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
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
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
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]
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
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
𝑿̇ First derivative of displacement vector 𝑿̈ 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 [°]
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 𝜀𝑡ℎ Thermal strain
𝜁 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
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
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
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.
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.
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.
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)
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.
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
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,
(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.
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):
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.
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)
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).
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)
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.
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)
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
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)
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 𝑟𝑏𝑥𝑖𝑛 is the radius of inner raceway.
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
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 = 𝑟𝑎𝑦𝑖𝑛 =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 = 𝑟𝑎𝑦𝑜𝑢𝑡 = 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
In turn the ellipticity parameter is defined as shown in equation 24 (Sopanen & Mikkola, 2003, p. 7)
𝑘𝑒 = [2𝜉 − 𝜁(1 + 𝑅𝑑) 𝜁(1 − 𝑅𝑑) ]
1
2 (24)
Where elliptical integrals 𝜉 & 𝜁 are as follows
𝜉 = ∫ [1 − (1 − 1
𝑘𝑒2) sin2𝜑]
−1 2 𝜋/2
0
𝑑𝜑 (25)
𝜉 = ∫ [1 − (1 − 1
𝑘𝑒2) sin2𝜑]
−1 2 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)
𝑘̅𝑒 = 1,0339 (𝑅𝑦 𝑅𝑥)
0,6360
(27) 𝜉̅ = 1,0003 + 0,5968𝑅𝑥
𝑅𝑦 (28)
𝜁̅ = 1,5277 + 0,6023𝑅𝑦
𝑅𝑥 (29)
𝐸′= 2
1 − 𝑣𝑎2
𝐸𝑎 +1 − 𝑣𝑏2 𝐸𝑏
(30)
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 = 𝜋𝑘̅𝑒𝐸′√ 𝑅𝜉̅
4,5𝜁̅3 (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)
2
3 + ( 1 𝐾𝑐𝑜𝑢𝑡)
2 3)
3
2 (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)
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)
The contact angle of a single bearing ball, 𝜙𝑗, is expressed as shown in equation 35
𝜙𝑗 = tan−1( 𝑒𝑗𝑡
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+ 𝑒𝑗𝑟− 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(𝛿𝑗𝑡𝑜𝑡)
3
2 (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
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))−1𝑸(𝑛) (44)
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
𝑲(𝑛)𝑇 =𝜕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.
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.
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.
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.
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.
𝜀𝑡ℎ = 𝛼(𝑇)(𝑇 − 𝑇𝑟𝑒𝑓) (50)
Where,
𝜀𝑡ℎ is the thermal strain,
𝛼 is the thermal expansion coefficient 𝑇𝑟𝑒𝑓 is the ambient temperature 𝑇 is temperature of the component
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)
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).
𝐌𝑿̈ + 𝑫𝑿̇ + 𝑲𝑿 = 𝑭(𝒕) (51)
Where
𝐌 is the mass matrix
𝑿̈ is the second derivative of displacement vector 𝑫 is the damping matrix
𝑿̇ 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
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)
𝐌𝑿̈ + (𝑫 + 𝝎𝑮)𝑿̇ + 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
𝑿̈ + 𝐌−𝟏(𝑫 + 𝝎𝑮)𝑿̇ + 𝐌−𝟏𝑲𝑿 = 𝟎 (53)
Which can be written in state-space form, as seen in equation 54
{ 𝒚𝟏= 𝑿̇ = 𝒚𝟐
𝒚𝟐 = 𝑿̈ = −𝐌−𝟏(𝑫 + 𝝎𝑮)𝑿̇ − 𝐌−𝟏𝑲𝑿 (54)
Given that states are 𝑦1 = 𝑋̇ and 𝑦2 = 𝑋̈
𝒚̇ = [ 𝟎 𝑰
−𝐌−𝟏𝑲 −𝐌−𝟏(𝑫 + 𝝎𝑮)] [𝒚𝟏
𝒚𝟐] = 𝑨𝒚 (53)
Where
𝐴 is the state matrix 𝑦 is the state vector
