = 975N
F = 975N
F = 565N
F = 565N
F = 565N
F = 565N
F = 1380N
F = 1380N
F = 883N
F = 883N
F = 883N
F = 883N
F = 1.5N
F = 1.5N
F = 1.5N
F = 1.5N
Fex = 4kN
Fex = 6kN
F = 2097N
F = 1752NF = 1752N
F = 2097N
F = 1464N
F
=
1182N
F = 1464N
F = 1182N
F = 1464N
F = 1182N
F = 1464N
F = 160N
F = 1182N
F = 66N F = 160N
F = 160N
F = 66N
F = 66N
F = 160N
F = 66N
Fex = 10kN
Fex = 8kN
Figure 4.1: Contact forces of the rolling elements in case of different radial loads, where
the shaded surface represents the spherical surface of the outer racel
12
10
Force [kN]
8
SRB, Table 1 parameters
BearinX
Gargiulo
6
4
2
0
20
22
24
26
28
30
32
34
Displacement [µm]
Figure 4.2: Radial force with respect to displacement
50
4.1.3
Clearance, Displacement, and Load
The third set of calculations established how SRB elastic deformation changes with load
as a function of diametral clearance cd . Figure 4.3 shows elastic deformation and radial
loading force plotted for four different values of clearance. The simulations demonstrate
that elastic deformation is not affected significantly by changes in cd . This conclusion is
supported by the BearinX prediction and the Gargiulo bearing radial deflection estimation.
Although elastic deformation seems to be insensitive to changes in cd , diametral clearance
does affect displacement between the bearing races as expected.
14
cd = 39 µm
cd = 41 µm
12
cd = 44 µm
Force [kN]
10
cd = 46 µm
BearinX
Gargiulo
8
6
4
2
0
0
2
4
6
8
10
12
14
Elastic Deformation [µm]
Figure 4.3: Effect of clearance on elastic deformation
of the bearing. In this case the
cd
elastic deformation is calculated by ey − 2
Figure 4.4 shows bearing stiffness behavior respect to radial load for four different values
of clearance.
4.1.4
Osculation Number, Displacement, and Load
The effect of osculation number on displacement for different levels of radial loading was
investigated with the fourth set of calculations. Figure 4.5 shows a family of displacementto-load curves representing four different values for osculation number. The prediction
reveals that osculation significantly affects bearing stiffness. An osculation number value
of Cos = 0.96 seems to correspond to the reference solution obtained using the BearinX
software.
51
8
4
x 10
3.5
Stiffness [N/m]
3
2.5
2
cd = 39 µm
1.5
cd = 41 µm
1
cd = 44 µm
cd = 46 µm
0.5
0
0
2
4
6
8
10
12
14
Force [kN]
Figure 4.4: Stiffness with respect to radial force
15
Cos = 0.98
Cos = 0.99
Cos = 0.97
10
Cos = 0.96
Force [kN]
BearinX
5
0
20
22
24
26
28
30
32
34
Displacement [µm]
Figure 4.5: Effect of osculation number on radial force-displacement relationship
52
4.1.5
Number of Rollers, Displacement, and Load
The fifth set of verification calculations investigated how displacement changes with radial
load and the number of bearing rollers. The results are presented in Figure 4.6. They
indicate that SRB load carrying capacity increases with its number of rolling elements. In
this case, the BearinX software predicts slightly higher displacement values.
12
N = 16
N = 18
N = 17
N = 14
BearinX
10
Force [kN]
8
6
4
2
0
20
22
24
26
28
30
32
34
Displacement [µm]
Figure 4.6: Effect of number of rollers on the radial force-displacement relationship
4.1.6
Angular Alignment of Side-by-Side Roller Arrays
The final set of verification calculations looked at bearing displacement with respect to
time for two separate arrangements of the caged side-by-side roller arrays; when they are
aligned and when they are staggered, as Figure 4.7-a illustrates. On the left in the figure,
the Type A arrangement shows the twin roller arrays in alignment. On the right, Type B
shows an 11.25° angular offset between them. For this pair of calculations, an external
force of Fy = -2000 N and an angular shaft velocity of ωin = 100 rad
s were applied. The
outer bearing races were fixed, that is to say, ωout = 0 rad
.Figure
4.7-b
plots the calculated
s
y displacements as a function of time assuming pure rolling motion between the bearing
rollers and the inner and outer raceways.
53
y
y
F ω
in
F ωin
x
x
One side shifted equal to 11.25o
Both side at the same level
Type A
Type B
(a)
25.6
Type A
Type B
Displacement in y direction [ µm]
25.58
25.56
25.54
25.52
25.5
25.48
25.46
25.44
25.42
25.4
0
5
10
15
20
25
30
35
40
45
50
Time [msec]
(b)
Figure 4.7: Effect of roller position on bearing force
The Type A bearing shows a varying compliance (VC) vibration at a frequency of 123.6
Hz. This is equal to the roller-pass-outer-ring frequency of the bearing. As expected,
bearing Type B vibrates at twice the roller-pass-outer-ring frequency. In the Type A
bearing, the displacement variation due to the VC effect is 0.34%. In contrast, the
54
displacement variation is 0.05% for the Type B bearing. The 11.25° angular shift between
the roller arrays seems to reduce the VC effect significantly.
4.2
Case 2: Rigid Rotor Supported by Ideal Spherical Roller Bearings
To demonstrate the application of the newly developed SRB model in a typical real world
analysis, a numerical simulation was carried out of a full rotor-bearing system comprising
a rigid rotor supported by SRBs on either end of the rotor axle.
4.2.1
Studied Structure
The rotor bearing system, shown in Figure 4.8, can be described with eight degrees
of freedom where the four-degree-of-freedom are defined for both bearings housing
dis-placements in the x and y directions, and the next four-degree-of-freedom for rotor
displacements that may be selected in many ways (see part 2.1.1, Rigid Rotor). One
possibility is to use the center of mass translations in the x and y directions and the two
rotations about those axes. Another possibility, which was selected for this work, is to use
the translational coordinates of two bearing locations as the system’s degree-of-freedom
y
yI
yII
mub
ψy
B
d1
ψx
eub
aub
d2
z
dI
CS
KS
dII
CS
KS
Figure 4.8: Rigid rotor supported with two spherical roller bearings
The bearing housings were connected to the ground using linear spring-dampers, whose
stiffness and damping coefficients are KS and CS , respectively. The angular velocity of
the rotor about the z-axis was assumed constant. Table 4.2 lists the dimensions and mass
properties for the modeled rotor-SRB system.
According to Equation ( 2.7 ), in this particular case, the individual force components can
be written with these equations.
55
Table 4.2: Dimensions of the rigid rotor
length
left bearing lacation
right bearing lacation
end part diameter
middle part diameter
density
mass of rotor
transverse moments of inertia
polar moment of inertia
L
dI
dII
d2
d1
ρ
mR
Ix = Iy
500
225
225
110
130
7850
49.138
0.9846
kg
kg m2
Iz
0.0993
kg m2
kg
m3

F ex
Fb
Fg
F ub

0
−250

= 
 0 
−250
 I
Fx
 I
 Fy 

= 
F II 
 x 
mm
mm
mm
mm
mm
FyII


0
− mr 
2 
= 
 0 
− m2r
(4.1)
(4.2)
(4.3)


cos (Ωt + θ0 )Lub
 sin (Ωt + θ0 )Lub 

= mub eub Ω2 
cos (Ωt + θ0 )Rub 
sin (Ωt + θ0 )Rub
(4.4)
where mub , eub are the mass and eccentricity, and θ0 is the phase angle of the unbalanced
mass, which is assumed zero. Lub and Rub are calculated as follows:
Lub =
Rub =
dII − aub
dII + dI
dII + aub
dII + dI
(4.5)
(4.6)
56
The unbalance mass parameters used in the rigid rotor model and the properties of the
supporting structure are listed in Table 4.3.
Table 4.3: Unbalance mass parameters of the rotor and support properties
unbalance mass
eccentricity of the unbalance mass
unbalance distance to center of mass of rotor
phase angle of the unbalance mass
mub
eub
aub
θ0
0.005
0.1
0.12
0
kg
m
m
rad
The properties of the supporting structure are listed in Table 4.4.
Table 4.4: the properties of the supporting structure
4.2.2
bearing mass
support stiffness
mS
KS
support damper
CS
11
1 · 107
5 · 103
kg
N
m
Ns
m
Simulation Results
Using Matlab-2011b, the dynamics of the rigid rotor and two SRBs were solved for
rotation speed of 3000 rpm. The differential equations of motion (Equation (2.10)) were
solved using an ode45 time integrator scheme. Figure 4.9 shows the predicted horizontal
and vertical translational rotor displacements at the A and B spherical roller bearing
locations as a function of time. After a brief initial transient vibration, the rotor settles
into steady-state harmonic vibration as a result of unbalanced forces. The rotor axis
deflects in an elliptical orbit. Because of the positioning of the unbalanced load, the
deflections for SRB A are greater than those for SRB B. Figure 4.10 illustrates the orbital
motion of the rotor axis.
15
SRB−A, x−disp
SRB−B, x−disp
Displacement (µm)
10
5
0
−5
−10
−15
0
0.1
0.2
0.3
0.4
0.5
time (s)
0
SRB−A, y−disp
SRB−B, y−disp
Displacement (µm)
−20
−40
−60
−80
−100
−120
−140
0
0.1
0.2
time (s)
0.3
0.4
0.5
Figure 4.9: Simulated horizontal and vertical displacements of the rotor
57
58
Orbit at SRB−A
Orbit at SRB−B
−78
−80
y (µm)
−82
−84
−86
−88
−90
−5
0
x (µm)
5
Figure 4.10: Simulated horizontal and vertical displacements of the rotor
Figure 4.9 and Figure 4.10 show that total rotor displacement in the y direction is about
84 µm on average. Much of this displacement is due to elastic compression of the support
springs. Theory predicts a value for support compression of about 60 µm. The additional
20 µm displacement is bearing clearance. Therefore, the elastic compression of the SRB
structure must be only a few micrometers in this case. However, the elliptical orbit of
rotor axis displacement (Figure 4.10) shows greater displacement in the x-axis direction
than in the y-axis direction, which implies differences in bearing stiffness. This difference
is the result of bearing clearance, which is taken up in the y-axis as the y-direction radial
load acts on the bearing, and not taken up in the x-axis direction.
4.3
4.3.1
Case 3: Single Spherical Roller Bearings with Localized Defects
Studied Structure
In this section, a series of numerical calculations are carried out for a single SRB subjected
to a simple radial load to verify the local defect analysis of the bearing. The investigated
spherical roller bearing type is the SKF 22216-EK roller bearing. The relevant dimensions and material properties of the studied spherical roller bearing are shown in Table
4.5 while the dimensions are depicted graphically in Figure 2.5. It should be noted that
some of the parameter values are based on manufacturer data and some are based on
experimental measurements on a real bearing. The measurements were carried out using a
three-dimensional coordinate measuring machine as shown in Figure 4.11. In the first set
4.3 Case 3: Single Spherical Roller Bearings with Localized Defects
59
of numerical calculations, the effect of outer ring defect on bearing force was calculated
and the second example explores the same effect for a bearing with inner ring defect.
Table 4.5: Dimensions and parameters of the spherical roller
bearing, 22216-EK
free contact angle
roller diameter1
inner raceway contour radius1
outer raceway contour radius1
bearing width
clearance
bearing pitch diameter
number of rows
Poisson’s ratio
φ0
dr
rin
rout
rr
B
cd
dp
nz
N
E
ν
8.25
15.544
63.2
63.2
61.49
33
41
132
2
21
206
0.3
degree
mm
mm
mm
mm
mm
µm
mm
GPa
-
[1] Based on experimental measurements on real bearing (Figure 4.11)
Figure 4.11: Bearing geometry parameters measurements
60
4.3.2
Single Bearing Load Analysis - Outer Ring Defects
The first set of calculations was performed to consider the vibration behavior of the
bearing with outer race defect. Figure 4.12 shows two types of defects (simple and
elliptical defect) on one side of the outer ring and hence only one row of rollers is in
contact with the defects. The dimension of the defects on the outer ring is presented in
Table 4.6. In both type of defects, the maximum depth of the defect is defined as equal to
0.05 mm and it is big enough to lose connection between the roller and the rings. The
only difference is that, for the simple defect, it happens whenever the roller enters a defect
area, but for the elliptical defect, it does not happen immediately. This model runs with a
constant displacement in the vertical direction which is equal to ey = -0.0324 mm and an
angular shaft velocity, ωin = 78.54 rad
s (750 rpm) for both type of defects that are shown
in Figure 4.12.
y
a
Dd
Elliptical defect
b
ey ωin
x
Dd
Wd
Simple defect
Figure 4.12: Dimension of defect on outer ring
In the following, Table 4.6 lists the dimensions of the outer ring defect.
Table 4.6: Dimensions of the outer ring defect
Elliptical defect
Simple defect
Ellipse major axis
Ellipse minor axis
Defect position in the outer ring
Defect macimum depth
Defect width
Defect depth
a
b
φ
Dd
φ
Wd
Dd
20
12
280
0.05
280
9.26
0.05
mm
mm
degree
mm
degree
mm
mm
Figure 4.13 shows the system response for both types of defects. The results presented
in the frequency domain clearly show peaks at the roller pass outer ring frequency, its
double frequency and also a minor peak at the cage frequency (See appendix B).
61
Figure 4.13: Force amplitude in time and frequency domain for outer race defect (a)
Elliptical defect-time domain; (b) Elliptical defect-spectrum; (c) Simple defect-time
domain; (d) simple defect-spectrum
The spectra shown in Figure 4.13 for both elliptical and simple defect contain peaks at the
outer ring defect frequencies. In this case, no side band frequencies are generated because
62
the outer ring is fixed to the bearing housing [70]. The second harmonic spectra for roller
pass outer ring frequency are also shown. Furthermore, it seems that in a spherical roller
bearing, the local defect shape (simple or elliptical) does not have significant effect on
dynamic behavior of system because of the large number of rollers in both side of the
bearing.
4.3.3
Single Bearing Load Analysis - Inner Ring Defects
In this section, the effect of inner ring defect on bearing vibration behavior was investigated. Similarly to the previous example, it is assumed that the bearing rotated with
a constant rotation speed equals to ωin = 78.54 rad
s (750 rpm) and the pre-defined
displacement in the vertical direction as ey = -0.0324 mm. Both elliptical and simple
defects are modeled on one rolling surface of the inner ring as shown in Figure 4.14.
y
Dd
a
ωin
b
Elliptical defect
ey
Dd
x
Simple defect
Figure 4.14: Dimension of defect on inner ring
Dimensions of the inner ring defect are shown in Table 4.7 Dimensions of the inner defect.
Contrary to the outer ring defect, the inner race defect rotates with the same frequency of
the rotor. Thus, it is assumed in this example that the initial angular position of the defect
is φ0 = 0.
Figure 4.15 shows the effect of inner race defect on bearing force in y direction.
63
Table 4.7: Dimensions of the inner defect
Elliptical defect
Simple defect
Ellipse major axis
Ellipse minor axis
Defect macimum depth
Defect width
Defect depth
a
b
φ0
Dd
φ0
Wd
Dd
20
12
0
0.05
0
9.26
0.05
mm
mm
degree
mm
degree
mm
mm
Figure 4.15: Force amplitude in time and frequency domain for inner race defect (a)
Elliptical defect-time domain; (b) Elliptical defect-spectrum (c) Simple defect-time
domain (d) simple defect-spectrum
64
The roller pass inner ring frequency and its multiple, as well as the shaft rotating frequency
and its multiples for both kind defects, are clearly visible in Figure 4.15. The roller pass
inner ring frequency and its harmonic have sidebands, which are made up of the difference
between RIF and shaft rotation frequencies.
4.4
4.4.1
Case 4: Rotor Supported with Spherical Roller Bearings with Localized Defects
Studied Structure
Figure 4.16 depicts the rigid rotor-bearing system used in the simulation and experimental
verification of the bearing model. The rotor-bearing system is supported by two SRBs
at both ends of the shaft. In general, the dynamic behavior of a rotor bearing system
containing bearing defects is a summation of steady-state harmonic vibration due to
harmonic unbalance forces and transient vibration caused by bearing defects. Therefore,
a time transient analysis using numerical integration is selected to predict the nonlinear
dynamic behavior of the rotor-bearing system.
l
l3
l2
l4
l5
l6
l1
y
d1
d2
CS
z
KS
d3
d4
CS
d5
KS
Figure 4.16: Schematics of the rotor supported with two spherical roller bearings
In simulations, the bearing housings are connected to the ground using linear springdampers, whose stiffness and damping coefficients are KS and CS , respectively. The
angular velocity of the rotor about the Z-axis was assumed constant. Table 4.8 lists the
dimensions for the modeled rotor-SRB system.
The rotor bearing system, shown in Figure 4.16, is modeled using the beam finite elements
that have four degrees of freedom per node. The system has 72 degrees of freedom where
four degrees of freedom are used for bearing housing displacements (i.e. translation
in the x and y directions), and the remaining 68 degrees of freedom are used for rotor
4.4 Case 4: Rotor Supported with Spherical Roller Bearings with Localized Defects65
Table 4.8: Dimensions of the rigid rotor
d1 = 70 mm
d2 = d4 = 80 mm
d3 = 88 mm
d5 = 70 mm
l = 600 mm
l1 = l6 = 100 mm
l2 = l5 = 130 mm
l3 = l4 = 160 mm
displacements (i.e. 17 nodes). It is also assumed that there are no displacements in the
rotor axial direction. In this example, the bearing houses are connected to the ground
using a linear spring-damper, whose stiffness and damping coefficients are KS and CS ,
respectively. The properties of the rotor bearing system, including its supporting structure,
are listed in Table 4.9.
Table 4.9: Properties of the rotor and supporting structure
4.4.2
Rotor mass
Rotor’s material density
Rotor Poisson ratio
Support mass
Support stiffness
mR
ρ
ν
mS
KS
Support damping
CS
24.299
7850
0.3
11
1 · 107
5 · 103
kg
kg
m3
kg
N
m
Ns
m
Measurement Setup
Two measurements setups for spherical roller bearings with elliptical inner and outer ring
defects, for which the dimensions are shown in Table 4.6 and Table 4.7, was built and
simulated with Matlab. Figure 4.17 shows defect machining on outer ring
By using numerical integrator (ode15s), the differential equation of the rotor bearing
system’s motion (Eq. ( 2.28 )) is solved. The measurement setup is shown in Figure 4.18.
The faulty bearing is located in right side of the rotor. The rotor was rotated by an external
rotational belt drive. The measurement data was recorded when system ran at a steady
rotation speed. In the measurement run, the steady rotation speed was 768 rpm measured
by contact tachometer. In this particular case, average external force equal to 500 ± 10 N
is applied in the vertical direction in the middle of the rotor. Applied external force was
measured by washer load cell (HC 2001 DS Europe) as shown in Figure 4.18. Moreover,
A ICP accelerometer with a sensitivity of 98 mV
g is used to measure the vibrations, which
is mounted on top of the housing of the test bearing. The accelerometer is connected to
the vibration analyzer (CSI 2130), the output of which is connected to a computer. The
signals are sampled at 1024 Hz for 4 seconds (i.e.4096 samples).High pass filter of 1 kHz
is used to remove the unwanted high frequency noise.
66
Figure 4.17: Defect machining on outer ring
Figure 4.18: Test setup for bearing defect experiments
4.4 Case 4: Rotor Supported with Spherical Roller Bearings with Localized Defects67
4.4.3
Simulation Results - Rotor with Outer Ring Defects
Figure 4.19 presents and compares the simulation result with measurements for the outer
ring defect in the frequency domain. In this case, by applying the Fast Fourier Transform
(FFT) analysis with the same FFT properties have used in measurement (signals are
sampled at 1024 Hz for 4 seconds and Hanning window applied to have better frequency
resolution), the magnitude of displacements in the y direction is calculated and presented
in the frequency domain. The spectrum for simulation results shows peaks exactly in
the roller pass outer ring frequency (118.1 Hz) that is listed in Table 4.10 and its second
harmonic spectra. Measurement results also have peaks around roller pass outer ring
frequency (115.1 Hz). Internal slipping between bearing components might be the reason
of this difference.
Figure 4.19: Spectrum of the vertical displacement for a bearing with outer ring defect
a) Simulation result; b) measurement result
68
4.4.4
Simulation Results - Rotor with Inner Ring Defects
Figure 4.20 depicts the comparison of the measured and simulated spectrums for the
bearing with inner ring defect. Spectrums are taken on vibration signals in the vertical
direction. To separate fault signals from extra noises, the bearing defect frequencies are
present in Table 4.10 (See appendix B).
Figure 4.20: Spectrum of the vertical displacement for bearing with inner ring defect a)
Simulation result; b) measurement result
Figure 4.20 shows peaks exactly equal to shaft rotation frequency and its multipliers for
both simulation and measurement results. Simulation results demonstrate the roller pass
inner ring frequency (RIF), and its side bands (RIF±SF) at 149.5 Hz. The same behavior
4.5 Case 5: Rotor Supported with Spherical Roller Bearings with Distributed Defect69
as explained in Figure 4.19, RIF value in measurement results has some difference with
the calculated value where the slipping between bearing components would be the reason
for this difference.
Table 4.10: Faulty bearing frequencies
Rotation Speed (rpm)
Shaft Frequency (Hz)
Cage Frequency (Hz)
Roller Pass Outer Ring Frequency (Hz)
Roller Pass Inner Ring Frequency (Hz)
4.5
763
12.72
5.61
117.96
149.08
Case 5: Rotor Supported with Spherical Roller Bearings with Distributed Defect
This study introduces resonances in the subcritical rotational speed range arising from
the bearing waviness. The waviness components cause excitations that are multiples of
the rotational speed. This might lead into a situation where the rotor starts to vibrate in
the subcritical rotational speed meaning the rotational speed below the operational speed.
In some applications, such as in paper machines, subcritical resonances are of practical
significance since they may be within the operating speed range and can influence the
quality of the final product.
This case study utilizes a finite element method to model a paper machine tube roll
system containing a symmetric tube and wavy spherical roller bearings. The waviness
of the bearing rings fall into the commonly used manufacturing tolerances. A transient
analysis is performed for the system and the results are converted into the frequency
domain in which the results are compared with the experimental ones. Results are
shown waterfall plots. Results indicate that the bearing waviness is dominating the
superharmonic frequencies and the uneven mass and stiffness distribution of the rotor
plays a minor role. However, in order to study the subcritical responses more accurately
the non-idealities of the rotor should be included in the finite element model.
4.5.1
Studied Structure
The rotor is modeled using a finite element method. The interactions between tube roll
and spherical roller bearings are described using contact force which is defined using
nonlinear Hertzian contact theory (Chapter 2). The finite element analysis is performed
in MATLAB software. The type of spherical roller bearings is 23026CCK/WW3. It
contains rolling elements in two rows and 25 rolls in a row. Table 4.11 gives the relevant
dimensions and parameters of the roller bearing,
70
Table 4.11: Dimensions and parameters of the spherical roller
bearing, 23026CCK/WW3
free contact angle
roller diameter1
inner raceway contour radius
outer raceway contour radius
bearing width
clearance
bearing pithc diameter
number of rows
poisson’s ratio
φ0
dr
rin
rout
rr
B
cd
dp
nz
N
E
ν
8.07
17.5
93.5
93.5
91.5
35
40
164
2
25
206
0.3
degree
mm
mm
mm
mm
mm
µm
mm
GPa
-
The element type is used in rotor model is Timoshenko beam element. The used matrices
are taken from the ANSYS Theory, referring namely to the element type BEAM4 (2001).
The finite element model comprised 21 nodes on the rotor having 4 DOFs per node
and additional mass elements to describe the support of the bearing housing. Axial
displacements and rotation around x-axis was constrained in all rotor nodes and only
radial translation of bearing housing was free resulting in a total of 88 DOFs. In transient
analysis, large matrices are computationally expensive. The number of DOFs can be
reduced using a reduction method such as Guyan reduction or the modal reduction used
in this analysis. The number of DOFs was decreased in the modal reduction using the
eigenmodes of the system [42]. The eigenvalue analysis was performed for the full finite
element model.
The rotor system under investigation consists of the tube roll supported by spherical roller
bearings in both ends. The parameters of the real bearings are emulated in the simulation
model of the system. This is achieved by measuring the non-idealities of the bearings
and incorporating them into the simulation model. The studied paper machine tube roll is
shown in Figure 4.21. The measurement equipment consists of a PC-based data collection
system, laser sensors fitted with amplifiers, a connection panel, and a guide trail where the
sensors are installed. The measurement results for the middle of the roll are considered
in this study. The locations of the measurement points are shown in Figure 4.21. The
angular velocity of the rotor is obtained using pulse sensor.
4.5.2
Simulation Results
The roller test rig is modeled using MATLAB software. The simulation model includes a
flexible symmetrical tube roll, the support of the spherical roller bearings, and nonlinear
Figure 4.21: The locations of the measurement points in millimeters. Cut view of the
tube roll is presented below the centerline
contact forces between the roll and the bearings. The parameters of the rotor model are
introduced in the Table 4.12. The stiffness of the bearing pedestals are measured from the
actual machine by applying static load and measuring the displacements. Damping ratios
are determined by simulating the force impulse and using the logarithmic decrement. A
more accurate description of the pedestal can be found in [27]. Even though the rotor
non-idealities are not modeled the subcritical vibrations can be studied since the waviness
of the bearings is contributing mainly for these vibrations.
Table 4.12: Properties of the finite element model
Material properties
Young’s modulus
Mass density
Poisson’s ratio
Dimensions of the rotor
Total length of the rotor
Length of the shell
Diameter of the shell
Thickness of the shell
Pedestal
Vertical stiffness
Horizontal stiffness
Damping ratios
Pedestal mass
2.07 · 1011
N
m
kg
m2
5
4
0.3225
18.72
m
m
m
mm
262 · 106
N
m
N
m
7801
0.3
63 · 106
2
127
%
kg
The waviness components of the spherical roller bearings inner rings in both rolling
rows in both ends are presented in Table 4.13. In the simulation model, attention is
paid to the waviness components of 2nd to 4th orders only because the amplitudes of
72
higher components are insignificantly small. The second order waviness has the highest
amplitudes. The phases of different orders of waviness are close in both rolling sides in
both tending and serving sides which increases the overall effect of waviness.
Table 4.13: Bearing inner ring waviness components of orders 2nd to 4th
Tending side
k
2
3
4
k
2
3
4
Inner rolling race
Amplitude ck [µm] Phase φk [°]
5.11
351.50
1.04
234.86
1.00
158.21
Inner rolling race
2.41
356.61
0.91
25.58
0.21
91.33
Outer rolling race
4.67
348.90
1.66
210.13
0.47
148.87
Outer rolling race
2.70
346.09
0.90
36.11
0.39
191.25
The simulation results are compared to measurements using spectrum maps of the
displacements. The spectrum maps include the frequency range of the twice-running
speed resonance between 14 and 18 hertz. The measured and simulated spectrum maps of
the horizontal displacements are shown in Figure 4.22. The corresponding spectrum maps
in vertical directions are shown in Figure 4.23. All the modeled excitations can be seen in
the simulated spectrum maps. The measured tube roll is balanced to meet the standards
but there is some unbalance left as can be seen from the first harmonic component of the
displacement. However, the amount of unbalance does not affect the higher harmonic
components, and thus, unbalance is not considered in this study. The amplitudes of
the horizontal second harmonics are greater than those of the vertical ones in all the
simulation cases as well as in the measurements. This is caused by the support of the roll
which stiffness in horizontal direction is less than in vertical direction. When simulated
and measured results are compared, it can be seen that the twice-running-speed resonance
can be found at the expected rotational speed in both horizontal and vertical direction.
The response amplitude in the horizontal direction is greater in simulated case. In vertical
direction, the measured amplitude is higher than the measured one in the resonance.
These relatively small differences can be caused by the rotational speed increment of 0.2
hertz which means that most likely the exact resonance frequency of the system is not hit
and the amplitude is diminished from the actual resonance peak in both simulation and
experiments. It can be also seen that the resonance of the FFT in simulation is less than in
measurements due to computational reasons. The running speed resonance is noticeable
in the measured spectrum maps. The running speed response arises from the unbalance
but also from the eccentricity of the bearings. Eccentricity component of the bearing is
not measured why it is not included in the waviness description of the bearing model.
The unbalance was not considered in the simulation why the rotating speed frequency
response cannot be seen. Three-times-running-speed resonance (3X-resonance) is visible
in the measured spectrum. It can be also weakly seen from the simulated spectrum
maps but the amplitudes of the responses are much smaller than the twice-running speed
resonance which is used in scaling the figure axes
a)
Amplitude [µm]
250
200
150
100
50
0
0
20
40
Frequency [Hz]
60
14
16
18
Speed [Hz]
16
18
Speed [Hz]
b)
300
Amplitude [µm]
250
200
150
100
50
0
0
20
40
Frequency [Hz]
60
14
Figure 4.22: Measured and simulated vibrations in horizontal direction.
74
a)
Amplitude [µm]
100
80
60
40
20
0
0
20
40
Frequency [Hz]
16
18
60
14
60
18
16
14 Speed [Hz]
Speed [Hz]
b)
Amplitude [µm]
250
200
150
100
50
0
0
20
40
Frequency [Hz]
Figure 4.23: Measured and simulated vibrations in vertical direction
C HAPTER 5
Conclusions
This doctoral dissertation introduces a comprehensive and computationally efficient,
three-degree-of-freedom, SRB model that was developed to predict the transient dynamic
behaviors of a rotor-SRB system. The simulation results demonstrate that the SRB model
is sufficiently accurate for typical rotor bearing systems. The new SRB model can be used
as an interface element between a rotor and its supporting structure in an analysis of rotor
dynamics. The model is simple and useable for either steady state or transient analyses.
Furthermore, local defect in inner and outer ring and also waviness as a distributed defect
are added as non-idealities to the SRB model, and rotor dynamic analysis has also done
with bearings non-idealities. The dynamic model describes contact forces as nonlinear
Hertzian contact deformations. It takes into account the influences of roller angular
position on bearing contact forces. The equations of motion were solved numerically.
To verify the new bearing model, a series of verifying numerical calculations were carried
out for a single ideal SRB subjected to a simple radial load. Physical parameters such as
contact force, bearing displacement, elastic deformation, diametral clearance, osculation
number, and the number and arrangement of bearing rollers were examined to verify the
model. The verification calculations supported or revealed the following.
• As theory predicts, roller contact forces change with increasing load. Fewer
rollers support lower applied radial loads, and more rollers come into play as load
increases.
• Bearing displacement increases with increasing load.
• Elastic deformation is not affected significantly by changes in cd . Although elastic
deformation seems to be insensitive to changes in cd , diametral clearance does
affect displacement between the bearing races.
75
76
5 Conclusions
• Osculation significantly affects bearing stiffness, and the force and displacement
responses are heavily dependent on bearing clearance and osculation number. These
parameters must be considered for an accurate assessment of system performance.
• SRB load carrying capacity increases with its number of rolling elements.
• Even an ideal spherical roller bearing experiences varying compliance (VC) vibration with a frequency equal to the roller-pass-outer-ring frequency of the bearing.
However, the VC effect can be reduced with an angular offset of the side-by-side
roller arrays. An 11.25° angular shift seems to reduce the VC effect significantly.
To demonstrate the application of the newly developed SRB model in a typical real
world analysis, a numerical simulation was carried out of a full rotor-bearing system
comprising a rigid rotor supported by SRBs on either end of the rotor axle. The governing
differential equations of motion for this specific rigid rotor-SRB system were solved
numerically. The predicted bearing displacements are consistent with the general theory
of rotor dynamics. After a brief initial transient vibration, the rotor settles into steady-state
harmonic vibration as a result of unbalanced forces. The rotor axis deflects in an elliptical
orbit. However, the elliptical orbit of rotor axis displacement shows greater displacement
in the x-axis direction than in the y-axis direction. This difference is the result of bearing
clearance, which is taken up in the y-axis as the y-direction radial load acts on the bearing,
and not taken up in the x-axis direction.
Proposed SRB dynamic simulation model, including an inner or outer race surface defect,
was verified by comparing measured-to-predicted behaviors. Four defect cases were run:
a simple surface cavity on the inner race, a simple surface cavity on the outer race, an
elliptical surface concavity on the inner race, and an elliptical surface concavity on the
outer race. The SRB model assumes that roller-to-race-surface contact over the elliptical
surface concavities is not lost immediately compared to simple defect. It means that in this
type of local defect, whenever the depth of defect is less than elastic compression between
rollers and races and also bearing internal clearance, the roller keeps its connection to
races and this roller contact force should also be taken into account to calculate the
bearing force.
Simulation results provided a catalog of behaviors as a function of defect type and
location, making it possible to identify an SRB defect by comparing the observed bearing
behavior with the various behaviors described by the catalog. Numerical results also
revealed that each local defect would result in vibration at a frequency equal to the
bearing defect cycling frequency. The predicted amplitude of vibration for the elliptical
defect is slightly greater than the vibration amplitude for an equivalent simple defect.
The measurements showed spectra peaking at different frequencies than predicted for
the bearing with the outer and inner ring defects. The percentage variation between
the theoretical and experimental defect frequency is found to be 1.7–2.7%. This may
77
be because of the internal slipping between rollers and inner and outer races which
effect on cage frequency and also roller pass inner and outer ring frequencies. However,
the real rotor bearing system has different unknown parameters which appeared in the
measurement results.
As a last numerical example, tube roll that is supported by spherical roller bearing in both
ends was studied. In this example bearings have waviness in the inner ring which are
measured and introduced in simulation model using Fourier cosine series. In the model
only 2nd to 4th order of waviness were considered. The dynamics of a paper machine
tube roll is analyzed in terms of twice-running-speed subcritical vibration. Final results
show that simulation and measurement results are comparable in both horizontal and
vertical direction.
Future Studies:
Study about spherical roller bearing and rotor-bearing systems can be improved by the
following suggestions:
• Performance of spherical roller bearing can be considers in non-axial rotation due
to shaft tilting and misalignment.
• Improve the contact theory for internal components connections
• Considering the effect of lubricant film damping in rolling contacts and centrifugal
force of rollers in dynamic behavior of bearing, might make the model more
realistic.
78
5 Conclusions
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87
APPENDICES
Appendix A: calculating the defect depth according to roller angular position
O
ψ1
βji
F1
a
ψ2
F2
P
δ + βji
Figure 1: Defect depth calculation
According to Fig.1 and the geometry properties of the ellipse, the following equations
can be written:
P F 1 + P F 2 = 2a
2
PF1
2
PF2
2
= OP +
2
= OP +
2
OF 1
2
OF 2
− 2OP · OF 1 · cos ψ1
− 2OP · OF 2 · cos ψ2
Finally, OP can be calculated to get defect depth depend on roller angular position.
(1)
(2)
(3)
88
APPENDICES
Appendix B: characteristic frequencies of bearing
The rotational frequency of the cage is calculated as follows
fcg
fR
=
2
dr
1+
cos φ0
dp
(4)
and roller pass outer ring frequency
frpof
fR
=N
2
dr
1−
cos φ0
dp
(5)
and also roller pass inner ring frequency
frpif
fR
=N
2
dr
1+
cos φ0 ,
dp
(6)
where fR is the rotor rotation frequency and dp is a bearing pitch diameter which is
defined as
dp = Rout + Rin
(7)
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dynamic analysis model of spherical roller bearings with

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