lappeenranta-lahti university of technology lut lut school
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Lappeenranta-Lahti University of Technology LUT
LUT School of Energy Systems
Degree programme in Mechatronic System Design
Iikka Martikainen
BEARING SYSTEM DESIGN IN OUTER ROTOR HIGH SPEED MOTOR
11.08.2020
Examiners: Prof. Jussi Sopanen
D.Sc. (Tech) Eerik Sikanen
TIIVISTELMÄ
Lappeenrannan-Lahden teknillinen yliopisto
LUT School of Energy Systems
LUT Kone
Iikka Martikainen
Laakerijärjestelmän suunnittelu suurnopeus ulkoroottorimoottoriin
74 sivua, 32 kuvaa, 9 taulukkoa ja 1 liite
Tarkastajat: Prof. Jussi Sopanen
TkT Eerik Sikanen
Hakusanat: Ulkoroottori, kestomagneettimoottori, roottoridynamiikka
Tämän opinnäytetyön tavoitteena oli löytää laakerisysteemin toimintaan vaikuttavat tekijä
sekä laakerin valintatyökalut ulkoroottorisovellukseen. Opinnäyte on tehty osana
protyyppisähkömoottoriprojektia, jonka roottoria opinnäytteessä tutkitaan. Sopivien
laakerien valinta ja dynaamisen vasteen tutkiminen on olennainen osa turvallisten pyörivien
laitteiden suunnittelua.
Laakerien kestoikälaskelmia käytettiin käyttökohteeseen sopivien laakerien valintaan. Kun
laakerit oli valittu, laakerien jäykkyys laskettiin kirjallisuudessa esitellyllä mallilla.
Laakerimalli oli jo ennalta implementoitu erikoisvalmisteiseksi koodiksi. Esikiristyksen
vaikutuksia laakerien jäykkyyksiin analysoitiin oikean esikiristysvoiman valitsemiseksi.
Myös lämpölaajenemisen vaikutuksia laakerijäykkyyteen tutkittiin.
Roottoridynaaminen analyysi suoritettiin elementtimenetelmää käyttäen, jotta voitiin
varmistaa moottorin turvallinen käyttö. Dynaamisen vasteen herkkyyttä laakerin sijainnille
analysoitiin, jotta voitaisiin saavuttaa optimaalinen laakerien sijoittelu.
Tukimuksissa todettiin, että roottorin dynaamiset ominaisuudet ovat turvallisia
käyttökierrosalueella. Huomattiin että tietyn kynnysarvon yläpuolella esikiristysvoimalla ja
laakerien välisellä etäisyydellä ei ollut merkittävää vaikutusta roottorin dynaamiseen
vasteeseen. Täten roottori ei ole kovinkaan herkkä muutoksille laakeriasetelmassa. Jotta
täysin ymmärrettäisiin moottorin dynaaminen käytös, tulisi staattorin ja tukirakenteiden
dynaamisia ominaisuuksia tutkia pidemmälle. Vaikka lämpölaajenemisen vaikutuksia
voitaisiin tutkia enemmän, saatiin lämpölaajenemisen vaikutukset rajattu turvalliselle tasolle
käyttämällä vakiovoima-esikiristysmenetelmää.
ABSTRACT
Lappeenranta University of Technology
LUT School of Energy Systems
LUT Mechanical Engineering
Iikka Martikainen
Bearing system design in outer rotor high speed motor
Master’s thesis
2020
74 pages, 32 figures, 9 tables and 1 appendix
Examiners: Prof. Jussi Sopanen
D.Sc. Eerik Sikanen
Keywords: Outer rotor, permanent magnet motor, rotor dynamics
The aim of this thesis was to figure out the factors effecting bearing systems operation and
find the tools needed for selecting bearings for an outer rotor application. The thesis was
made as a part of electric motor prototype project and rotor of the prototype was studied.
Selecting correct bearings and studying dynamic response is vital part of designing safe
rotating machine.
Bearing lifetime calculations were used to select suitable bearings for the application. When
bearings were selected, the bearing stiffnesses were calculated with model presented in
literature. The bearing model was already implemented in a custom code. Effect of preload
on bearing stiffness was analysed to select correct preload values for the system. Also the
effects of thermal expansion to stiffness were studied.
Rotor dynamic analysis was conducted with finite element method to ensure that motor could
be operated safely. Sensitivity of the dynamic response to the bearing location was analysed
to achieve optimal bearing locations.
Rotor dynamic properties of the rotor were found to be safe for operation within the
operating speed range. It was found that above certain threshold the preload force and
distance between the bearings did not have significant effect on the dynamic response of the
rotor. Thus, the system is not very sensitive to changes in bearing arrangement. To fully
understand dynamic behaviour of the machine, dynamics of the stator and support structures
would need to be studied further. Although thermal expansion effects could be studied
further, thermal expansions effect on bearings can be limited to safe level, by using constant
force preloading method.
ACKNOWLEDGEMENTS
I am greatly grateful to my examiner professor Jussi Sopanen and examiner D.Sc. Eerik
Sikanen for offering this very interesting subject and for their guidance throughout the
writing process of this master’s thesis. Their guidance has been more than beneficial and
their vast knowledge in field of rotor dynamics has sparked an interest to the field in me. I
have also been lucky to be able to work with practical matters of eMAD project with talented
people, which has been excellent opportunity to improve my professional skillset in
mechanical engineering and manufacturing considerations.
Last, but not least, I want to thank all of the many friends I have made during my studies
here in Lappeenranta. With help of friends I have made it through even the hardest times of
studies. Five years have passed faster than I could ever imagined.
Iikka Martikainen
Lappeenranta 11.8.2020
5
TABLE OF CONTENTS
TIIVISTELMÄ
ABSTRACT
ACKNOWLEDGEMENTS
TABLE OF CONTENTS
LIST OF SYMBOLS AND ABBREVIATIONS
TIIVISTELMÄ .................................................................................................................... 2
ABSTRACT .......................................................................................................................... 3
ACKNOWLEDGEMENTS ................................................................................................ 4
TABLE OF CONTENTS .................................................................................................... 5
LIST OF SYMBOLS AND ABBREVIATIONS ............................................................... 7
1 INTRODUCTION ..................................................................................................... 11
1.1 eMAD –project .................................................................................................... 12
1.2 Objectives ............................................................................................................ 14
1.3 Scope .................................................................................................................... 14
2 THEORY BEHIND BEARING SYSTEM DESIGN .............................................. 15
2.1 Rolling bearings in electric motors ...................................................................... 15
2.2 Loads on an outer rotor ........................................................................................ 17
2.2.1 Dynamic unbalance .................................................................................. 19
2.2.2 Equivalent dynamic bearing load ............................................................ 20
2.3 Bearing selection criteria ..................................................................................... 21
2.3.1 Selection based on basic rating life .......................................................... 21
2.3.2 Improving estimation accuracy with modified rating life ....................... 22
2.3.3 Other selection aspects ............................................................................. 27
2.4 Selecting internal clearance and preload ............................................................. 28
2.4.1 Calculating bearing stiffness based on preload ........................................ 28
2.4.2 Preloading methods .................................................................................. 40
2.4.3 Thermal expansion effects on bearing preload ........................................ 42
2.5 Bearing arrangement effects on rotor dynamics .................................................. 44
3 RESULTS ................................................................................................................... 48
3.1 Selecting bearings for eMAD motor .................................................................... 48
6
3.2 Bearing preload effects on rotor dynamics .......................................................... 52
3.2.1 RoBeDyn beam and mass disk model ..................................................... 60
3.3 Bearing arrangement optimisation for rotor dynamics ........................................ 62
3.4 Thermal expansion effects ................................................................................... 64
3.4.1 Effects of thermal expansion of the bearing components ........................ 64
3.4.2 Compensating for axial thermal expansion ............................................. 67
4 DISCUSSION ............................................................................................................. 69
5 CONCLUSIONS ........................................................................................................ 71
REFERENCES ................................................................................................................... 72
APPENDIX
Appendix I: Basic information for application
7
LIST OF SYMBOLS AND ABBREVIATIONS
𝑎𝑒 Semimajor axis of the ellipse [mm]
𝑎𝐼𝑆𝑂 Life modification factor for systems approach
𝑎1 Life modification factor for reliability
𝑏𝑒 Semiminor axis of the ellipse [mm]
𝐶 Basic dynamic load rating [N]
𝒄𝑏 Damping matrix of the bearing
𝑐𝑑 Diametral clearance [m]
𝐶𝑢 Fatigue load limit [N]
𝑑 Bearing ball diameter [mm]
𝑫 Damping matrix
𝐷ℎ Bearing housing diameter [m]
𝑑𝑗 Distance along contact line between the outer and inner raceways.
𝐷𝑖 Inner raceway diameter
𝑑𝑚 Pitch diameter [m]
𝐷𝑜 Outer raceway diameter
𝑑𝑠 Bore diameter
𝐸𝑎 Modulus of elasticity for material a [Pa]
𝐸𝑏 Modulus of elasticity for material b [Pa]
𝑒𝐶 Contamination factor
(𝑒𝑝𝑒𝑟 · 𝜔) Numerical value for selected balance quality grade [mm/s]
𝐸′ Effective modulus of elasticity [Pa]
𝑒𝑥 Relative displacement in x direction
𝑒𝑦 Relative displacement in y direction
𝑒𝑧 Relative displacement in z direction
𝒆(𝑛) Matrix containing the displacements of the bearing on iteration step n
𝒆(𝑛+1) Matrix containing the displacements of the bearing on iteration step 1+n
𝐹𝑎 Axial load on the bearing [N]
𝐹𝑐 Unbalance force [N]
𝐹𝑗 Forces acting on a single bearing ball [N]
𝐹𝑟 Radial load on the bearing [N]
8
𝑭(𝒕) External force vector as function of time
𝐹𝑋 Bearing force in x direction [N]
𝐹𝑌 Bearing force in y direction [N]
𝐹𝑍 Bearing force in z direction [N]
𝑔 Gravitational acceleration [m/s]
𝐺 Gyroscopic matrix
𝑲 Stiffness matrix
𝒌𝑏 Tangent stiffness matrix
𝐾𝑐 Contact stiffness coefficient
𝐾𝑐𝑡𝑜𝑡 Total stiffness of a bearing ball
𝐾𝑐𝑖𝑛 Contact stiffness between bearing ball and inner raceway
𝐾𝑐𝑜𝑢𝑡 Contact stiffness between bearing ball and outer raceway
𝐾𝑐𝑜𝑛𝑣 Convergence criterion factor
𝑘𝑒 Ellipticity parameter
𝒌𝑙𝑖𝑛 Linearized stiffness of the bearings [N/mm]
𝑲𝑇(𝑛)
Tangent stiffness matrix vector on iteration step n
𝐿𝑛𝑚 Modified rating life [millions of revolutions]
𝐿10 Basic rating life [millions of revolutions (at 90% reliability)]
𝑚 Mass of the rotor [kg]
𝐌 Mass matrix
𝑛 Rotational speed [rpm]
𝑁1 Support force of the first bearing [F]
𝑁2 Support force of the second bearing [F]
𝑃 Equivalent dynamic bearing load in [N]
𝑃𝑟 Equivalent radial load [N]
𝑸(𝑛) Matrix containing the forces acting on bearing on iteration step n
𝑸𝑏(𝑛)
Matrix of bearing forces
𝑸𝑒𝑥𝑡(𝑛)
Matrix of external forces
𝑅 Curvature sum
𝑟𝑎𝑥 Radius of the surface in elliptical contact conjunction
𝑟𝑎𝑦 Radius of the surface in elliptical contact conjunction
𝑟𝑏 Radius of the bearing ball
9
𝑟𝑏𝑥 Radius of the surface in elliptical contact conjunction
𝑟𝑏𝑦 Radius of the surface in elliptical contact conjunction
𝑟𝑏𝑥𝑖𝑛 Radius of inner raceway
𝑟𝑏𝑥𝑜𝑢𝑡 Radius of outer raceway
𝑟𝑏𝑦𝑖𝑛 Groove radius of bearing inner ring,
𝑟𝑏𝑦𝑜𝑢𝑡 Groove radius of bearing outer ring,
𝑅𝑑 Curvature difference
𝑟𝑖𝑛 Inner bearing ring groove radius
𝑟𝑜𝑢𝑡 Outer bearing ring groove radius
𝑅𝑖𝑛 Radius of the inner raceway
𝑅𝑜𝑢𝑡 Radius of the outer raceway
𝑇 Temperature of the component [K]
𝑇𝑟𝑒𝑓 Ambient temperature [K]
𝑇𝑋 Bearing moment around x axis [Nm]
𝑇𝑌 Bearing moment around y axis [Nm]
𝑈 Unbalance [kgm]
𝑈𝑝𝑒𝑟 Permissible residual unbalance [gmm]
𝑣 Kinematic viscosity
𝑣𝑎 Poisson’s ratio for material a
𝑣𝑏 Poisson’s ratio for material b
𝑣1 Reference kinematic viscosity
𝑋 Radial load factor for the bearing
𝑿 Displacement vector
�̇� First derivative of displacement vector
�̈� Second derivative of displacement vector
𝑥1 Axial distance from first bearing to rotors centre of gravity [m]
𝑥2 Axial distance between the bearings [m]
𝑌 Axial load factor for the bearing
𝑧 Number of bearing balls
𝛼 Thermal expansion coefficient
𝛽𝑗 Attitude angle of bearing ball j [°]
10
𝛾𝑥 Relative misalignment of the inner and outer raceway about x axis [°]
𝛾𝑦 Relative misalignment of the inner and outer raceway about y axis [°]
𝛿𝑗𝑡𝑜𝑡 Total elastic deformation of a single bearing ball
휀𝑡ℎ Thermal strain
휁 Elliptical integral
𝜅 Viscosity ratio
𝜉 Elliptical integral
𝜙 Contact angle of the bearing,
𝜑 Auxiliary angle [°]
𝜙𝑗 Contact angle of a single bearing ball [°]
𝜔 Rotational speed [rad/s]
DE Drive end
DOF Degree of freedom
FEM Finite Element Method
NDE Non drive end
UBR Unbalance response
11
1 INTRODUCTION
Electric motors have wide range of use cases from industrial applications to household items
– from machine tools, robots and medical devices to consumer electronics, electric vehicles
and kitchen appliances. It is estimated that 45% of electricity used in the world today is
consumed by electric-motor-driven systems. For some parts of the world, such as China, the
estimation is even higher, about 60% of produced electricity is used by electric motor driven
systems. (Tong, 2014, p. 1) In Figure 1 electric motor of an Audi e-tron electric car is
presented as an example of use case for electric motor.
Figure 1. Exploded view of the Audi e-tron motor (AUDI AG., 2020)
Within the electric motor, the bearings act as connection between stator and rotor. In terms
of bearing types, there is a lot of options to choose from: Journal bearings, rolling bearings,
air bearings, magnetic bearings, etc. Bearings must locate the components accurately in
every operating condition to ensure trouble free operation. Since a lot of electric motor
failures can be traced down to bearing failure, it is important that all factors that affect
bearing life have been taken into consideration when selecting bearings. Such factors include
12
bearing type, bearing arrangement, operating speed, load and lubrication. (Tong, 2014, p.
309) The same applies to many other factors of application, so careful engineering design is
needed when specifying bearing system for an application.
To ensure lifetime of bearings and rotating equipment they are installed in, it is elementary
to understand the importance of dynamic behaviour of rotors, which is also known as rotor
dynamics. By analysing rotor dynamics of system, we can make sure that vibrations within
operating range of the system stay in acceptable limits. Excessive vibrations can cause
premature wear in machine components (such as bearings) or even premature failure. Since
bearings are the connecting feature between rotor and ground (via frame and machine’s
mounting), it is obvious that properties of bearings have impact on rotor dynamics of the
system. Bearings stiffness and damping can vary depending on the operating speed, which
makes it even more important to consider bearing properties when ensuring safe operation
within the operating range of the machine. (Friswell, et al., 2010, pp. 1-2)
1.1 eMAD –project
LUT University and LAB University of Applied Sciences are working on a project to create
high power density high speed outer rotor permanent magnet motor. More detailed
description of the project can be found in the appendix I.
The project is looking to meet following specifications:
• Maximum speed: 10 000 rpm
• Output power: 1 MW
• Torque output: 1200 Nm
• Operating life: 5000 h
• Power density: 15 kW/kg
As written in the description of the eMAD-project, an electric motor with such high power
density could have market on high performance automotive application. Current electric
motors are not able to operate at peak power for continuous period of time, so the eMAD-
project is aiming to produce a motor that can fit in limited space and operate almost
constantly on its peak power, either accelerating or decelerating. These properties would
13
make the motor ideal for motorsport applications. In Figure 2 a simplified cross section of
the eMAD outer rotor motor is presented.
Figure 2. Cross section of eMAD outer rotor motor.
To enable continuous peak power operation sufficient cooling is needed. eMAD-project
continues a line of electric machines with cooling tubes within stator coils, a cooling solution
designed in LUT. The cooling of the stator coils is patented (United States Patent
US9712011) and this structure allows effective cooling of stator coils, increasing power
capabilities of an electric machine. The cooling of permanent magnet motors and direct
stator coil cooling has been studied by Mariia Polikarpova et al. (2014) and Ilya Petrov et al.
(2019). Although the specifications of the eMAD motor corresponds to requirements of high
power automotive application, at this phase the eMAD motor is not designed for any specific
application, but is meant to be manufactured for testing in laboratory conditions.
14
1.2 Objectives
This master’s thesis was done as a part of larger development process of outer rotor
permanent magnet motor. Focus of this study is the bearing arrangement of the outer rotor
motor. Thermal expansion and its effects on bearing clearance and preload were studied to
ensure satisfactory operation across the operating temperature range. Rotor dynamics of the
electric motor were studied to demonstrate the effects of bearing arrangement to the rotor
dynamics.
The aim of this study is to find tools for bearing selection and for finding optimal bearing
arrangement. Effects of thermal expansion will be studied to determine what actions need to
be taken to ensure problem free operation through the operating temperature range. Rotor
dynamics of the electric machine will be analysed to demonstrate the effects of bearing
arrangement to the rotor dynamics. The information gathered will be used to find optimal
bearing solution for outer rotor motor in question.
Research questions:
• What factors should be taken into account when selecting bearings for electric
motor?
• What is optimal bearing arrangement for outer rotor motor?
• How thermal expansion needs to be accounted for in bearing system?
1.3 Scope
Since this study aims to optimise bearing system for very specific application, it will not
present complete universal guide for optimising bearing system for any application. Here the
application will be a high speed outer rotor permanent magnet motor and only ball bearings
will be studied. This study also will not cover design of shaft, bearing housing or any other
related components.
15
2 THEORY BEHIND BEARING SYSTEM DESIGN
Bearings are meant to support rotating or sliding machine elements. Depending on how
bearings are loaded they are divided into radial and axial bearings. Radial bearings carry
mainly radial loads and axial respectively axial loads. Based on structure of the bearing the
bearings can be further divided into rolling bearings, which carry the loads via rolling
elements and journal bearings where lubricant film between outer and inner race of the
bearing carries the load. (Airila, et al., 1995, p. 417) In this study we will focus on rolling
bearings that accommodate mostly radial load, since that is the dominating bearing type in
electric motor production (Tong, 2014, p. 313).
2.1 Rolling bearings in electric motors
Rolling bearings consist of inner and outer race, rolling elements (balls or rollers depending
on the type of bearing) and cage, whose task is to keep the rolling elements relative position
to each other constant. The loads are transmitted between the raceways via the rolling
elements. Depending on the shape of the raceways and rolling element arrangement, the
bearing can accommodate either only radial or axial load or both to some extent. (Tong,
2014, p. 313)
Deep groove ball bearing is example of bearing commonly used in electric motors and of
bearing that is capable to carry loads in radial directions as well as minor loads in axial
direction. Geometry of the raceway grooves and balls affect the frictional, fatigue life and
stress properties of the bearing. This geometry also determines the contact angle of the
bearing and thus how much axial load bearing can carry. Angular contact ball bearing is
example of bearing, where raceway geometry is designed so that the bearing can carry much
higher loads in axial direction than regular deep groove ball bearing. Angular contact
bearings can carry axial load only in one axial direction, so they must be used as a pair to
accommodate axial loads in both directions. (Tong, 2014, pp. 313-316) In Figure 3 a
sectioned view of an angular contact bearing is presented.
16
Figure 3. Partial section view of an angular contact bearing
Bearings have been identified as the most crucial component for reliability of electric
machines. Failure of the bearing can lead to displacement of the rotor and this can cause
further damage within the electric motor. Bearing damages can occur in number of different
ways and it is not rare that bearing do not reach their life predicted with material fatigue.
Often other kind of failure modes happen well before end of predicted bearing life. For
example insufficient lubrication, excessive loading, improper handling and installation or
manufacturing defects in the bearing can cause premature failure within predicted bearing
life. (Tong, 2014, pp. 343-345)
17
2.2 Loads on an outer rotor
Here loads on the outer rotor will be explained. Focus will be on loads that are carried
through bearings of the rotor. Below is listed different loads that affect on electric motor
shaft (Tong, 2014, p. 159)
Typical loads on electric motor shaft (Tong, 2014, p. 159)
• Torsional load
• Transverse load
o Gravitational
o Gears, pulleys, etc.
o Unbalance magnetic pull (when nonuniform air gap)
• Axial load
• Bearing preload
• Shock loads
Considering loads carried by bearings, the torsional loads can be neglected. Residual
unbalance of the rotor is not mentioned by Tong (2014), but is most certainly relevant load,
especially in high speed application.
The gravitational loads on the bearings can be calculated using basic static analysis of the
rotor when mass properties of rotor and bearing locations are known. Same can be done for
gear and pulley connections where torque transmission results to transversal loads on the
shaft. What comes to unbalanced magnetic pull, it is more complex phenomenon and needs
more sophisticated analysis. For simplicity it will not be covered in this study. The subject
of magnetic pull has been covered already in studies by H. Kim, et al. (2020) and H. Kim,
et al. (2019).
In Figure 4 is presented a cross section of example outer rotor with support loads on bearings
and gravitational load from shaft. The main components of the rotor are named in the figure.
18
Figure 4. Gravitational load of the shaft and structure of the rotor
The radial loads caused by the gravitational load of the shaft can be thus calculated using
basic static equations (1) and (2)
Σ𝐹𝑦 = 𝑚𝑔 − 𝑁1 − 𝑁2 = 0 (1)
Σ𝑀 = 𝑚𝑔𝑥1 − 𝑁2𝑥2 = 0 (2)
Where
𝑚 is the mass of the rotor in kg,
𝑔 is the gravitational acceleration in m/s,
𝑁1 is the support force of the first bearing,
𝑁2 is the support force of the second bearing,
𝑥1 is the axial distance from first bearing to rotors centre of gravity,
𝑥2 is the axial distance between the bearings.
𝑁1 𝑁2
𝑚𝑔
𝑥1
𝑥2
𝑟𝑜𝑡𝑜𝑟 𝑐𝑜𝑟𝑒
𝑓𝑙𝑎𝑛𝑔𝑒
𝑠ℎ𝑎𝑓𝑡
19
In most cases a shaft of electric motor is mainly loaded in radial direction and only minor
loads are present in axial direction (Tong, 2014, p. 159). These loads might be due to helical
gear installed on to the shaft or other loads originating from device that the motor is
connected to. Other origin of axial load might be axial bearing preload in applications where
bearings must be axially preloaded in order to ensure proper operation conditions.
Estimating or calculating shock loads is difficult without knowledge of the application. In
best case shock loads could be measured in application where the electric motor is used
during operation, but if measuring is not possible, assumption could be used.
2.2.1 Dynamic unbalance
Due to manufacturing inaccuracies there is always some residual unbalance in the rotor.
Unbalance can be caused also by uneven heating (and thus uneven expansion) of the rotor
or uneven magnetic field in the active parts of the motor. In all cases mentioned above the
centre of mass will not be in line with rotational axis of the rotor. This causes vibrations in
the rotor that translate to variable load in the bearings. (Tong, 2014, pp. 338-340) Below is
presented the equation 3 for calculating the force caused by mass unbalance.
𝐹𝑐 = 𝑈𝜔2 (3)
Where
𝐹𝑐 is the unbalance force
𝜔 is rotational speed in radians per second,
𝑈 Is the unbalance in kgm.
According to ISO standard the permissible residual unbalance on rotor can be calculated
with equation 4 (ISO 1940-1, 2003, p. 10).
𝑈𝑝𝑒𝑟 = 1000(𝑒𝑝𝑒𝑟 ∙ 𝜔) ∙ 𝑚
𝜔 (4)
Where
𝑈𝑝𝑒𝑟 is permissible residual unbalance in gmm,
20
(𝑒𝑝𝑒𝑟 · 𝜔) is the value for selected balance quality grade in mm/s
To combine static gravity load and dynamic unbalance load we need to calculate mean load
using equation 5. The factor 𝑓𝑚 can be obtained from diagram 15 in SKF rolling bearings
book. (SKF Group, 2013, p. 86)
𝐹𝑚 = 𝑓𝑚(𝐹𝑔 + 𝐹𝑐) (5)
Where
𝐹𝑚 is the mean radial load
𝑓𝑚 is the mean load factor
𝐹𝑔 is gravitational load
2.2.2 Equivalent dynamic bearing load
Equivalent dynamic bearing load is a tool used to simplify varying loads radial and axial
loads into one single constant magnitude radial load. Equivalent dynamic bearing load is
used in bearing life calculations and it will yield same bearing life as the actual loads on the
bearing will. (SKF Group, 2013, p. 85) When axial and radial loads are acting on the bearing,
the equivalent radial load for single bearing can be calculated according to equation 6 as
presented in ISO 281 standard:
𝑃𝑟 = 𝑋𝐹𝑟 + 𝑌𝐹𝑎 (6)
Where
𝑃𝑟 is the equivalent radial load
𝐹𝑟 is the radial load
𝐹𝑎 is the axial load
𝑋 is the radial load factor for the bearing
𝑌 is the axial load factor for the bearing
Standard ISO 281 provides table for calculating factors 𝑋 and 𝑌. The values can be found in
standard ISO 281 table 3.
21
2.3 Bearing selection criteria
The calculated loads will give us a foundation for selecting a bearing that will last the desired
lifetime or service interval of the machine. One way to complete selection process is depicted
in following chapters. The process consists of following steps can be done iteratively until
suitable bearing is found:
1. Initial selection of bearing is made using basic rating life equation.
2. Modified rating life is calculated to ensure bearings lifetime
3. Selected bearing is assessed by other relevant criteria (e.g. minimum load, maximum
speed)
Depending on the application there might be restriction on the type or size of the bearing.
These aspects are design specific and will not be covered in this study, instead they are left
for the designer to consider during the design process.
2.3.1 Selection based on basic rating life
Basic rating life is a mathematical estimation of a lifetime that 90% of bearing manufactured
from high quality material of good manufacturing quality will reach under conventional
operating conditions (ISO 281, 2007, p. 2). The definition is not completely unambiguous,
because it is not clear what is high quality material and what are conventional operating
conditions. Nevertheless, the basic rating life gives some prediction of the lifetime for most
use cases and in situations where the operation conditions are hard to estimate.
The basic rating life equations use basic dynamic load rating to describe bearings ability to
carry load. Basic dynamic load rating is described as theoretical maximal load value that
bearing can carry for million revolutions (ISO 281, 2007, p. 2). It must be noted that the
equation does not consider failure modes caused by very small loads and it does not work
well with very high loads where equivalent dynamic bearing load is more than half of the
basic dynamic load rating of the bearing (ISO 281, 2007, p. 10).
Basic rating life of ball bearings can be calculated with equation 7 (ISO 281, 2007, p. 10):
22
𝐿10 = (𝐶
𝑃)
3
(7)
Where
𝐿10 is basic rating life in millions of revolutions (at 90% reliability),
𝐶 is basic dynamic load rating in N ,
𝑃 is equivalent dynamic bearing load in N
The equation can be used to estimate lifetime of bearing when basic dynamic load rating is
known. But it might be even more useful to solve equation for basic dynamic load rating and
calculate the minimum basic dynamic load rating when the desired lifetime or service
interval of the machine is known. When minimum basic dynamic load rating is known it is
easy to search bearing manufacturers databases for suitable bearings.
2.3.2 Improving estimation accuracy with modified rating life
As described above, the basic rating life gives estimation of the lifetime for use cases defined
as conventional, the modified rating life can be modified to specific use case and application.
Modified rating life can be calculated for 90% or any other reliability, bearing fatigue load
and it can take into account special bearing properties, contamination in lubricant and
operation condition considered non-conventional (ISO 281, 2007, p. 2). Since there are a lot
more factors considered compared to basic rating life, it is quite clear that modified rating
life will yield much more accurate estimation of the bearing’s lifetime. It has been noted that
when bearings are operated in desirable lubrication and contact stress conditions the lifetime
of bearing can even exceed the calculated basic rating life (ISO 281, 2007, p. 20)
Modified rating life can be calculated with equation 8 (ISO 281, 2007, p. 20):
𝐿𝑛𝑚 = 𝑎1𝑎𝐼𝑆𝑂𝐿10 (8)
Where
𝐿𝑛𝑚 is the modified rating life in millions of revolutions
𝑎1 is life modification factor for reliability,
𝑎𝐼𝑆𝑂 is life modification factor for systems approach.
23
The life modification factor for reliability, 𝑎1, is used to adjust the reliability percentage of
the lifetime. Values for factor 𝑎1 are presented in Table 1.
Table 1. Life modification factor for reliability (ISO 281, 2007, p. 21)
The life modification factor for systems approach, 𝑎𝐼𝑆𝑂, is used to consider the varying
operating conditions of the bearings. When the contact stress between components of the
ball bearing does not exceed the fatigue stress limit of the material, the bearing can have
virtually infinite life. All of the operating conditions that reduce the lifetime of a bearing can
ultimately be boiled down to be affecting either fatigue stress limit or contact stress. For
example, indentations on the bearing raceway increase contact stresses and high temperature
reduces fatigue stress limit. (ISO 281, 2007, p. 21)
24
Since calculation of contact stresses and fatigue stress limit is not straightforward, a more
practical method has been developed. Here the fatigue stress limits and contact stress are
substituted with fatigue load limit 𝐶𝑢 and equivalent bearing load. (ISO 281, 2007, pp. 22-
23) The fatigue load limit can be determined by calculating with equations provided in
ISO 281, but bearing manufacturers often provide value for fatigue load limit in their bearing
catalogues.
The modified rating life considers also environmental effects, such as lubrication,
contamination level and particle size, sealing solution. So, in addition to fatigue load limit,
a contamination factor 𝑒𝐶 and viscosity ratio 𝜅 are needed to calculate the life modification
factor for systems approach. The life modification factor for systems approach is expressed
as function of the factors mentioned above as described by equation 9 (ISO 281, 2007, p.
23):
𝑎𝐼𝑆𝑂 = 𝑓 (𝑒𝐶𝐶𝑢
𝑃, 𝜅) (9)
Where
𝑎𝐼𝑆𝑂 is the life modification factor for systems approach,
𝑒𝐶 is the contamination factor,
𝐶𝑢 is the fatigue load limit,
𝑃 is the equivalent bearing load
𝜅 is the viscosity ratio
A value for the contamination factor 𝑒𝐶 can be selected from Table 2. 𝐷𝑝𝑤 is the pitch
diameter of the ball set within the bearing (diameter of the ball centrelines).
25
Table 2. Contamination factor (ISO 281, 2007, p. 24)
The viscosity ratio is defined as ratio between kinematic viscosity, 𝑣, and reference
kinematic viscosity, 𝑣1, as described in equation 10 (ISO 281, 2007, p. 25):
𝜅 =𝑣
𝑣1 (10)
Where
𝑣 is the kinematic viscosity
𝑣1 is the reference kinematic viscosity
The calculation of viscosity ratio is based on mineral oils. The kinematic viscosity 𝑣 is
lubricant specific value for the operating temperature. The reference kinematic viscosity can
be calculated with equations 11 & 12 (ISO 281, 2007, p. 25):
𝑣1 = 45000𝑛−0.83𝑑𝑚−0.5, 𝑓𝑜𝑟 𝑛 < 1000𝑟/𝑚𝑖𝑛 (11)
𝑣1 = 4500𝑛−0.5𝑑𝑚−0.5, 𝑓𝑜𝑟 𝑛 ≥ 1000𝑟/𝑚𝑖𝑛 (12)
26
Where,
𝑛 is the rotational speed in rpm,
𝑑𝑚 is the pitch diameter of the ball set within the bearing.
When fatigue load limit 𝐶𝑢, equivalent bearing load 𝑃, the contamination factor 𝑒𝐶 and
viscosity ratio 𝜅 are known the life modification factor for systems approach can be
estimated for ball bearings using Figure 5.
Figure 5. Life modification factor for radial ball bearings (ISO 281, 2007, p. 27)
After acquiring basic rating life, life modification factor for reliability and life modification
factor for systems approach, the modified lifetime of the bearing can be calculated.
27
Calculated modified lifetime can be used to make sure that the bearing lasts the required
lifetime or service interval.
2.3.3 Other selection aspects
As already mentioned, the basic rating life does not cover the failure modes that occur with
very light loads and the equation 6 does not give very accurate estimations with extremely
high loads. Since the basic rating life is used as factor in modified rating life equation it can
be conducted that even equation 7 does not cover these cases very well. That is why it is
necessary to study these cases separately to ensure proper operation of the bearings.
In applications where bearings are only lightly loaded and there are rapid changes in
magnitude of the rotational speed there is increased risk of slipping between rolling elements
and raceways of the bearing. The slipping is even more prevalent in applications where
operating speed exceed 50% of the maximum rated speed. To prevent risk associated with
light loading, a guideline has been set for ball bearings to have a minimum load of 1% of the
basic dynamic load rating. (SKF Group, 2013, p. 86)
The speed of bearing is restricted by thermal and mechanical factors. Increase of speed of
bearing causes the heat generated in the bearing to increase. Dependent of external heat and
heat conducted away from the bearing the bearing will reach its operating temperature limit.
Since the heat flow of the bearing system is highly dependent on application, there have been
made a standardized reference values for heat flow rate. The heat flow values can be found
in ISO 15312 standard. With standardised heat flow have been calculated reference speeds
for each bearing, which enables the comparison of bearings. The reference speed describes
the maximum speed bearing can reach within its operating temperature range, when it is
subjected to the standardized heat flow. Bearing manufacturers also present limiting speed,
after which the mechanical integrity of the bearing starts to deteriorate. (SKF Group, 2013,
p. 118 & 126)
There are a lot of different lubrication options for ball bearings, such as grease, oil bath etc.
and thus a consideration of best lubrication method for application should be conducted. The
chosen lubrication method can affect the number of available bearing options, since not all
bearing types are compatible with all lubrication methods. (SKF Group, 2013, p. 240)
28
2.4 Selecting internal clearance and preload
Internal clearance means the distance bearing raceways can move in relation to each other
either in radial (radial internal clearance) or axial (axial internal clearance) direction. The
internal clearance of bearing can vary dramatically from its room temperature uninstalled
state to being in operating temperature and installed with interference fits. Factors affecting
operating bearing internal clearance include difference of the raceways’ thermal expansion,
effects of the interference fit to bearing housing and to the shaft. The amount of internal
clearance has a huge effect on the operation of the bearing, such as friction and fatigue life.
(SKF Group, 2013, pp. 149 & 212-217)
A negative bearing internal clearance is called bearing preload. Introducing suitable bearing
preload can increase the stiffness of the bearing and even slightly increase bearings lifetime,
but it will also increase the bearings frictional moment. Because friction is increased with
bearing preload, the temperature of the bearing might increase, which can lead to thermal
expansion, further increasing the preload – A process that can ultimately destroy the bearing.
(SKF Group, 2013, pp. 149 & 212-217) Oswald, et al. found in their study that decreasing
the internal clearance below zero clearance (i.e. applying preload) will load more of the
rolling elements and to a limit increase the lifetime of the bearing. After certain point the
preload increase will reduce the lifetime of the bearing. (Oswald, et al., 2012, p. 11)
2.4.1 Calculating bearing stiffness based on preload
Within this study the preload of the bearings is selected based purely on required stiffness
of the bearings. The other aspects of the preload selection are neglected to keep focus on the
rotor dynamics. Bearing stiffnesses are calculated using Rotor-Bearing Dynamics
(RoBeDyn) toolbox for MATLAB -software. RoBeDyn toolbox is developed in LUT
university and is used to simulate rotor and bearing dynamics. Package includes function to
calculate bearing stiffnesses and thus is useful in this study. The bearing stiffness function
is based on Hertzian contact theory and results are calculated using Newton-Raphson
iteration. The stiffness calculation method is described in detail in following paragraphs.
In Figure 6 main dimensions of a ball bearing are presented. Where 𝑑 is ball diameter, 𝑟𝑜𝑢𝑡
is the outer groove radius, 𝑟𝑖𝑛 is the inner groove radius, 𝑐𝑑 is the diametral clearance, 𝐷ℎ is
29
the bearing housing diameter, 𝑑𝑚 is the pitch diameter, 𝑑𝑠 is the bore diameter, 𝐷𝑖 is the
inner raceway diameter and 𝐷𝑜 is the outer raceway diameter. (Kurvinen, et al., 2015, p. 13)
Figure 6. Main dimensions of a ball bearing (Kurvinen, et al., 2015, p. 13)
As described in article by E. Kurvinen et al. the elliptical contact conjunction is used to
calculate contact stiffness between rolling elements and bearing raceway. Figure 7 the
geometry of elliptical contact conjunction is shown with the normal force that acts on the
solids. It is noteworthy that radius of contact conjunction is defined negative when surface
is concave. (Kurvinen, et al., 2015, p. 11)
Figure 7. Elliptical contact conjunctions (Kurvinen, et al., 2015, p. 11)
30
Contact deformation are defined with aid of curvature sum, 𝑅 , and curvature difference, 𝑅𝑑,
as shown in equations 13 & 14.
1
R=
1
Rx+
1
Ry (13)
𝑅𝑑 = 𝑅 (1
𝑅𝑥−
1
𝑅𝑦) (14)
Where
1
𝑅𝑥=
1
𝑟𝑎𝑥+
1
𝑟𝑏𝑥 (15)
1
𝑅𝑦=
1
𝑟𝑎𝑦+
1
𝑟𝑏𝑦 (16)
𝑟𝑎𝑥, 𝑟𝑏𝑥, 𝑟𝑎𝑦 and 𝑟𝑏𝑦 are radiuses of the surfaces as shown in Figure 7. (Kurvinen, et al., 2015,
p. 12)
In case of angular contact bearing, the contact angle must be considered. Figure 8 shows a
cross section of angular contact bearing. In Figure 9 the cross section is taken in plane that
is colinear with the contact angle line. In Figure 8 𝜙 is the contact angle of the bearing, 𝑟𝑏𝑦𝑖𝑛
is the groove radius of bearing inner ring, 𝑟𝑏𝑦𝑜𝑢𝑡 is the groove radius of bearing outer ring, 𝑟𝑏
is the radius of the bearing ball, and 𝑑𝑗 is the distance along contact line between the outer
and inner raceways. In Figure 9 𝑟𝑏𝑥𝑜𝑢𝑡 is the radius of outer raceway and 𝑟𝑏𝑥
𝑖𝑛 is the radius of
inner raceway.
31
Figure 8. Cross section of angular contact ball bearing geometry.
Figure 9. Contact plane cross section of angular contact ball bearing
𝑟𝑏𝑦𝑜𝑢𝑡
𝑟𝑏𝑦𝑖𝑛
𝜙
𝑟𝑏
𝑑𝑗
𝑟𝑏𝑥𝑖𝑛
𝑟𝑏𝑥𝑜𝑢𝑡
32
When contact angle is considered, the radiuses of the inner race are calculated as shown
equation 17-19 (Kurvinen, et al., 2015, p. 12)
𝑟𝑎𝑥𝑖𝑛 = 𝑟𝑎𝑦
𝑖𝑛 =𝑑
2= 𝑟𝑏 (17)
𝑟𝑏𝑥𝑖𝑛 =
𝑑𝑚 − (𝑑 +𝑐𝑑
2 ) 𝑐𝑜𝑠𝜙
2𝑐𝑜𝑠𝜙 (18)
𝑟𝑏𝑦𝑖𝑛 = −𝑟𝑖𝑛 (19)
For outer raceway the corresponding equations are 20-22 (Kurvinen, et al., 2015, p. 12)
𝑟𝑎𝑥𝑜𝑢𝑡 = 𝑟𝑎𝑦
𝑜𝑢𝑡 =𝑑
2= 𝑟𝑏 (20)
𝑟𝑏𝑥𝑜𝑢𝑡 =
𝑑𝑚 + (𝑑 +𝑐𝑑
2 ) 𝑐𝑜𝑠𝜙
2𝑐𝑜𝑠𝜙 (21)
𝑟𝑏𝑦𝑜𝑢𝑡 = −𝑟𝑜𝑢𝑡 (22)
When stiffness of angular contact ball bearing is calculated equations 17-22 must be used in
formulas 13 & 14 to retrieve curvature sum, 𝑅 , and curvature difference, 𝑅𝑑, that in turn,
will be used in later calculations.
When a load is applied to the solids, contact point is expanded to an ellipse. The ellipticity
parameter, 𝑘𝑒, is described as shown below, in equation 23 (Sopanen & Mikkola, 2003, p.
7).
𝑘𝑒 =𝑎𝑒
𝑏𝑒 (23)
Where
𝑎𝑒 is the semimajor axis of the ellipse
𝑏𝑒 is the semiminor axis of the ellipse
33
In turn the ellipticity parameter is defined as shown in equation 24 (Sopanen & Mikkola,
2003, p. 7)
𝑘𝑒 = [2𝜉 − 휁(1 + 𝑅𝑑)
휁(1 − 𝑅𝑑)]
12
(24)
Where elliptical integrals 𝜉 & 휁 are as follows
𝜉 = ∫ [1 − (1 −1
𝑘𝑒2
) sin2 𝜑]−
12
𝜋/2
0
𝑑𝜑 (25)
𝜉 = ∫ [1 − (1 −1
𝑘𝑒2
) sin2 𝜑]−
12
𝜋/2
0
𝑑𝜑 (26)
Where
𝜑 is an auxiliary angle.
As stated by Sopanen & Mikkola, the elliptical integral requires an iterative process. To
simplify calculation following approximated values can be used (Kurvinen, et al., 2015, p.
12) (Sopanen & Mikkola, 2003, p. 8)
�̅�𝑒 = 1,0339 (𝑅𝑦
𝑅𝑥)
0,6360
(27)
𝜉̅ = 1,0003 + 0,5968𝑅𝑥
𝑅𝑦 (28)
휁̅ = 1,5277 + 0,6023𝑅𝑦
𝑅𝑥 (29)
𝐸′ =
2
1 − 𝑣𝑎2
𝐸𝑎+
1 − 𝑣𝑏2
𝐸𝑏
(30)
34
Where
𝐸′ is the effective modulus of elasticity
𝐸𝑎 is the modulus of elasticity for material a
𝐸𝑏 is the modulus of elasticity for material b
𝑣𝑎 is the Poisson’s ratio for material a
𝑣𝑏 is the Poisson’s ratio for material b
Using equations 27-30 shown above, the contact stiffness coefficient 𝐾𝑐 can be calculated
with equation 31
𝐾𝑐 = 𝜋�̅�𝑒𝐸′√𝑅𝜉̅
4,5휁3̅ (31)
Total stiffness of a single bearing ball can be calculated when contacts with inner and outer
race are combined with equation 32 as shown below
𝐾𝑐𝑡𝑜𝑡 =
1
((1
𝐾𝑐𝑖𝑛)
23
+ (1
𝐾𝑐𝑜𝑢𝑡)
23
)
32
(32)
Where
𝐾𝑐𝑡𝑜𝑡 is the total stiffness of a bearing ball
𝐾𝑐𝑖𝑛 is the contact stiffness between bearing ball and inner raceway
𝐾𝑐𝑜𝑢𝑡 is the contact stiffness between bearing ball and outer raceway
The ball bearing forces and moments can be calculated using the relative displacements of
the raceways. The displacements of each bearing balls can be evaluated in radial direction
using equation 33 and in axial direction with equation 34 (Kurvinen, et al., 2015).
𝑒𝑗𝑟 = 𝑒𝑥𝑐𝑜𝑠𝛽𝑗 + 𝑒𝑦𝑠𝑖𝑛𝛽𝑗 (33)
𝑒𝑗𝑡 = 𝑒𝑧 − (−𝛾𝑥𝑠𝑖𝑛𝛽𝑗 + 𝛾𝑦𝑐𝑜𝑠𝛽𝑗)(𝑅𝑖𝑛 + 𝑟𝑖𝑛) (34)
35
Where
𝑒𝑥 is relative displacement in x direction
𝑒𝑦 is relative displacement in y direction
𝑒𝑧 is relative displacement in z direction
𝛽𝑗 is the attitude angle of ball bearing j
𝛾𝑥 is the relative misalignment of the inner and outer raceway about x axis
𝛾𝑦 is the relative misalignment of the inner and outer raceway about y axis
The displacements of raceways and bearing balls are visualised in Figure 10.
Figure 10. Axial and transverse cross-section in the A-A plane of illustrative structure of a
ball bearing (Kurvinen, et al., 2015, p. 14)
36
The contact angle of a single bearing ball, 𝜙𝑗, is expressed as shown in equation 35
𝜙𝑗 = tan−1 (𝑒𝑗
𝑡
𝑅𝑖𝑛 + 𝑟𝑖𝑛 + 𝑒𝑗𝑟 − 𝑅𝑜𝑢𝑡 + 𝑟𝑜𝑢𝑡
) (35)
Where,
𝑅𝑖𝑛 is the radius of the inner raceway
𝑅𝑜𝑢𝑡 is the radius of the outer raceway
𝑟𝑖𝑛 is the radius of the inner raceway groove
𝑟𝑜𝑢𝑡 is the radius of the outer raceway groove
the distance between raceways along the contact line of the bearing 𝑑𝑗 is expressed by
equation 36. (Kurvinen, et al., 2015, p. 16)
𝑑𝑗 = 𝑟𝑜𝑢𝑡 + 𝑟𝑖𝑛 −𝑅𝑖𝑛 + 𝑟𝑖𝑛 + 𝑒𝑗
𝑟 − 𝑅𝑜𝑢𝑡 + 𝑟𝑜𝑢𝑡
𝑐𝑜𝑠𝜙𝑗 (36)
The total elastic deformation of a single bearing ball, 𝛿𝑗𝑡𝑜𝑡, is calculated with equation 37
(Kurvinen, et al., 2015, p. 16)
𝛿𝑗𝑡𝑜𝑡 = 2𝑟𝐵 − 𝑑𝑗 (37)
Where 𝑟𝐵 is the radius of the bearing ball.
Then the forces acting on a single bearing ball, 𝐹𝑗, can be calculated using equation 38
(Kurvinen, et al., 2015, p. 16).
𝐹𝑗 = 𝐾𝑐𝑡𝑜𝑡(𝛿𝑗
𝑡𝑜𝑡)32 (38)
The forces of each individual bearing ball can be combined to form bearing forces of the
whole ball bearing using equations 39 – 43 (Kurvinen, et al., 2015, p. 16). Sum of forces and
37
moments should include only components where bearing balls are under compression
(Sopanen & Mikkola, 2003, p. 12).
𝐹𝑋 = − ∑ 𝐹𝑗𝑐𝑜𝑠𝜙𝑗𝑐𝑜𝑠𝛽𝑗
𝑧
𝑗=1
(39)
𝐹𝑌 = − ∑ 𝐹𝑗𝑐𝑜𝑠𝜙𝑗𝑠𝑖𝑛𝛽𝑗
𝑧
𝑗=1
(40)
𝐹𝑍 = − ∑ 𝐹𝑗𝑠𝑖𝑛𝜙𝑗
𝑧
𝑗=1
(41)
𝑇𝑋 = − ∑ 𝐹𝑗(𝑅𝑖𝑛 + 𝑟𝑏)𝑠𝑖𝑛𝜙𝑗𝑠𝑖𝑛𝛽𝑗
𝑧
𝑗=1
(42)
𝑇𝑌 = − ∑ 𝐹𝑗(𝑅𝑖𝑛 + 𝑟𝑏)𝑠𝑖𝑛𝜙𝑗(−𝑐𝑜𝑠𝛽𝑗)
𝑧
𝑗=1
(43)
Where
𝐹𝑋 is the bearing force in x direction
𝐹𝑌 is the bearing force in y direction
𝐹𝑍 is the bearing force in z direction
𝑇𝑋 is the bearing moment around x axis
𝑇𝑌 is the bearing moment around y axis
𝑧 is the number of bearing balls
In the RoBeDyn toolbox for MATLAB the bearing displacements are calculated using
Newton Raphson iteration with equation 44.
𝒆(𝑛+1) = 𝒆(𝑛) − (𝑲𝑇(𝑛)
)−1
𝑸(𝑛) (44)
38
Where
𝒆(𝑛+1) is matrix containing the displacements of the bearing on iteration step 1+n
𝒆(𝑛) is matrix containing the displacements of the bearing on iteration step n
𝑲𝑇(𝑛)
is the tangent stiffness matrix vector on iteration step n
𝑸(𝑛) is matrix containing the forces acting on bearing on iteration step n
bearing forces 𝑸(𝑛) include bearing and external forces as described in equation 45
𝑸(𝑛) = 𝑸𝑏(𝑛)
− 𝑸𝑒𝑥𝑡(𝑛)
(45)
Where
𝑸𝑏(𝑛)
is the matrix of bearing forces
𝑸𝑒𝑥𝑡(𝑛)
is the matrix of external forces
External forces include bearing preload forces and thus amount of preload is factor when
calculating bearing stiffness.
The stiffness matrix for each iteration step, 𝑲𝑇(𝑛)
, is presented in equation 46
𝑲𝑇(𝑛)
=𝜕𝑸(𝑛)
𝜕𝒆(𝑛) (46)
Newton Raphson iteration is continued until convergence criterion, shown in equation 47, is
met
|𝑸| < 𝐾𝑐𝑜𝑛𝑣 ∙ |𝑸𝒆𝒙𝒕| (47)
Where
𝐾𝑐𝑜𝑛𝑣 is the convergence criterion factor
Bearing stiffness matrix 𝒌𝑏 is the tangent stiffness matrix from last converged iteration step.
39
From E. Krämer’s book we can retrieve an estimation for damping factor in rolling element
bearings. The damping matrix of the bearing is estimated as shown in equation 48. (Krämer,
1993)
𝒄𝒃 = (0,25 … 2,5) ∙ 10−5𝒌𝒍𝒊𝒏 (48)
where
𝒄𝒃 is the damping matrix of the bearing
𝒌𝑙𝑖𝑛 is the linearized stiffness of the bearings in N/mm
A constant is chosen within the limits shown in equation 48. For calculating damping
matrices, the equation 49 is used.
𝒄𝒃 = 2,5 ∙ 10−5𝒌𝒃 (49)
Similar calculation method specifically for angular contact bearings has been introduced by
Noel, et al. in their study (2013). In their study this method has proven good results with not
considerable increase in calculation effort. (Noel, et al., 2013) Since RoBeDyn -toolbox was
available to use, the effort of implementing analytical method was considered to be out of
scope of this study.
The procedure of bearing stiffness calculation using RoBeDyn MATLAB toolbox is
presented in flowchart in Figure 11.
40
Figure 11. Flowchart bearing stiffness calculation using RoBeDyn-toolbox
2.4.2 Preloading methods
The preload can be applied to the bearing system in various ways but three methods are most
common: Fixed position preload, stiff spring preload and constant preload. In fixed position
preload the raceways are displaced in relation to each other by locating features such as
precision ground spacers. The displacements of the raceways stay constant, until thermal
expansion takes effect and it can severely change the relative displacements and thus the
preload in the bearings.
41
To reduce effects of the thermal expansion a spring can be used to allow movement of the
bearing. The preload method is called stiff spring preload when the stiffness of the spring
cannot be ignored, and the spring force and thermal expansion must be considered when
calculating the preload. Preload method can be considered as constant preload system when
the springs are so soft that thermal displacements effect on spring force can be neglected.
The selection between these preloading methods is balancing act: fixed position preload can
provide rigidity of the bearing system, whereas springs allow for thermal expansion and can
prevent failures related to excessive preload caused by thermal expansion. The study shows
that the rigid preload method produces more stiff bearing arrangement and is preferred if the
preload effects of thermal expansion can be controlled by other means. (Cao, et al., 2011, p.
872) In Figure 12 an assembly, in which constant force preload has been implemented using
coil springs, is presented. On left side of the figure the assembly is presented before
tightening the axle nut and spring is on its free length. On the right-side, the axle nut is
tightened, and the spring is under tension.
Figure 12. Constant force preload method
In Figure 13 an assembly with constant displacement preload is presented. The axial gap
between inner bearing ring and axle corresponds to the desired constant axial preload. The
axle nut is tightened to close the gap and thus axial preload is applied.
42
Figure 13. Constant displacement preload method
2.4.3 Thermal expansion effects on bearing preload
Thermal expansion on the components of a bearing system can affect the internal clearances
and preload of the bearing. For example, inner bearing ring that expands radially will
decrease the radial internal clearance. Similarly, if shaft expands axially it will affect location
of bearing inner rings and thus affect the axial internal clearance of the bearing. In most basic
form the thermal expansion of the components related to the bearing system (i e. shaft,
housing) can be calculated using thermal strain as presented in equation 50.
휀𝑡ℎ = 𝛼(𝑇)(𝑇 − 𝑇𝑟𝑒𝑓) (50)
Where,
휀𝑡ℎ is the thermal strain,
𝛼 is the thermal expansion coefficient
𝑇𝑟𝑒𝑓 is the ambient temperature
𝑇 is temperature of the component
43
Given that the above calculation method for thermal expansion is simple to use and
understand, it will not be adequate for complex structure as bearing system. For this reason
more sophisticated evaluation method is needed.
As stated in article by S.M. Kim & S.K. Lee, the thermo-mechanical system that consists of
shaft, bearings and housing, has a closed loop interaction. This means that e.g. increase in
temperature can increase the preload, which in turn increases heat generation within the
bearing and thus increases the temperature. This interaction means that in transient state the
preload of the bearing can rise higher than the steady state of the preload is. This is caused
by pseudo thermal inertia, which enables the preload of the bearing keep rising past its steady
state value until it settles to the steady state value. (Kim & Lee, 2005, p. 1063) Figure 14
visualises closed loop interactions within the bearing system.
Figure 14. Closed loop thermo-mechanical interaction of bearing system (Kim & Lee, 2005,
p. 1064)
44
Holkup, et al. described in their article how the thermo-mechanical system can be modelled
using Finite Element Method (FEM) in ANSYS software. The model presented considers
the closed loop interaction described above and transient state changes in bearing stiffness.
The bearings are modelled using Jones’ theory (1960), which considers the centrifugal forces
acting on the rolling elements and Hertzian contact between rolling elements and the
raceways. The other mechanical components are modelled using 2D axisymmetric elements.
Thermal modelling is based on Fouriers’s law of conduction and first law of
thermodynamics. (Holkup, et al., 2010, p. 365) The subject has been also studied considering
the surface waviness on drop down bearing by Neisi, et al. (2019).
2.5 Bearing arrangement effects on rotor dynamics
Rotor dynamic studies the dynamic behaviour of rotating systems. Goal is to ensure problem
free operation throughout the operating speed range. One way of achieving this goal is to
avoid resonant frequencies completely, if that is not possible, the response of the system
(vibration) should be kept within acceptable limits.
In rotor dynamics the structure to be analysed must be divided into elements which each
have certain degrees of freedom (DOF) in which they can vibrate. Elements can be flexible
beam or solid, or rigid mass elements and they each have different number of DOFs. By
combining the all the DOFs of the structure we get the total number of system DOFs. The
equation of motion for system with multiple degrees of freedom is presented in equation 51
(Matsushita, et al., 2017, p. 43).
𝐌�̈� + 𝑫�̇� + 𝑲𝑿 = 𝑭(𝒕) (51)
Where
𝐌 is the mass matrix
�̈� is the second derivative of displacement vector
𝑫 is the damping matrix
�̇� is the first derivative of displacement vector
𝑲 is the stiffness matrix
𝑿 is the displacement vector
𝑭(𝒕) is the external force vector as function of time
45
When the gyroscopic effects of the rotor are considered the equation of motion takes form
as presented in equation 52. (Friswell, et al., 2010, pp. 96-97)
𝐌�̈� + (𝑫 + 𝝎𝑮)�̇� + 𝑲𝑿 = 𝑭(𝒕) (52)
Where
𝑮 is the gyroscopic matrix
In the equation of motion, the stiffness and mass matrices include not only stiffness and
damping values related to the geometry and material of the rotor but also stiffness and
damping values of supporting structures e.g. bearings. The natural frequencies and mode
shapes can be solved from the equation of motion using damped eigenvalue problem.
To solve damped eigenvalue problem the equation of motion (52) has to have zero external
forces and then it can be multiplied by 𝐌−𝟏 yielding equation 53
�̈� + 𝐌−𝟏(𝑫 + 𝝎𝑮)�̇� + 𝐌−𝟏𝑲𝑿 = 𝟎 (53)
Which can be written in state-space form, as seen in equation 54
{𝒚𝟏 = �̇� = 𝒚𝟐
𝒚𝟐 = �̈� = −𝐌−𝟏(𝑫 + 𝝎𝑮)�̇� − 𝐌−𝟏𝑲𝑿 (54)
Given that states are 𝑦1 = �̇� and 𝑦2 = �̈�
�̇� = [𝟎 𝑰
−𝐌−𝟏𝑲 −𝐌−𝟏(𝑫 + 𝝎𝑮)] [
𝒚𝟏
𝒚𝟐] = 𝑨𝒚 (53)
Where
𝐴 is the state matrix
𝑦 is the state vector
46
To get the equation to standard eigenvalue problem form, it is given that 𝑦 = 𝒛𝑒𝜆𝑡, where 𝒛
is constant vector and 𝜆 is a scalar value. Thus equation 54 can be obtained
𝑨𝒛𝒊 = 𝝀𝒊𝒛𝒊 (54)
Solving the eigenvalue problem gives complex conjugate pair of results where the imaginary
part represents the damped natural frequency. By solving the damped natural frequencies
throughout the operating speed range of the machine we can plot the Campbell diagram of
the rotor. The Campbell diagram visualizes how natural frequencies change over the
operating speed range and shows the possible resonant operating points where the operating
speed is equal to a natural frequency. (Friswell, et al., 2010, p. 229)
Looking even at a simply supported beam, distance between supports influences natural
frequency and mode shapes. The analogy is accurate also for shaft supported by bearings.
Below is presented how changing the location of bearings affects the rotor dynamic
properties of a Jeffcott rotor. The results and visualisations are produced with RoBeDyn-
toolbox for MATLAB. In Figure 15 the effects changing bearing location have on
modeshapes is shown using a simple Jeffcott rotor as an example. Green dots mark the
position of bearings, on the right side the bearings are closer to each other.
Figure 15. Modeshapes of a Jeffcott rotor with varied bearing location
47
To achieve acceptable dynamic properties for the electric motor, besides the effects of
bearing locations, the arrangement of angular contact bearings (face-to-face or back-to-back)
should be considered. The two arrangements differ in their ability to carry tilting loads. The
back-to-back arrangement can carry tilting loads and is relatively stiff, whereas the face-to-
face allows more misalignment but is not as stiff and as able to carry tilting loads (SKF
Group, 2013, p. 386). In Figure 16 the two arrangement options are demonstrated. The
dashed lines going through bearing balls demonstrate the contact line of bearing balls.
a) b)
Figure 16. demonstration of angular contact bearing mounting methods. in subfigure a),
face-to-face arrangement and on b), back-to-back configuration.
In their paper Helfrich and Wagner studied a shaft with two bearing and two rotors. They
used PERMAS FEM software to optimise the shaft properties to maximize the first natural
frequency of the shaft. They optimised dimensions of the shaft, location of disks and
bearings as well as stiffness and damping of the bearings. In their findings the size of the
disk is reduced to minimum and the stiffness of the shaft is maximised. In their optimisation
the bearings optimal location is as far as possible from each other. (Helfrich & Wagner,
2015) Since, in case of eMAD-project motor, most of the factors are predetermined,
optimisation is focused on bearing location and stiffness (via preload).
48
3 RESULTS
The eMAD electric motor is designed in outer rotor permanent magnet configuration as seen
in Figure 4. The rotor has a shaft which is supported to the stator with pair of bearings. The
construction differs from conventional inner rotor structure, where the active parts of the
rotor are closely connected to the shaft. In outer rotor construction more complex structure
is needed to connect the active parts of the rotor to the supporting shaft. This means that
structural and dynamical analysis of the rotor is even more important than in inner rotor
structure. In Table 3 some of the design parameters for the eMAD motor are presented.
Abbreviations DE and NDE are used to differentiate drive end and non-drive end
components from each other.
Table 3. Design requirements for eMAD motor
eMAD properties
Property value unit
Maximum rotational speed 10000 rpm
1047,2 rad/s
Maximum rotor mass 40 kg
Distance from DE bearing location to center of gravity of the rotor 0,024 m
Distance between bearing locations 0,145 m
Lifetime 5000 h
Diameter of DE bearing housing 130 mm
Diameter of NDE bearing housing 110 mm
3.1 Selecting bearings for eMAD motor
In eMAD motor, many of the design parameters of the bearings are predetermined by other
components and design factors, some of which are shown in the Table 3. Just the diameters
of bearing housings limit the number of bearing options. Factoring in the operational speed
and requirement for minimal maintenance, the selection narrows down further. In practise
this means that the bearing should be sealed construction and grease lubricated while
achieving up to 10 000 rpm operating speeds. In order to select correct bearings for loads of
the application, following calculations are conducted.
49
The gravitational loads on the bearings can be calculated using equations 1 & 2.
{
40 kg ∙ 9.81 m/s − 𝑁1 − 𝑁2 = 040 kg ∙ 9.81 m/s ∙ 0.024 m − 𝑁2 ∙ 0.145 m = 0
Which can be solved for 𝑁1 & 𝑁2
{𝑁1 = 327 N
𝑁2 = 65 N
Balance quality grades are used to describe the tolerance of unbalance in the finished rotor.
According to ISO 1940-1 (2003) standard the recommended balancing quality grade for
electric motors with shaft height above 80 mm and operating speed over 950 rpm is G 2,5.
Using equation 4 and the constants found in the Table 3 the permissible residual unbalance
can be calculated.
𝑈𝑝𝑒𝑟 = 10002,5 mm/s ∙ 40 kg
1047.21s
= 95.5 gmm
Now the loads caused by dynamic unbalance can be calculated with equation 3.
𝐹𝑐 = 95.5 ∙ 10−6 kgm ∙ (1047.21
s)
2
= 104.7 N
Since the DE bearing is more critical in this application it will be selected first, thus the
gravitational load 𝐹𝑔 = 𝑁1 = 327 𝑁. Since it is not known how the unbalance load will be
distributed to the two bearings the full amplitude will be used. From SKF rolling bearings
(2013, p. 86) a value for mean load factor is retrieved 𝑓𝑚 = 0.8125 .
𝐹𝑚 = 𝑓𝑚(𝐹𝑔 + 𝐹𝑐) = 350.8 N
50
The next step is to calculate the equivalent radial load using equation 6. The equation
requires some bearing specific values, so estimated values are used at this point. Following
estimations are used: Axial load 𝐹𝑎 = 900 𝑁, 𝑋 = 0.44 and 𝑌 = 1.2
𝑃𝑟 = 𝑋𝐹𝑟 + 𝑌𝐹𝑎 = 0.44 ∙ 350.8 N + 1.2 ∙ 900 N = 1234 N
Basic rating life is useful tool when initially selecting a bearing for an application. By solving
the equation 7 for basic dynamic load rating 𝐶, we can calculate the requirement for dynamic
load rating when loads and expected lifetime is known. The lifetime is calculated on
assumption that the machine will run on maximum speed for its whole operating life, giving
𝐿10 = 3000 million revolutions
𝐶 = 𝑃 ∙ √𝐿103 = 1234 N ∙ √3000
3= 17797 N
Looking for angular contact ball bearings that fulfill requirements mentioned above and with
high enough operating speed and basic dynamic load rating above the required value
calculated above, bearing S71919 CE/P4A seems like suitable selection.
Table 4 Specifications of S71919 CE/P4A bearing
S71919 CE/P4A specifications
Property value unit
Maximum attainable speed with grease lubrication
14 000 rpm
Basic dynamic load rating 30.7 kN
Fatique load limit 0.98 kN
Pitch diameter 0.1125 m
Kinematic viscosity of lubricant at operating temperature
13 mm^2/s
Using the values presented in Table 4 the basic rating life can be calculated.
𝐿10 = (30,7 𝑘𝑁
1.234 𝑘𝑁)
3
= 15 398 𝑚𝑖𝑙𝑙𝑖𝑜𝑛 𝑟𝑒𝑣𝑜𝑙𝑢𝑡𝑖𝑜𝑛𝑠
51
To calculate the modified rating life the estimated value for contamination factor 𝑒𝐶 = 0,7
is extracted from Table 2. Reference kinematic viscosity can be calculated using equation 12.
𝑣1 = 4500 · 10 000−0.5 · 112.5−0.5 = 4.24 𝑚𝑚2/𝑠
Viscosity ratio can be calculated with equation 10
𝜅 =13 𝑚𝑚2/𝑠
4.24 𝑚𝑚2/𝑠= 3.07
Now that all required factors are known the life modification factor can be estimated using
Figure 5. It seems that the value goes out of bounds of the figure, in which case ISO 281
instructs to use 𝑎𝐼𝑆𝑂 = 50. Life modification factor 𝑎1 = 0.25 (99% reliability) is selected
from Table 1. Now the modified rating life can be calculated using equation 8.
𝐿𝑛𝑚 = 0.25 · 50 · 15 398 = 192 475 𝑚𝑖𝑙𝑙𝑖𝑜𝑛 𝑟𝑒𝑣𝑜𝑙𝑢𝑡𝑖𝑜𝑛𝑠
The modified rating life is well over the lifetime requirements, so no iterations are needed.
Similar calculations are conducted for NDE bearing and S71916 CE/P4A is selected. The
unbalance forces, viscosity ratio and life modification factor are same for DE and NDE
bearing. The NDE bearing specifications are shown in Table 5.
Table 5. Specifications of S71916 CE/P4A bearing
S71916 CE/P4A specifications
Property value unit
Maximum attainable speed with grease lubrication
16 500 rpm
Basic dynamic load rating 22,5 kN
Fatique load limit 0,75 kN
Pitch diameter 0,095 m
The results of bearing calculations for NDE bearing are shown in Table 6.
52
Table 6. Bearing calculation results of S71916 CE/P4A bearing
S71916 CE/P4A results
Property value unit
Mean load factor 0,765
Equivalent radial load 1225 N
Basic rating life 6191 million revolutions
Modified rating life 77395 million revolutions
3.2 Bearing preload effects on rotor dynamics
In order to determine the correct axial preload for the bearings the effects of bearing preload
on rotor dynamics must be known. First the preload effects on the bearing stiffness were
studied. The bearing stiffness of the bearings was calculated as described in the chapter 2.4.1
using RoBeDyn toolbox for MatLab software. The variables used in bearing stiffness
calculation can be found in
Table 7.
Table 7. Variables used in bearing stiffness calculation
Property Bearing Unit
location DE NDE
model S71919 CE/P4A
S71916 CE/P4A
Bore diameter 95 80 mm
Outer diameter 130 110 mm
Pitch diameter 112,5 95 mm
Ball diameter 11,112 9,525 mm
Number of bearing balls
25 24
Contact angle 15 15 deg
Inner race conformity ratio
0,52 0,52
Elastic modulus of bearing ring material
207 207
GPa
Poisson's ratio of bearing ring material
0,3 0,3
Elastic modulus of bearing ball material
315 315
GPa
Poisson's ratio of bearing ball material
0,26 0,26
Gravitational load 327 65 N
53
To verify the model, axial stiffness values are compared to values provided by bearing
manufacturer SKF in Table 8.
Table 8. Axial stiffness of the bearings
Bearing Preload [N]
SKF value for stiffness [N/µm]
RoBeDyn Stiffness at 0 rpm [N/µm]
SKF S71919 CE/P4A
166 68 54,3
500 107 84,2
995 147 113,6
SKF S71916 CE/P4A
123 56 56,2
370 89 86,4
740 123 114,8
To analyse the effects of axial preload to the stiffness of bearings, the stiffness values of the
bearings are calculated with RoBeDyn-toolbox and plotted in respect to preload values. The
relation is visualised in Figure 17. In the figure labels with Kx correspond to the axial
stiffness of the bearing and Ky and Kz refer to radial directions respectively.
Figure 17. Preloads effect on bearing stiffness at 10 000 rpm
54
From Figure 17 it can be clearly seen that axial stiffnesses of the bearings are far lower than
radial stiffnesses. It can also be noted that radial stiffnesses curves rise relatively quickly in
the lower end of the preload and curves flatten in the higher range of preload. Interesting is
that above preload of 750 N the radial stiffness values are very close to each other, thus
stiffnesses are close to isotropic.
The stiffness of the angular contact ball bearings could also be calculated analytically as
presented in study by Noel, et al. (2013), but due to scope limitations and time constraints,
it will not be covered in this study.
To analyse the rotor dynamics of the rotor, a simplified 3D model of the rotor was imported
into the ANSYS FEM software. First a modal analysis was conducted to find out the critical
speeds and mode shapes of the rotor. In Figure 18. Campbell diagram of rotor bearing system
with 900 N preload the Campbell diagram is shown.
Figure 18. Campbell diagram of rotor bearing system with 900 N preload
As can be seen from the Figure 18, the rotor has three resonant frequencies (around 0, 2000
and 10000 rpm) within its operating range. By looking at deformation shapes of these three
modes it can be concluded that, since bearings do not have axial stiffness in ANSYS, first
critical speed corresponds to axial rigid body mode and second critical speed corresponds to
torsional deformation mode. Thus, only third critical speed is relevant when evaluating UBR.
The mode shape of third critical speed is shown in Figure 19. From the figure it can be seen,
55
that the outer parts of the rotor start to whirl backwards (opposite to rotating direction) while
the shaft in the center stays relatively undeformed.
Figure 19. Mode 3 of the rotor
The speed dependent stiffness values calculated with RoBeDyn toolbox were used as design
points in the ANSYS software and the maximum displacements of the rotor core were
calculated in each design point. This means that analysis considers also the speed dependent
bearing stiffness. With the help of the design points, the unbalance response (UBR) diagrams
can be plotted. When calculating UBR values, the variables shown in Table 7 and residual
unbalance 𝑈 = 95.5 𝑔𝑚𝑚 were used.
56
Figure 20 Figure 20 and Figure 21 the UBR plots of rotor bearing system are shown. Here
the unbalance is located in the NDE of the rotor core and deformations correspond to mode
3 shown in Figure 19.
Figure 20. UBR diagram of rotor bearing system with 800 N and 900 N preloads
0,00E+00
5,00E-04
1,00E-03
1,50E-03
2,00E-03
2,50E-03
3,00E-03
0
50
0
10
00
15
00
20
00
24
00
30
00
36
00
41
00
46
00
49
00
55
00
60
00
65
00
70
00
75
00
80
00
85
00
90
00
95
00
10
00
0
10
50
0
11
00
0
11
50
0
12
00
0
mm
RPM
UBR of eMAD rotor
UBR 900 N UBR 800 N
0,00E+00
5,00E-04
1,00E-03
1,50E-03
2,00E-03
2,50E-03
3,00E-03
0
50
0
10
00
15
00
20
00
24
00
30
00
36
00
41
00
46
00
49
00
55
00
60
00
65
00
70
00
75
00
80
00
85
00
90
00
95
00
10
00
0
10
50
0
11
00
0
11
50
0
12
00
0
mm
RPM
UBR of eMAD rotor
UBR 900 N UBR 800 N
57
Figure 21. UBR diagram of rotor bearing system with various preloads
As seen in the Figure 20 and Figure 21 the UBR of the rotor increases steadily as operating
speed increases, although according to Figure 18 there is critical speed at around 10 000 rpm
when the bearings are preloaded with 900 N. Since the analysis was conducted with design
points in 500 rpm interwall the shapes of the plots are not perfect. As explained by Friswell,
et al., if the stiffness of bearings is isotropic, the rotor orbits will be circular. And if stiffness
is anisotropic, the rotor orbits will be elliptical. (Friswell, et al., 2010, p. 117) The orbits
actually consist of forward and backward component and when orbit is circular the both
components are circular. When forward whirling component is dominating the whirl mode
is forward and vice versa. Since the unbalance excites only forward components, in case of
isotropic bearing only the forward modes are excited. (Chen & Gunter, 2005, p. 46 & 62)
Fact that The mode 3 is backward whirling mode and bearings are nearly isotropic with
preload over 750 N, explains why there is no notable resonance with higher preload values.
It can be also noted that different preloads have slight effect on the UBR, although in all
cases the amplitudes are quite small, only few micrometres. In Figure 22 the 400 N preload
UBR with 50 rpm design point interwall can be seen. Closer inspection reveals that the UBR
has a spike around 9500 rpm.
0
0,0005
0,001
0,0015
0,002
0,0025
0,003
0,0035
0,004
0,0045
0,005
5500 6000 6500 7000 7500 8000 8500 9000 9500 10000 10500 11000 11500 12000
mm
RPM
UBR of eMAD rotor
UBR 400 N UBR 600 N UBR 800 N UBR 900 N UBR 1000 N UBR 1200 N
58
Figure 22. 400 N preload UBR with 50 rpm interwall
In Figure 23 similar 50 rpm interwall analysis with 900 N preload is shown. There are no
significant spike, because with isotropic bearings the backward mode is not excited.
Figure 23. 900 N preload UBR with 50 rpm interwall
To check systems sensitivity to increasing the unbalance, analysis with higher unbalance
was conducted. When analysed with 10 times the allowed residual unbalance, the UBR
results become roughly tenfold but still there is no notable difference between the different
preload forces. This can be seen from results presented in Figure 24.
0
0,001
0,002
0,003
0,004
0,005
0,006
0,0078
50
0
85
50
86
00
86
50
87
00
87
50
88
00
88
50
89
00
89
50
90
00
90
50
91
00
91
50
92
00
92
50
93
00
93
50
94
00
94
50
95
00
95
50
96
00
96
50
97
00
97
50
98
00
98
50
99
00
99
50
10
00
0
mm
RPM
UBR of eMAD rotor
UBR 400 N
0
0,0005
0,001
0,0015
0,002
0,0025
95
00
95
50
96
00
96
50
97
00
97
50
98
00
98
50
99
00
99
50
10
00
0
10
05
0
10
10
0
10
15
0
10
20
0
10
25
0
10
30
0
10
35
0
10
40
0
10
45
0
10
50
0
10
55
0
10
60
0
10
65
0
10
70
0
10
75
0
10
80
0
10
85
0
10
90
0
10
95
0
mm
RPM
UBR of eMAD rotor
UBR 900 N
59
Figure 24. UBR with tenfold residual unbalance
In result described above, it was assumed that the unbalance would be located in the outer
perimeter of the rotor. This is quite reasonable assumption since most of the mass (rotor
core, magnets) are located in the outer perimeter of the rotor and thus it is harder to ensure
perfect balance of these areas. In Figure 25 deformation of the rotor is shown, visualising
how the results would change if the residual unbalance would be located in the most central
part of the rotor, the shaft. Amplitude of deformations in this case are even lower than
described above, about 100 times less than when the unbalance is located in the outer parts
of the rotor, thus this scenario is not significant concern for longevity of the rotor.
0
0,01
0,02
0,03
0,04
0,05
5500 6000 6500 7000 7500 8000 8500 9000 9500 10000 10500 11000 11500 12000
mm
RPM
UBR of eMAD rotor with 10 times the residual unbalance
UBR 400 N UBR 1200 N
60
Figure 25. Rotor deformation shape with unbalance located in the shaft
3.2.1 RoBeDyn beam and mass disk model
In addition to the solid element model made with ANSYS software, the rotor was modelled
with RoBeDyn toolbox using beam elements and a mass disk. The shaft of the rotor was
modelled with beam elements, while the flange along with rotor core were modelled as a
mass disk element. Mass properties, center of mass and moments of inertia of the mass disk
where imported from the 3D-model and a set of torsional springs were used to connect the
mass disk to the shaft. Torsional springs allow the otherwise nondeformable mass disk to
deflect, mimicking deformation in the flange, as seen in the mode 3 shown in Figure 19. To
tune the model, the stiffness values of the torsional springs were adjusted until the
frequencies of mode 3 in the Campbell diagram calculated for RoBeDyn model matched to
the one shown in Figure 18.
In Figure 26 the beam and mass disk model of the rotor is presented. Since the RoBeDyn
toolbox is not designed for outer rotors, the visualization of the rotor core does not
correspond to the real structure.
61
Figure 26. Visualisation of the beam and mass disk rotor model
By extracting rotation of torsional springs and calculating the resulting deflection of the outer
parts of the rotor, a 900 N preload UBR plot shown in Figure 27 can be obtained.
62
Figure 27. UBR of Mass disk model
When Figure 27 and Figure 23 are compared, the clear difference in UBR values obtained
with ANSYS-software and RoBeDyn-toolbox can be noted. Firstly, the UBR calculated with
RoBeDyn gives much lower amplitudes and secondly there is clear spike at resonant
frequency 10050 rpm. Although the results are somewhat similar, the discrepancy in results
signals that assumption of rigid rotor creates notable error to the results.
3.3 Bearing arrangement optimisation for rotor dynamics
Standard form of RoBeDyn -toolbox does not take into consideration whether the angular
contact ball bearings are installed in back-to-back or face-to-face -arrangement. It was
decided that code development for purpose of numerical comparison between these two
arrangement options was beyond the scope of this thesis. For the prototype motor the back-
to-back arrangement was selected due to its higher resistance of tilting of the rotor.
To optimise the location of bearings the effects of bearing location was analysed. The
location of the DE bearing was kept constant and distance between DE and NDE bearings
were altered. The effects of bearing distance can be seen in Figure 28.
63
Figure 28. The effects of distance between bearings to UBR of the rotor, DE stationary
From the Figure 28 it can be seen that decreasing the distance between bearings seems to
increase the UBR slightly, most notable effect occurs in distance below 70 mm. Similar
analysis was done when NDE bearing was kept in its original location and DE Bearing was
moved closer to NDE bearing. Results of the analysis can be found in Figure 29.
Figure 29. The effects of distance between bearings to UBR of the rotor, NDE stationary
0
0,0005
0,001
0,0015
0,002
0,0025
0,003
0,0035
d=142 d=136 d=130 d=124 d=118 d=112 d=106 d=100 d=94 d=88 d=82 d=76 d=70 d=64 d=58
mm
Bearing distance effects on UBR of eMAD rotor
11500 rpm 10000 rpm 8000 rpm 6000 rpm
0
0,0005
0,001
0,0015
0,002
0,0025
0,003
0,0035
0,004
0,0045
0,005
d=132 d=114,5 d=108,5 d=102,5 d=96,5 d=90,5 d=84,5 d=78,5
mm
Bearing distance effects on UBR of eMAD rotor
11500 rpm 10000 rpm 8000 rpm 6000 rpm
64
As can be seen, the response stays relatively constant until a threshold is reached and
response grows radically in higher operating speeds. It is noteworthy that the threshold is
reached much sooner than when NDE bearing is moved.
3.4 Thermal expansion effects
Since eMAD project electric motor does not have a specific design point where it will be
operated for extended periods of a time, the bearing system must be designed so that motor
can operated at any speed within the operating speed range (0-10000 rpm). This means that
a large portion of the motor’s operation will happen in transient state, and thus it is desirable
to minimise the “thermal inertia” of the system and to keep the preload of the bearing as
stable as possible. Also as seen from Figure 21, the rotor dynamic of the motor changes
noticeably when preload changes significantly, so in order to keep the rotor dynamics of the
system within acceptable limit, the preload should stay relatively constant. A constant force
preload method with springs could be one way to achieve this goal.
3.4.1 Effects of thermal expansion of the bearing components
Since calculating a complete thermal model of the bearing system is not within the scope of
this study, assumptions and simplifications had to be used. First of all, it is assumed that
bearing housing does not restrict the expansion of the bearing outer race. Secondly, since the
stator has liquid cooling, the temperature of the outer ring of the bearing is estimated to stay
constant, while the temperature of inner ring and rolling elements are varied. To further
simplify the calculation the conformity ratios of the bearings are assumed to stay constant.
Calculating the temperature of bearing elements during operation is complex task and
requires accurate information of heat generation and heat dissipation to the surrounding
structures. Since such information is not readily available for this application, a sensitivity
analysis was conducted. The stiffness of bearing was studied within reasonable operating
temperature range to check if there are significant effects on the stiffness of the bearings.
The effects of bearing temperature were studied by applying the thermal expansion equation
50 to components of the bearing. First the thermal expansion effects are applied to the
bearing balls. For thermal expansion coefficient of bearing balls, a typical value for silicon
nitride was used: 2.5·1e-6/K. The bearing balls increase in diameter and thus reduce the
65
internal clearance of the bearing. Axial preload of 900 N was used in analysis. In Figure 30
the effects of the thermal expansion of the bearing balls to the stiffness of the bearings are
visualised.
Figure 30. Effects of bearing ball’s temperature to the stiffness of the bearings
As stated above, it is assumed that the outer ring temperature remains constant, but
temperature of the inner ring changes and thermal expansion effects should be studied. For
thermal expansion coefficient of bearing inner ring, a typical value for carbon steel was used:
10.8·1e-6/K. The thickness of the bearing inner ring was used as a base dimension when
calculating the thermal expansion. The effects of thermal expansion of the bearing balls and
inner ring of the bearing is shown in Figure 31.
66
Figure 31. Effects of bearing temperature to the stiffness of the bearings
Internal clearance reduces as result of the thermal expansion as shown in Figure 32.
Figure 32. Effects of bearing temperature to the internal clearance of the bearings
As can be seen from Figure 30 & Figure 31, the stiffness of the bearings increases steadily
as the temperature increases. This corresponds to the linear reduction of internal clearance,
shown in Figure 32, as the temperature is increased.
67
3.4.2 Compensating for axial thermal expansion
The bearing manufacturer SKF recommends 900 N of preload for CE-design bearings with
bore diameter between 80-95 mm, when the bearings are preloaded with constant force
(SKF). A series of 24 coil springs are used to provide desired preload to NDE bearing with
assembly as shown in Figure 12. The springs must be selected so that the preload force
changes are minimal when thermal expansion occurs. This means that springs with low
spring rate are favoured, but physical limitations of assembly and ease of installation must
be kept in mind.
Springs are available in multiple lengths and spring preload values and calculations are
needed to determine the proper spring arrangement. Due to space limitations and to make
bearing system assembly easier, following limitations were applied:
• Approx. 8 mm outer diameter coil springs were selected
• compressed length of the springs in assembly is 28 mm
• free length of the springs should not be more than 10 mm longer than compressed
springs
With limitations described above, springs shown in Table 9 were selected.
Table 9. Properties of preload springs
Springs
code C03000421380M C03000451500M
Free length [mm] 35,05 38,1
Spring constant [N/mm] 3,8 4,64
When 12 pieces of each spring are used, the total preload force can be calculated by applying
Hooke’s law as follows
F = 12 ∙ (35,05 − 28) ∙ 3,8 + 12 ∙ (38,1 − 28) ∙ 4,64 = 883,8 𝑁
68
If the compressed length of the spring is changed by 0,2 mm, due to e.g. thermal expansion
the preload can be calculated
F−0,2 = 12 ∙ (35,05 − 28,2) ∙ 3,8 + 12 ∙ (38,1 − 28,2) ∙ 4,64
= 863,6 𝑁
With this combination of springs, the change in preload is only 20 N, which is neglectable,
thus the preload force remains nearly constant.
By replacing some, or all of the springs, the total spring rate of the preloading assembly can
be altered, resulting in different preload force. This feature gives possibility for adjustment
of preload with inexpensive coil springs. Same effect can be achieved by adding spacers in
the preload assembly to decrease the value of compressed spring length.
69
4 DISCUSSION
Due to scope limitations of the study, not all factors affecting the bearing system were
covered. One of these factors is the misalignment of the bearings. Misalignment causes the
internal clearances of the bearing to vary throughout the perimeter of the bearing. As
discussed in chapter 2.4.1 the internal clearances affect the stiffness of the bearing and thus
misalignment in the bearing could alter the stiffness of the bearings. Other manufacturing
inaccuracies, such as surface waviness (covered by Neisi, et al. (2019)), can also affect the
operation of the bearing system. Since the effects of these manufacturing inaccuracies were
not studied, related tolerances for manufacturing were specified according to bearing
manufacturer’s instructions. It would be beneficial to study effects of manufacturing
tolerances, if it would lead to looser tolerances and thus reduction in manufacturing costs of
the components. Lowered manufacturing costs would yield considerable benefits in series
production, but since the project is in prototype stage, it was not justifiable use of engineering
resources.
Simplifications were made when studying the rotor dynamics of the system. Simplifications
can be justified, because the goal was to study the effects of the bearing stiffness to the rotor
dynamics. Motor was simplified so that vibrations of only the rotor were studied, when the
deformations in the stator and other support structures were ignored. Dynamical analysis of
the stator and support structures would be important to conduct, since due to outer rotor
construction of the motor, the stator is supported only from the NDE side of the motor. This
makes the support structures of the rotor essentially like overhung beam, which gives the
rotor less rigid support than could be achieved with traditional inner rotor construction. In
complex structure such as outer rotor motor, a complete understanding of dynamic behaviour
of the system cannot be achieved until the dynamics of the support structure are studied.
As seen in Figure 21, the UBR of the rotor are quite low and notable increase in response
can be achieved only with relatively low preload values, such as 400 N. The increase in
response correlates with anisotropy in bearing stiffness as seen in Figure 17. This means that
the system is not particularly sensitive to preload value, when the preload is above the
threshold of 750 N. It seems that geometry of the rotor and especially the rigidity of the
70
flange are more critical to the response of deflection mode 3 (shown in Figure 19), as long
as the stiffness of the bearing remains above the threshold.
Similar patterns can be noticed from analysis of distance between the bearings. There are no
evident effects until distance between bearings goes below certain threshold. It can also be
noted that NDE bearing can be moved much closer to DE bearings original location
compared to moving DE bearing away from its original position. Seems that location of DE
bearing is more critical, which is understandable, since majority of the loads are carried by
the DE bearing. Since effects of preload force and bearing location are minimal, they do not
have to be strictly tolerated and focus in engineering work can be shifted to more impactful
factors, such as rigidity of the support structure.
Unfortunately, with reasonable effort, it was not possible to study the effects of face-to-face
or back-to-back arrangements to the rotor dynamics of the system. The results found in the
effects of distance between the bearings would suggest that bearing arrangement does not
have huge impact on the rotor dynamics.
Thermal analysis of the bearing yielded only rough results, because the thermal model of the
bearing was very crude. The temperature distribution within the bearing and thermal
expansion was overly simplified to make analysis less time consuming. In order to better
understand the temperature’s effect on the stiffness of the bearing, a FEM model of the
bearing system, as described in chapter 2.4.3, should be used. Then the complex geometry
of the bearing, heat generation and heat conduction could be better accounted for and the
analysis would give more accurate results. Providing that bearing systems reaction to thermal
expansion is not well known, using constant force preloading is justified. The constant force
preloading method ensures that the preload, and thus bearing stiffness, stays relatively
constant, even though there are notable thermal expansion in axial direction. The preloading
assembly with multiple individual springs also gives the ability to adjust the preload easily,
if found necessary.
71
5 CONCLUSIONS
This study was conducted to find tools for bearing selection, find optimal bearing
arrangement and find out effects of thermal expansion to the bearing system in outer rotor
permanent magnet motor. Bearings are vital component of any rotating equipment and that
is why understanding bearing systems and making correct engineering choices is important.
The bearing selection for eMAD motor was conducted systematically considering the
operating conditions and required lifetime as related literature suggests. Lifetime
calculations were conducted to ensure that the bearing will survive the required lifetime of
the motor.
As shown in results, the stiffnesses of bearings were calculated successfully using the
RoBeDyn-toolbox and the values were verified with stiffness values provided by the bearing
manufacturer. Effects of preload and bearing temperature were analysed and operating
points, where the bearing stiffnesses are acceptable, were found.
The rotor dynamics of the rotor were studied and found that dynamic responses of the rotor
can be considered safe within the operating speed range. It was found that above certain
threshold the distance between bearings does not increase the dynamic response
significantly. Thus, the system is not particularly sensitive to changes in bearing
arrangement.
Due to limitations in study scope a full understanding of bearing system was not achieved.
Thermal modelling should have been done more accurately to understand the effects of
bearing temperature and dynamics of the supporting structures needs to be studied further in
order to ensure safe operation of the whole machine.
72
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Appendix I, 1
BASIC INFORMATION FOR APPLICATION
Record: 6414/31/2019
Name of organisation: Lappeenrannan-Lahden teknillinen yliopisto LUT
Organisation's business reg. no.: 0245904-2
Funding service: New knowledge and business from research ideas
PERSONS
Responsible person
Juha Pyrhönen
Email: [email protected], Telephone number: +358405711645
Contact person
Ilya Petrov
Email: [email protected], Telephone number: +358465517885
Other persons
Riitta Nylund
Email: [email protected], Telephone number: +35850 322 3267
Samuli Nikkanen
Email: [email protected], Telephone number: +358405009677
Application reference
Programme
Public description in English
This description is required for public research projects, corporate projects belonging to
Business Finland programmes and projects for which funding has been applied from the
European Regional Development Fund (ERDF). Carbon dioxide is considered as one of the
most significant environmental emissions. The industrial sector has already taken various
measures to limit the level of CO2 emissions. Transport is the main target of CO2 emission
limits. Due to new transport CO2 emission regulations and tax treatment, hybrid (HEV) and
Electric (EV) vehicles have become very popular in Europe in recent years. As demand
grows, the supply of EV and HEV cars from all major international car manufacturers is
increasing and growing all the time. This has increased competition in the electronic vehicle
market. However, there is one area in this market where we can talk about specialized
product segments, namely sports, super and rally cars. The market for the maximum
Appendix I, 2
performance of the special cars is small, but the price of vehicles compared to the standard
cars is remarkably high. This market segment is also quite young and there is no clear market
leader(s) in the market.
The popularity of electric vehicles and the inevitable disappearance of the internal
combustion engine (ICE), superseded by the electrical power lines still requires a lot of work
and numerous challenges are ahead. e-MAD (Electric Motor Active Drive) responds to at
least one significant challenge in this transformation; the new electric motor which is capable
to operate in active driving style without losing its prowess even in continuously repetitive
acceleration and braking (during which the vehicle battery is charged). Unlike contemporary
e-motors, which can only operate at peak power for short periods, the e-MAD can operate
at peak power almost without time restrictions. It offers significant advantages over
conventional solutions, especially when a vehicle requires a high-performance specification
within the limited space.
PROJECT GOALS
What are the project's goals and expected concrete results?
The goal of the project is to commercialize the product that compiles set of technologies
developed in LUT University. The product is an electrical motor for very demanding
applications (e.g. electrical super cars, hyper cars, and racing cars) which require top level
performance components especially for the power elements. The development phase of the
technologies (applied in the proposed motor) have a deep backgrounds from different
research projects at LUT (e.g. development of high-speed technology) which were recently
(2017) strengthened by Academy of Finland 3-years project to develop and finalize the
motor package that accumulates all the technological achievements (with especial highlights
to the cooling approach) and to create the most competitive high-power density motor.
The market of super cars is the high cost and luxury market and the companies in this market
are always seeking for high-end solutions for their cars. This makes them very technology
oriented companies which are trying to utilize the latest and the most promising
technological achievements. Therefore, our main goal is to prove the high level competence
of our solution and offer our motor to these companies. As a result of the project, we are
expecting to start a business in one of possible ways discovered during the project (in market
analysis and potential customer negotiations phases). Along with that we would like
strengthen our position as a mature team with a strong competence to be recognized in the
Appendix I, 2
field of demanding transportation sector (super cars, heavy machinery). This would help us
to be a noticeable players with highly competitive solution offered in the aimed niche.