Doctoral Thesis in Machine Design
A Comparative Investigation of Gear Performance Between Wrought and Sintered Powder Metallurgical Steel
Utilizing In-situ Surface Profi le Measurements to
Investigate the Initiation and Evolution of Micropitting and Pitting Damage
EDWIN BERGSTEDT
ISBN 978-91-7873-821-2 TRITA-ITM-AVL 2021:13
KTH ROYAL INSTITUTE OF TECHNOLOGY
EDWIN BERGSTEDT A Comparative Investigation of Gear Performance Between Wrought and Sintered Powder Metallurgical SteelKTH
A Comparative Investigation of Gear Performance Between Wrought and Sintered Powder Metallurgical Steel
Utilizing In-situ Surface Profile Measurements to
Investigate the Initiation and Evolution of Micropitting and Pitting Damage
EDWIN BERGSTEDT
Doctoral Thesis in Machine Design KTH Royal Institute of Technology Stockholm, Sweden 2021
Academic Dissertation which, with due permission of the KTH Royal Institute of Technology, is submitted for public defence for the Degree of Doctor of Engineering on Friday, June 4th, online via Zoom, 2021 at 10:00 AM.
© Edwin Bergstedt ISBN 978-91-7873-821-2 TRITA-ITM-AVL 2021:13
Printed by: Universitetsservice US-AB, Sweden 2021
Abstract
Vehicle electrification is a strong trend that introduces new challenges, such as increased input speed of the transmission and increased power density. Also the noise emittance of the gearbox is of increasing importance, as the sound of the gearbox is no longer masked by the internal combustion engine. Pressed and sintered powder metallurgical steel could be an interesting alternative to wrought steel; the internal porosity has a dampening effect on the noise, and gears can be made in a fast and efficient process. However, current manufactur- ing of powder metallurgical steel has significant performance limitations. The Nanotechnology Enhanced Sintered Steel Processing project aims to reduce the gap in performance between conventional steel and powder metallurgical steel.
One of the potential benefits is that with the inclusion of nano-powder the density can be increased. To validate the new material, its performance needs to be compared to the performance of current generation powder metallurgical materials and also to wrought steel. It is therefor crucial to be able to test and evaluate different materials and gears. This thesis has developed methods for testing, comparing, and evaluating the performance of gears. Powder metal- lurgical steel has been tested and compared to wrought steel; the efficiency as well as pitting life have been investigated in an FZG test rig. Also the effects of different surface finishing operations have been evaluated. The gear flanks were measured in-situ in the gearbox using a stylus instrument; an optimisation routine was created to fit the measurements to the theoretical involute profile.
This enabled an in-depth analysis of surface wear and presented an opportunity to investigate micropitting initiation. It was found that the damage mecha- nisms of wrought steel and powder metallurgical steel are similar and related to the surface finishing method. However, the powder metallurgical steel was also susceptible to sub-surface cracks. Superfinished gears can be negatively influenced by the lack of tip relief as cracks initiate in the surface layer of the root, rapidly destroying the tooth.
Keywords
Gear testing, Micropitting, Pitting, Efficiency, Surface transformation
Sammanfattning
Den p˚ag˚aende elektrifieringen st¨aller nya krav p˚a transmissioner och kugghjul.
F¨or att minska f¨orluster b¨or elmotorn anv¨andas vid h¨oga varvtal, dessutom
¨ar ljudniv˚an allt mer viktig d˚a f¨orbr¨anningsmotorns ljud inte l¨angre d¨oljer det vinande ljudet fr˚an transmissionen. Pressade och sintrade komponenter av pulvermetall ¨ar ett intressant alternativ till konventionellt st˚al, d˚a processen ¨ar snabb och effektiv, dessutom d¨ampar porerna inne i materialet ljud d˚a ljudv˚agor inte kan propagera lika fritt genom gas som genom solidt st˚al. Dagens pulver- metallurgiskamaterial har dock vissa begr¨ansningar, s˚a som l¨agre styrka. SSF projektet Nanotechnology Enhanced Sintered Steel Processing jobbar mot att f¨orb¨attra dagens pulvermetall material. Genom att blanda in nano-partiklar s˚a kan densiteten ¨okas och d¨armed f¨orb¨attras materialets egenskaper.
F¨or att kunna utv¨ardera nya kugghjul och materialkombinationer s˚a beh¨over prestandan kartl¨aggas f¨or dagens material. Det ¨ar d¨armed viktigt att hitta en metod f¨or att kunna testa och g¨ora relevanta j¨amf¨orelser.
Denna avhandling presenterar metoder f¨or att testa samt utv¨ardera pre- standan f¨or olika material och d¨armed generera underlag f¨or att kunna j¨amf¨ora de olika materialen. Genom att genomf¨ora effektivitets samt pittingprov i en FZG testrig, har prestandan f¨or dagens pulvermetallmaterial kunnat j¨amf¨oras mot konventionellt st˚al, ut¨over materialskillnader har ett antal olika slutbear- betningsmetorder har ocks˚a utv¨arderats. Kuggflankerna har m¨atts p˚a plats i v¨axell˚adan fortl¨opande under testningen med ett sl¨apn˚alsinstrument, en metod f¨or att optimera positionen av de m¨atta profilerna mot den teoretiska kuggpro- filen har ocks˚a utvecklats. Genom denna metod ¨ar det m¨ojligt att direkt j¨amf¨ora olika m¨atningar f¨or att se hur slitage p˚averkar profilen. D¨armed kan man stud- era hur mikropitting initieras och ¨aven f¨orst˚a hur skademekanismerna p˚averkas av material och slutbearbetningsmetod. Vid samma slutbearbetningsmetod s˚a uppvisade pulvermetallmaterialen liknande ytinitierade skademekanismer som konventionellt st˚al. En skillnad ¨ar att pulvermetallmaterialet ¨aven uppvisade skador som initierats inuti materialet. Kugghjul med superfinerad yta uppvisade tidigt omfattande skador i pittingtesten. Detta ¨ar kopplat till avsaknaden av toppavl¨attning (en parameter som modifierar kuggprofilens utseende) p˚a kugg- profilen, kraftiga slag ger sprickbildning i roten och n¨ar tillr¨acklig m¨angd sprickor ansamlats s˚a b¨orjar kuggflanken flagna, d¨arefter propagerar skadan snabbt mot toppen av tanden.
Nyckelord
Kugghjulstestning, Micropitting, Pitting, Effektivitetsm¨atning, Yttransformationer
Preface
The work conducted that is the foundation to this thesis was carried out at KTH Royal Institute of Technology in Stockholm, at the Department of Machine Design between January 2017 and December of 2020.
I am grateful for the opportunity given to me to pursue a doctor´s de- gree; without the funding from Swedish Foundation for Strategic Research this project would not have been possible. I would also like to thank the persons that have supported and guided me through out the endeavour leading to my disputation, especially my main supervisor Ulf Olofsson, and my co-supervisors:
Per Lindholm, Ellen Bergseth, and ˚Asa Kassman Rudolphi. I am also grateful for the support from H¨ogan¨as AB, and Michael Andersson.
I would like to give special appreciation to my co-author Jiachun Lin of Beijing University of Technology; during your time as a guest researcher in Sweden we had a really good collaboration. And I am glad that we could maintain our collaboration even though you went home to China.
There are also persons working at the Department of Machine Design that are deeply appreciated; Peter Carlsson and Thomas ¨Ostberg was always there for me to make my life easier.
Many thanks are also directed to Minghui Tu and Yezhe Lyu and my other co-workers at Machine Design; you made the experience really memorable and fun.Finally, I would like to thank my family and friends. With a special thank you to my beloved wife Linn Bergstedt for her love and support. Before starting to work towards a PhD we had no children; now we have two wonderful kids, Nils and Signe, who fill our lives with joy every day.
As I look back to the code I first wrote when I started my PhD, I often find myself reflecting on this quote:
When I wrote this code, Only God and I knew what i did, Now only God does.
- Unknown
Tullinge, March 2021
Edwin Bergstedt
/
J
List of appended papers Paper A
Bergstedt E., Holmberg A., Lindholm P., and Olofsson U. ”Influence of the Din 3962 Quality Class on the Efficiency in Honed Powder Metal and Wrought Steel Gears”. Tribology Transactions. Accepted 13th of July, 2020
Paper B
Lin J., Bergstedt E., Lindholm P., and Olofsson U. ”In Situ Measurement of Gear Tooth Profile During FZG Gear Micropitting Test”. IOP Publishing Sur- face Topology: Metrology and Properties. Accepted 11th of February, 2019
Paper C
Bergstedt E., Lin J., and Olofsson U. ”Influence of Gear Surface Roughness on the Pitting and Micropitting Life”. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science. Accepted 9th of May, 2020
Paper D
Lin J., Teng C., Bergstedt E., Li H., Shi Z, and Olofsson U. ”A Quantitative Dis- tributed Wear Measurement Method for Spur Gears During FZG Micropitting Test”, Tribology International. Accepted 26th of December, 2020
Paper E
Bergstedt E., Lin J., Andersson M., Bergseth E., and Olofsson U. ”Gear Micro- pitting Initiation of Ground and Superfinished Gears: Wrought versus Pressed and Sintered Steel”, Tribology International. Accepted 19th of April, 2021
Division of work between authors Paper A
CRediT authorship contribution statement
Edwin Bergstedt: Data curation, Investigation, Formal analysis, Visualisa- tion, Writing - original draft. Anders Holmberg: Resources, Writing - review
& editing. Per Lindholm: Supervision, Writing - review & editing. Ulf Olof- sson: Conceptualisation, Supervision, Project administration, Funding acquisi- tion, Writing - review & editing.
Paper B
CRediT authorship contribution statement
Jiachun Lin: Conceptualisation, Methodology, Visualisation, Writing - original draft, Funding acquisition. Edwin Bergstedt: Data curation, Writing - review
& editing, Investigation. Per Lindholm Supervision, Writing - review & edit- ing. Ulf Olofsson: Supervision, Project administration, Funding acquisition, Writing - review & editing.
Paper C
CRediT authorship contribution statement
Edwin Bergstedt: Conceptualization, Data curation, Investigation, Formal analysis, Visualisation, Writing - original draft. Jiachun Lin: Conceptualisa- tion, Methodology, Visualisation, Writing - original draft, Funding acquisition.
Ulf Olofsson: Supervision, Project administration, Funding acquisition, Writ- ing - review & editing.
Paper D
CRediT authorship contribution statement
Jiachun Lin: Conceptualisation, Methodology, Visualisation, Writing - original draft, Funding acquisition. Chen Teng: Methodology, Software, Writing - review & editing. Edwin Bergstedt: Data curation, Writing - review & editing, Investigation. Hanxiao Li: Formal analysis, Visualisation, Writing - review &
editing. Zhaoyao Shi: Funding acquisition, Writing - review & editing. Ulf Olofsson: Supervision, Project administration, Funding acquisition, Writing - review & editing.
Paper E
CRediT authorship contribution statement
Edwin Bergstedt: Conceptualisation, Data curation, Investigation, Formal analysis, Visualisation, Writing - original draft. Jiachun Lin: Methodology, Software, Funding acquisition, Writing - review & editing. Michael Andersson:
Resources, Writing - review & editing. Ellen Bergseth: Supervision, Writing - review & editing. Ulf Olofsson: Conceptualisation, Supervision, Project administration, Funding acquisition, Writing - review & editing
Contents
1 Introduction 1
1.1 Swedish Foundation for Strategic Research - SSF . . . 2
1.2 Sustainability . . . 3
1.3 Thesis outline . . . 3
1.4 Thesis objective . . . 4
1.5 Research questions . . . 4
2 Gear manufacturing and surface failures 5 2.1 Gear manufacturing . . . 5
2.1.1 Wrought steel gears . . . 5
2.1.2 Pressed and sintered powder metal steel gears . . . 8
2.2 Gear surface finishing . . . 9
2.2.1 Grinding . . . 9
2.2.2 Honing . . . 9
2.2.3 Lapping . . . 9
2.2.4 Shaving . . . 10
2.2.5 Roll finishing . . . 10
2.2.6 Superfinishing . . . 10
2.2.7 Shot peening . . . 10
2.3 Gear terminology . . . 11
2.4 Gear profile evaluation methods . . . 11
2.5 Gearbox efficiency . . . 12
2.6 Gear surface failures . . . 13
2.6.1 Micropitting . . . 13
2.6.2 Pitting . . . 14
3 Gear performance evaluation methodology 15 3.1 Test equipment . . . 15
3.1.1 FZG Test rig . . . 15
3.1.2 In-situ tooth profile measurements . . . 16
3.2 Gear specimen . . . 18
CONTENTS
3.2.1 Materials and surface finish . . . 18
3.3 Test procedures . . . 19
3.3.1 Efficiency test . . . 19
3.3.2 Pitting test . . . 19
3.4 Calculations . . . 21
3.4.1 Gear efficiency calculation . . . 21
3.4.2 Profile measurement optimisation and fitting . . . 23
3.4.3 Film thickness calculation . . . 30
4 Summary of appended papers 31 5 Discussion 35 5.1 Research questions . . . 35
5.2 Other aspects of the thesis results . . . 39
6 Conclusions 41
7 Future Work 43
Nomenclature
Abbreviations
EV Electric Vehicle
F ZG Forschungsstelle f¨ur Zahnr¨ader und Getreibebau GR Ground (Surface)
HIP Hot Isostatic Pressing HO Honed (Surface)
ICE Internal Combustion Engine LS Load Stage
P AO Polyalphaolefin (Lubricant)
P M Powder Metal (Pressed and Sintered) SF Superfinished (Surface)
SSF Swedish Foundation for Strategic Research W Wrought (Steel)
Efficiency Parameters
ηGear−M esh Gear mesh efficiency [-]
ηT otal Total efficiency [-]
ω1,2 In-going angular speed of the pinion/ gear [m/s]
n Rotations per minute [rpm]
T1 The applied load in the inner power loop [Nm]
TBearings Torque loss of the bearings [Nm]
NOMENCLATURE
TGear−M esh Torque loss of the gear mesh [Nm]
TLoad−Dependent Load dependent torque loss [Nm]
TLoad−Independent Load independent torque loss [Nm]
TST A1,2 KTH model load-dependent torque loss [Nm]
TT otal Total loss torque [Nm]
u Gear ratio [-]
Film thickness Parameters
ρnY The normal radius of relative curvature at point Y GM The material parameter
hY The local lubricant film thickness KA The application factor
KV The dynamic factor
pH,Y,A The local nominal Hertzian contact stress,calculated with a 3D load distribution program
Ra The effective arithmetic mean roughness value SGF,Y The local sliding parameter
UY The local velocity parameter WY The local load parameter Gear Parameters
α Pressure angle [°]
β Helix angle [°]
a Centre distance [mm]
b Face width [mm]
da1,2 Tip diameter [mm]
dw1,2 Working pitch diameter [mm]
m Module [-]
NOMENCLATURE
x1,2 Profile shift factor [-]
z1,2 Number of teeth [-]
Measurement Parameters λS Cut off length [mm]
σ20I,II Cost function performance index N The normal to point P
P Any point on the involute profile
P0 Start of the involute profile, on the base circle rB Base circle [mm]
rm The measured tooth profile, coordinate vector
rϑ Positional vector that describes the location P using an angle ϑ [mm]
ϑa The roll angle where the tip break starts [°]
ϑF The roll angle at the start of the involute [°]
a Fitting parameters ag Form fitting parameter ap Position fitting parameter ar Rotational fitting parameter
B The point where the normal N intersects the base circle
dmin The minimum distance between the measured profile and the optimised theoretical profile
I The identity matrix PTP The weighting matrix R Rotational matrix
Wi Cumulative wear, the difference compared to the initial profile wi Stage wear, the difference compared to the previous profile XY Z Local coordinate system
NOMENCLATURE
xyz Global machine coordinate system
r0 The optimal position of the theoretical involute after fitting to the measured profile rm
X0 The optimal minimum position points Subscripts
1 Pinion
2 Gear
Chapter 1
Introduction
The invention of gears has enabled much of the technology we know today. The first gear-like mechanism consisted of a crude system of interlinking wooden pins. In its time it was truly revolutionary, suddenly there was a way to transfer power and change the direction of power. Also, by changing the gear ratio, the speed of the input and output shafts can be adjusted to better suit the application. Gears provide a means to harness the energy from, for example, a water wheel. The energy can be transferred and manipulated, enabling the use of heavy equipment e.g. to mill or to hammer wrought steel. Modern gear applications are faced with a completely different set of challenges. Fierce competition and demand for cost savings spurs the interest in alternative gear manufacturing methods. Also the power density of the entire drive train is increasing.
To reach the stipulated environmental goal and minimise the effect of global warming [1], a severe reduction in the volume of emitted greenhouse gases is needed. Therefore, the efficiency and gear mesh losses are increasingly impor- tant as tougher emission legislations are passed. The modern gear has to be produced cheaply, be sufficiently strong and durable for its application. Fur- thermore, the losses and sound emitted should be kept to a minimum. An interesting alternative to the traditionally machined gears are gears made from pressed and sintered powder metal (PM). This PM material can be shaped into near-net shape with significantly less waste material [2], [3] and with signifi- cantly shorter cycle time compared to traditional gear generating methods [4].
Another potential benefit of the PM material is that it can dampen vibration and reduce the emitted noise [5], this is due to the internal porosity preventing the sound waves to propagate freely. The current generation of sintered PM materials can reach a density of roughly 7.3 g/cm3 after compaction and sin- tering. The maximum density that is achievable is dependent on many factors, such as the size and shape distribution of the metal powder, and the proper- ties of the additives. The main issue is the exponential increase in compaction
CHAPTER 1. INTRODUCTION
pressure needed to compress the powder particles before sintering [6]. The density also effects mechanical properties such as the Young’s modulus, tensile strength, and hardness [7], [8].
Today, sintered PM gears are mainly used in low loaded applications as the gears are weaker than the wrought steel counterpart. If the strength of the PM gears can be increased, the PM gears would be an interesting option to consider as there are several benefits in choosing a PM material for gears. The man- ufacturing process is fast and efficient, with hardly any wasted metal powder.
Another benefit of using PM technology in manufacturing gears is the possi- bility for creating complex shaped gears as the limitations of the conventional gear generation methods do not apply [9]. Applications could be optimised root geometry for decreasing the local load concentrations and also creating gears with complex shapes, e.g. holes for weight/ material reduction. However, the making of complex-shaped gears requires a specialised tool which is more ex- pensive than for the standard gear. This can be compensated by a large volume and the materials saved per gear manufactured.
1.1 Swedish Foundation for Strategic Research - SSF
The Swedish Foundation for Strategic Research (SSF) is a foundation that grants funding to research projects in science, engineering, and medicine. The goal is to ensure that Sweden can maintain its strong position in research and innovation and remain competitive in the future.
This PhD thesis is part of the ”Nanotechnology Enhanced Sintered Steel Processing” project funded by SSF, Grant No. GMT14-0045. The project is a collaboration between Chalmers University of Technology, Lund Univer- sity, KTH Royal Institute of Technology, and Uppsala University. H¨ogan¨as is involved as a industrial partner and supports the project with resources and technical knowledge. Chalmers University is responsible for manufacturing the nano powder, creating material samples and evaluating the materials on a lab scale level. KTH and Uppsala are responsible for evaluating the materials´ per- formance tribologically, as well as conducting metallographic analyses. Lund University is responsible for analysing the project´s potential from a cost and sustainability perspective in comparison to traditional gear manufacturing.
The ”Nanotechnology Enhanced Sintered Steel Processing” project is de- voted to exploring the possibilities with mixing in ultra-fine nano-sized powder particles into the regular powder mix used for sinter steel. There are several potential benefits with a nano enhanced material. The density can be increased
1.2. SUSTAINABILITY
as the nano powder can fill voids between normal powder particles. Even a small increase in density could potentially be of great importance as closed porosity, is then achieved i.e. the pathways in between pores are closed. With closed porosity the material can be run through a hot isostatic pressing (HIP) process without the need to first be capsuled in a sealed enclosure [10]; thus a fully dense material can be made at relatively low cost. Another potential benefit of the nano-enhanced material is that the small particles will have a lower melting point, thus initiate the necking process in between the regular particles and increasing the initial diffusion rates.
1.2 Sustainability
The research conducted in the scope of this thesis could potentially increase the sustainability. By finding better materials and surface treatment methods, the gear mesh efficiency can be increased, thus lowering energy consumption.
Both conventional internal combustion engines (ICE) and electric vehicles (EV) benefit from increased efficiency, i.e. lower fuel consumption for the ICE and smaller battery pack size and thus less weight for the EV. Another challenge for the electrification is that in order to increase efficiency of the electric motors, the operating speed needs to be several times higher than the normal operat- ing speed of an ICE engine [11], [12]. This poses new challenges, as higher speed results in far more contacts, thus increasing the surface fatigue damage.
Manufacturing gears from metal powder also has potential to increase sustain- ability, as the process has fewer processing steps and can utilise the material more efficiently, i.e less waste material. The PM process also has another in- teresting property: a gear made with a complex shape and with holes to reduce weight [9] is more sustainable as less powder is used. The main disadvantage to the PM technology is that it requires high volume to compensate for the initially higher tool cost [13], also the strength of the material is lower than for wrought steel. However, the strength and performance can be compensated for and the surface can be densified to obtain a hybrid material with a porous core and a dense surface layer.
1.3 Thesis outline
This Chapter aims to give an introduction to the subjects discussed in this thesis, and the research questions that are to be answered. An overview of the research project of which this doctoral thesis is a part is also presented. The sustainability impact of the work in this thesis can also be seen in this chapter.
Chapter 2 provides a brief overview of gears, such as the gear manufacturing process, both for wrought steel as well as gears made from pressed and sintered
CHAPTER 1. INTRODUCTION
materials. Also, some gear surface finishing techniques, a basic introduction to gear micro geometry, and gear flank damage are presented. Chapter 3 contains the methods used in conducting the research, such as the test procedures, test equipment, and calculation methods. Chapter 4 summarises the appended papers, and in Chapter 5 the research questions are discussed and answered.
Chapter 6 summarises the most important findings for the reader´s convenience.
1.4 Thesis objective
This thesis seeks to increasing knowledge on how to evaluate the performance of both conventional gears as well as sintered and pressed powder metallurgical gears. Research and development of methods for comparing and assessing profile changes during gear testing can contribute to a deeper understanding of how different surface finishing operations affect the pitting life and gear efficiency.
1.5 Research questions
This thesis seeks to explore the subject of gears. The objective is to achieve a deeper understanding and further knowledge in testing and evaluating gear performance. In order to achieve this the a number of research questions were formulated and presented below. The research questions are discussed further in Section 5.1.
• Can the gear mesh efficiency be directly related to the DIN 3962 gear quality class index?
• Does the gear mesh efficiency differ significantly between honed PM steel gears and honed gears made from wrought steel?
• Can micropitting initiation mechanisms be evaluated using surface profile measurements during an FZG pitting test?
• How can the gear surface finishing method affect the surface damage mechanism for wrought steel gears?
• How do the principal surface damage mechanisms compare between wrought steel and PM steel gears?
Chapter 2
Gear manufacturing and surface failures
2.1 Gear manufacturing
This section is meant to give the reader a basic introduction to gears in terms of manufacturing, surface finishing processes, gear measurement, and gear surface failure.
2.1.1 Wrought steel gears
In order to make a gear from a piece of wrought steel, first the teeth are cut from the gear blank. This leaves a rough surface. The next step is to use a finishing process and apply a heat treatment to harden the gear, the order of these steps can be chosen to best suite the products needs. In the finishing process the gear profile is finalised to achieve the desired geometrical shape, surface texture, and surface roughness.
There are several methods for making gears, these methods can be di- vided into two sub categories; generating methods, and forming methods. The main distinction is that the tool used for gear generating can produce gears with various number of teeth while forming method incorporates tools that are specifically made for one specific gear, i.e. a set number of teeth, module, and pressure angle.
Generating methods
In gear manufacturing with a pinion type cutter, the cutter is made to the image of a the mating gear that one wants to generate. The gear blank and tool is then locked in rotation as a pair of mating gears would. The tool is positioned above the work piece and at a distance so that the tool barely touches the gear blank. The tool is then moved down over the gear flank cutting the surface, the tool is backed away from the cut and moved back up to make a new cut.
CHAPTER 2. GEAR MANUFACTURING AND SURFACE FAILURES
The Maag generating method, shown in Figure 2.1, uses a rack cutter, this can be thought of as involute gear of infinite size.
Figure 2.1: Illustration of gear generation using the Maag method with a rack cutter, the cutting rack is positioned above the gear blank and moved down in a cutting stroke. The tool is then moved away from the gear blank and up to the initial position, the gear is rotated a bit for the next cut to be performed
The Fellows method uses a cutting tool that is round, or in contrary to the Maag method has a finite radius. In Figure 2.2 one example of the Fellows generating method can be seen. One benefit compared to the Maag method is that the Fellows method is also suitable for cutting internal gears.
Another common generating method is hobbing as can be seen in Figure 2.3.
The hob tool is at first glance a bit awkward in shape, almost like a rolling pin for making flat bread with small knobs all over. Upon further inspection one can see that there are some important differences. The gear hob is not straight as the rolling pin, it is in fact a single tooth worm gear that has been cut perpendicular to the rolling direction at several positions, this create the cutting edges of the gear hob. The result can be seen as a collection of rack cutters mounted on a cylinder, but with the helical shape of the worm gear.
By rotating the hob in sync with the gear blank and moving the hob over the width of the gear the teeth are generated.
Forming methods
Gear forming is different from gear generating, for gear forming the gear blank is fixed in position and the material in-between two adjacent teeth are milled away
2.1. GEAR MANUFACTURING
Figure 2.2: Illustration of gear forming using the Fellows generating method with a pinion type cutter, the tool and gear blank is rotated together, the pinion cutter is positioned above the gear blank and moves down in a cutting stroke, then returns to the initial position and rotated a bit for the next cut
Figure 2.3: Illustration of gear forming using a hob cutter, the hob and gear blank rotates in sync and the hob is moved down to perform the cut
CHAPTER 2. GEAR MANUFACTURING AND SURFACE FAILURES
in a milling machine. The gear blank is rotated by a distance corresponding to one tooth for the next cut, the process repeats until the gear is completed. It is important to notice that only spur gears can be made using this method.
2.1.2 Pressed and sintered powder metal steel gears
Manufacturing components by pressing and sintering powder metal is a conve- nient and fast mean of production. The process of pressing the metal powder can be seen in Figure 2.4 [10]. The powder metal gears are made by filling a gear shaped cavity with a metal powder mixed with additives [14]. Then by using a set of punches the powder is compacted under high load to a semi-solid component, a green body, where the individual powder particles have bonded mechanically, but are not fused together.
The whole filling and compaction process is quick and only takes a few sec- onds per gear. Afterwards the green body gears are sintered, that is subjecting the gears to specially designed heat cycles. The heat fuses the individual pow- der particles together resulting in a solid material, although with reminiscent porosity. The process shrinks the gear as the density increases. Even tough the compaction process seems simple at first glance, it is still possible to create complex shaped gears such as helical gears.
Die fill stage Compaction Part ejection
Die Powder
Green body part
Upper punch
Lower punch
Figure 2.4: Die pressing of metallic powders
2.2. GEAR SURFACE FINISHING
2.2 Gear surface finishing
The use of finishing operations are crucial to obtain the correct geometrical property and surface finish on the gears. After the machining operations the surface finish and micro geometry is usually not adequate for the needed appli- cation. Furthermore, if the gear have been subjected to a hardening process, the gears will distort to some degree by the heat. The surface finishing op- erations remove the outermost surface layer and ensures the correct shape of the gear profile. There are several available methods for gear surface finishing e.g. grinding, honing, lapping, shaving, and roll finishing. Superfinishing is an additional process that can further enhance the surface finish.
2.2.1 Grinding
There are two main methods of gear grinding, form grinding and generation grinding. The former uses a grinding disc wheel that is dressed to the shape of the involute profile and runs in the space in-between two teeth. The latter is either a single straight edge grinding wheel or multiple grinding wheels, the flanks mimic a toothed rack and the it rolls over the reference circle of the gear. The grinding disc spins and is moved over the surface to grind the teeth to the involute profile shape. The benefit of grinding is that it can satisfy high tolerance requirements, it is also possible to grind hardened gear surfaces.
The downside is that the process generates heat, and that the process is time consuming.
2.2.2 Honing
Honing of gears is a hard grinding process where a honing tool is moved over the gear flank [15]. The honing stone is resin matrix containing abrasive particles, the tool is moulded to a external gear and dressed using a diamond wheel for the specified gear parameters. The gear is rotated against the honing tool, resulting in a surface texture that are almost parallel to the tooth at the tip and root, and perpendicular to the tooth at the pitch.
2.2.3 Lapping
Lapping is a mechanical polishing process where a paste containing abrasive particles are used in between a set of mating gears [16]. The gears are revolved, and quickly reciprocated along the gear face at a controlled pressure. Thus conforming the surfaces to one another. One way is to use a master lapping gear, this ensures that the production gear can conform with high accuracy to the form of the master gear.
CHAPTER 2. GEAR MANUFACTURING AND SURFACE FAILURES
2.2.4 Shaving
Gear shaving can only be used on non hardened gear surfaces, the accuracy is thus limited as distortions can occur during the heat treatment cycle [17]. The shaving process uses a tool shaped like a gear with serrations forming numerous of cutting edges [16]. The tool and gear is positioned with crossed axes, a motor rotates the tool driving the gear which can rotate freely. The centre distance is reduced in small increments until the final form is achieved. The process removes waviness and cutter marks from previous machining. One benefit of shaving is that the process generates low heat in comparison to grinding.
2.2.5 Roll finishing
Gear rolling does not remove any material, it is purely a yield process where the surface is conformed to the shape of the counter surface. The gear is mounted and meshed against a tool, by applying pressure and rotating the gear the metal flows smoothing the surface, also good dimensional control is possible. As no material is removed with the roll finishing process the excess material will flow and form lips at the tip and sides of the gear. The rolling process is specially beneficial for PM components as the rolling compresses the surface and closes pores, reducing the chances of sub-surface fatigue damage.
2.2.6 Superfinishing
Superfinishing is an additional treatment that can be performed to enhance the surface further. It is a type of polishing that can be mechanical, chemical, or a combination of both. The theory is the same regardless, the polishing process removes the surface peaks leaving a mirror-like surface finish. The mechanical process uses a extremely fine grit abrasive, the abrasive is either moved over the surface while rotating or oscillating creating a cross pattern on the surface [18].
The chemical process etches the surface, the peaks will etch more than the base material as the surface area in contrast to the volume is high. One important downside to the superfinishing process is that it is a slow and costly process, often only suitable for high performance applications, i.e. helicopter gears etc..
2.2.7 Shot peening
Shot peening is a method of enhancing the surface properties of a material and can be used on gears. Shoot peening strikes the surface with a high number of small circular objects e.g. glass, metal, or ceramic. The velocity is high enough to cause plastic deformation in the surface layer, which introduces a compressive residual stress. The treatment makes the gears less susceptible for surface damage such as cracks.
2.3. GEAR TERMINOLOGY
2.3 Gear terminology
In Figure 2.5 some of the most important gear terminology can be seen. There are several important regions of the gear tooth, represented by circles originating from the centre of the gear. At the root circle the tooth begins, and the base circle is the start of the involute profile. The pitch circle is the point where the pinion and wheel, in theory, have a pure rolling contact. Finally the addendum circle denounces the end of the involute profile at the tip of the gear tooth.
The addendum and dedendum regions is the name of the involute profile above and below the pitch circle respectively.
Root Circle Pitch Circle Base Circle Addendum
Dedendum
Addendum Circle
Figure 2.5: Illustration of a gear with important gear terminology marked
2.4 Gear profile evaluation methods
The gear surface profile is usually measured in a gear coordinate measurement machine as can be seen in Figure 2.6. The gear is mounted and positioned in the device and indexed according to the gear teeth. A ball probe then measures the position of the surface, the gear surface profile is usually measured in a grid shaped pattern, the number of points to probe can be selected, however a large number of probing points will take a significant amount of time to measure.
CHAPTER 2. GEAR MANUFACTURING AND SURFACE FAILURES
The coordinate measuring machine is suitable for measuring the form of the gear tooth, however it is not suitable for measuring the surface roughness. It is important to note that the coordinate measuring machine is also used for measuring distance between teeth, inner diameter of the gear, as well as other gear parameters.
Figure 2.6: Gear profile measurement using a ball probe
2.5 Gearbox efficiency
Gearbox efficiency is a measure of how much losses a set of gears have in a gearbox. There are several factors contributing to the total losses, and they can be divided in to load dependent and load independent losses [19]. Load- independent losses are losses related to the rotation of the gears, such as oil churning losses and losses from the bearing seals. Load-dependent losses are losses that are influenced of the applied load, such as bearing losses and gear mesh losses. The gear mesh efficiency is important as a slight increase in efficiency could have a large impact of the total energy consumption of the motor.
2.6. GEAR SURFACE FAILURES
2.6 Gear surface failures
As the gears rotate the teeth are constantly subjected to both rolling and sliding along the involute profile. At the pitch the contact is mostly rolling and at the tip and in the root the sliding speed is high. Pitting damage is a contact fatigue damage that can occur due to the rolling and sliding on the gear surface. Pitting damage can be divided into two categories based on the appearance of the damage, micropitting or macropitting. There are also other types of damage that can occur on gears such as scuffing, where the surfaces bond due to e.g.
failure of the lubricant.
2.6.1 Micropitting
Micropitting or gray staining is usually found in high loaded and hardened gears, the damage is caused by the interaction between surface asperities. The appearance of a micropitted surface is dull as the surface is filled with micro- cracks, dispersing and scattering the light, hence the name gray staining [20].
By observing micropitted surfaces in a scanning electron microscope it was concluded that the damage mechanism is the same as for pitting, the scale is only smaller [21]. As the micro-cracks grow in number and size, the surface is undermined with cavities with a size roughly equal to the asperities. Mallipeddi et al. [22] found one type of micropitting initiation. They found plastically deformed regions below asperities down to a depth of 15µm when studying micropitting in an FZG test rig. The plastic deformation forced dislocations to move in slip bands inside the grains of the material. The pileup of dislocations in grain boundaries enabled cracks to nucleate, thus initiating the micropitting damage.
Both the gear micro geometry and surface finish are important to mitigate micropitting, a superfinished surface protects against micropitting, and also the use of tip relief on the gear profile can prevent micropitting from occuring [23].
CHAPTER 2. GEAR MANUFACTURING AND SURFACE FAILURES
2.6.2 Pitting
Macropitting or pitting is damage that occur on or below the pitch in a lu- bricated contact, the repeated contacts and high contact pressure affects both the surface and a region below the surface [21]. The contact initiates cracks that propagate until small pieces of the surface is separated, the shape of the damage can either be pin-holes or spalls. Pin-holes are small circular holes in the surface where the material have been lost, while spalls are a v-shaped dam- age that initiate in a point on the surface [24], the cracks then propagate at an angle in a v-shape and also down into the material, the damage grows below the surface until the critical crack length is achieved and a piece of the surface is removed. The resulting damage is shaped like a clam-shell, which is also a common name for the damage.
Chapter 3
Gear performance evaluation methodology
3.1 Test equipment
3.1.1 FZG Test rig
The FZG back-to-back test rig was designed by the Gear Research Centre (Forschungsstelle f¨ur Zahnr¨ader und Getreibebau) at the Technical University of Munich. The FZG test rig uses a circulating power loop that is loaded me- chanically using lever arms and weights. This makes the test rig efficient, as the electric motor only needs to supply energy to account for the losses in the power loop. The FZG test rig can be used in different configurations, in this work two main setups were used, a setup to measure efficiency and one for conducting pitting tests. For conducting efficiency measurements, the test rig is configured according to Figure 3.1.
The second configuration can be seen in Figure 3.2. The FZG test rig consists of two gearboxes, (1) and (3), containing one pinion and one gear, which are connected with two shafts forming a circulating power loop. One of the shafts is fitted with a load clutch (2) used for applying a pre-load into the power loop. Finally an electric motor (5) drives the power loop. The difference between the efficiency and pitting setups is at positions (3) and (4).
In the efficiency test gearbox (1) and slave gearbox (3) are identical, but for the pitting test, the gears in the slave gearbox (3) are replaced with another gearbox with wider helical gears. This is done to promote pitting only in the test gearbox (1). At position (4), there is a torque sensor for the efficiency test, and for the pitting test setup a speed reducer is fitted. The speed reducer can run either a 1:1 or 25:1 gear ratio.
CHAPTER 3. GEAR PERFORMANCE EVALUATION METHODOLOGY
Figure 3.1: Schematic of the FZG back-to-back test rig in the efficiency measurement configuration. (1) Test gearbox, (2) Load clutch, (3) Slave gearbox, (4) Torque sensor, (5) Motor. Source: The figure was created by Edwin86bergstedt and is not altered. The figure is licensed under the Creative Commons Attribution-Share Alike 4.0 International licence,
https://creativecommons.org/licenses/by-sa/4.0/deed.en
1 2 3 4 5
Figure 3.2: Schematic of the FZG back-to-back test rig in the pitting test configuration. (1) Test gearbox, (2) Load clutch, (3) Slave gearbox, (4) Reduction gearbox, (5) Motor. Source: The figure was created by Edwin86bergstedt and is not altered. The figure is licensed under the Creative Commons Attribution-Share Alike 4.0 International licence,
https://creativecommons.org/licenses/by-sa/4.0/deed.en
3.1.2 In-situ tooth profile measurements
A methodology for measuring gears in-situ in the gearbox was developed at KTH by Sosa et al. [25]. A Taylor Hobson Intra 50 stylus instrument was mounted on a bracket attached to the test gearbox with bolts and guide pins.
Figure 3.3 shows the measurement device mounted on the gearbox, and also the probe position in the root of the gear. The in-situ measurement method has a couple of advantages compared to traditional methods of evaluating wear in gears, i.e. weighing or measuring them in a coordinate gear measuring ma- chine. The gears can be measured without disassembling the test rig. This is convenient for the operator, and it also reduces the risk of influencing the test results. With the bracket mounted on the gearbox, a high positional accuracy can be obtained, which enables repeatable measurements that can accurately
3.1. TEST EQUIPMENT
track profile changes during the course of a pitting test. Three factors mainly affect the quality of the measurement. The gears´ angular measurement posi- tion, the position along the width of the tooth, and the calibrated start position of the stylus instrument. The gears´ angular position is aligned using a spirit level placed on top of the gear. The accuracy of the spirit level was stated as 15 min of arc. The position along the tooth width is controlled by a micrometer screw gauge with an accuracy of ±5 µm. The starting position of the mea- surement can change slightly due to limitations of the measurement device; the shift is usually below 20 measurement points or ±10 µm.
In order to minimise errors due to local variations, three teeth evenly spaced around the gear (teeth number 1, 9, and 17) were measured. At each tooth six parallel traces were measured, starting in the centre of the tooth width and spaced 0.1mm apart. Profile measurements were conducted initially before the pitting test commenced, after running-in, and after the finish of each consec- utive test. In total 18 measurements were recorded for each load tested, and as the tests were repeated two times, a total of 36 measurements are available per tested load stage.
Figure 3.3: The Taylor Hobson stylus instrument mounted on the test gearbox, the position of the probe in the root of the gear is also visible in the figure
CHAPTER 3. GEAR PERFORMANCE EVALUATION METHODOLOGY
3.2 Gear specimen
The gears used in Papers A to E are standard FZG C-Pt spur gears without any profile modifications i.e. tip/ root relief or crowning. Same gear type is used for both the efficiency test procedure (Section 3.3.1) and for the pitting test (Section 3.3.2). The data of the gears can be seen in Table 3.1
Table 3.1: Gear parameters for the tested C-Pt gears
Symbol Unit C-Pt
Centre distance a mm 91.5
Number of teeth Pinion z1 - 16
Gear z2 - 24
Module m mm 4.5
Pressure angle α ° 20
Helix angle β ° 0
Face width b mm 14
Profile shift factor Pinion x1 - 0.1817
Gear x2 - 0.1715
Working pitch diameter Pinion dw1 mm 73.2
Gear dw2 mm 109.8
Tip diameter Pinion da1 mm 82.5
Gear da2 mm 118.4
Material - 16MnCr5
Heat treatment - Case carburized
Surface roughness Ra µm 0.5 ± 0.1
3.2.1 Materials and surface finish
In Papers A to E, several materials and surface finishing operations are utilised.
Two material types, wrought steel and pressed and sintered powder metallurgical steel, were tested. The wrought steel is a common commercial gear steel, 16MnCr5. The two PM steels used, Distaloy™ AQ and Astaloy™ Mo, were supplied by H¨ogan¨as. The chemical composition of the materials tested is presented in Table 3.2.
Three surface finishing methods were tested experimentally: honing, grind- ing, and superfinishing. The superfinishing process was performed as an addi- tional step on the ground surface.
3.3. TEST PROCEDURES Table 3.2: The chemical composition of the wrought steel and powder metal materials
Chemical composition (weight %)
Fe Mn Cr Ni Mo C S P Si
16MnCr5 96.95-98.78 1-1.3 1.1 - - 0.14-0.19 ≤ 0.035 ≤ 0.025 0.4
Distaloy™ AQ 98.8 0.5 - 0.5 - 0.2 - - -
Astaloy™ Mo 98.3 - - - 1.5 0.2 - - -
3.3 Test procedures
3.3.1 Efficiency test
The efficiency measurement test procedure was developed at KTH and has effectively been used in a wide range of research projects, see e.g. [19], [26]–
[31]. The efficiency tests required a new set of gears for each test. In order to change the test gears in both the test and slave gearbox, the test rig was dismounted. The top and side panels of the gearbox were removed. Both the motor and torque sensor were moved to change gears in the slave gearbox. The reassembly was performed following a strict procedure, as Andersson et al. [29]
concluded that a rebuild of the test rig can influence the efficiency results.
The gearboxes were filled with 1.5 L of a Polyalphaolefin (PAO) lubricant, up to the centre of the shaft. The specified nominal viscosity of the PAO lubricant was 64.1 mm2/s (cSt) at 40°C and 11.8 mm2/s(cSt) at 100°C.
The efficiency test starts with a running-in of the gears for four hours, using load stage (LS) 5 corresponding to a pitch line torque of 94.1 Nm, and with a pitch line velocity of 0.5 m/s. The efficiency test starts by running a baseline test without any load applied, this is to isolate the load independent losses. The loss torque is measured at five-minute intervals in order to reach a steady state for the losses. A series of eight speeds were tested: 0.5, 1, 2, 3.2, 8.3, 10, 15, and 20 m/s. The test series is then repeated at three additional load stages, 4, 5, and 7, in order to calculate the load-dependent losses. The pitch line torque for the load stages is shown in Table 3.3. Each efficiency test was repeated three times using new gears in both gearboxes. During the tests, the speed, oil temperature, and loss torque were recorded at a sample rate of 1 Hz. The oil temperature in the gearboxes was kept at a constant 90°C (-1 to +4°C).
3.3.2 Pitting test
In Papers B to E, pitting tests were performed in the FZG test rig. The pitting test procedure used was based upon the DGMK [32] short pitting test pro- cedure. The DGMK test consists of a run-in for 1.3 × 105 contacts at LS 3 corresponding to a pitch line torque of 35.3 Nm, followed by the pitting test
CHAPTER 3. GEAR PERFORMANCE EVALUATION METHODOLOGY
which was run at intervals of 2.1 × 106 contacts. The speed of the pinion was 2250 RP M and the oil temperature was kept constant at 90°C. There were a few alterations made to the procedure to account for more load stages, thus enabling the gathering of surface profile data in a wider range. The DGMK method uses a run-in period, one run at LS 7 (183.4 Nm), and then the test continues at LS 10 (372.7 Nm) until a certain profile deviation is reached.
In the altered procedure, all load stages from LS 3 to LSmax were tested in sequence, where LSmaxis set to LS 9 and LS 10, for the pressed and sintered PM material and wrought steel respectively. When the test reaches the max- imum load level LSmax, the test continues at this level until either a pitting damage greater than 5 mm2 is observed, or run-out is reached at 4.0 × 107 contacts. The load stages and corresponding pitch line torque are presented in Table 3.3. The oil temperature had to be lowered from 90°C to 80°C, as the cooling system of the FZG test rig used had difficulties with maintaining a constant temperature at 90°C. A flowchart overview of the pitting test and measuring procedure can be found in Figure 3.4.
Table 3.3: FZG Load stage and corresponding pitch line torque in Nm
LS 3 4 5 6 7 8 9 10
Torque [Nm] 35.3 60.8 94.1 135.3 183.4 239.3 302.0 372.7
3.4. CALCULATIONS
Start
Running-in 1.3·105 contacts
LS=3
Test 2.1·10⁶ contacts
LS
Measure profile
Pitting?
Abort test
LS = LSmax? LS = LS + 1
Yes Yes
No No
Measure profile
Measure profile
Run-Out?
40·106 contacts at LSmax
No
Yes
Figure 3.4: A flowchart of the pitting test and surface measurement
procedure. LSmax is 9 and 10, for the PM and the wrought steel respectively.
Source: The figure was created by Edwin86bergstedt and is unaltered except the text font.
The figure is licensed under the Creative Commons Attribution-Share Alike 4.0 International licence, https://creativecommons.org/licenses/by-sa/4.0/deed.en
3.4 Calculations
3.4.1 Gear efficiency calculation
The losses for the FZG test rig operating in efficiency mode (Figure 3.1) can be assumed to be equal to the torque supplied by the electric motor to keep the test rig at a constant velocity. The pre-loaded inner loop maintains the power within the loop and the electric motor therefore needs to supply enough torque to overcome the total losses TT otal. The total losses can be divided into load-dependent TLoad−Dependent and load-independent TLoad−Dependent
CHAPTER 3. GEAR PERFORMANCE EVALUATION METHODOLOGY
losses:
TT otal= TLoad−Dependent+ TLoad−Independent (3.1) Load-dependent losses are all losses related to the applied load, i.e. gear mesh losses, and losses in the bearings is given by:
TLoad−Dependent= TBearings+ TGear−M esh (3.2) The load-independent losses are losses that are not affected by the applied load, oil churning losses[33], and losses from the bearing seals is given by:
TLoad−Independent= TOil−Churning+ TBearing−Seal (3.3) To calculate the gear mesh loss torque, Equation 3.2 is substituted into Equation 3.1 giving the following expression:
TGear−M esh = TT otal− TLoad−Independent− TBearings (3.4) The load independent losses can be obtained by performing tests at each speed without any load applied in the power loop. The gearbox efficiency for one gearbox can be calculated using the following expression:
ηT otal= 1 −1
2 ·TT otal
uT1 (3.5)
Where u is the gear ratio and T1 is the nominal torque transferred by the pinion. T1is equal to the load applied to the inner power loop and was assumed to remain constant throughout the experiment. Given the assumption that the gearboxes contribute equally to the losses, the efficiency for one gearbox can be obtained by multiplying the ratio by 12.
There are several models available for calculating the bearing losses. One commonly used method for NJ 406 cylindrical roller bearings used in the FZG test rig was developed by SKF Industries inc. Researchers at KTH have de- veloped another empirical bearing model named STA [34]. The STA bearing model is shown below:
TST A1,2= An +B
n + C (3.6)
Where the parameters A, B, and C (Appendix A) were determined empiri- cally and depend on the load, temperature, lubricant, and bearing type.
The loss torque of the bearings can be calculated using Equation 3.7, where ω1,2 is the in-going angular speed of the pinion and gear shafts.
TBearings= 4 TST A1· ω1+ TST A2· ω2
ω2
(3.7)
3.4. CALCULATIONS
The gear mesh loss can be obtained by using the bearing losses, the mea- sured total loss and the measured load-independent loss into the following ex- pression:
ηGear−M esh= 1 − 1
2· TGear−M esh
uT1 (3.8)
Finally, the gear mesh efficiency can be calculated using Equation 3.8.
3.4.2 Profile measurement optimisation and fitting
The measured gear involute profiles will not be able to fit on top of each other in the as-measured state. As the positioning of the gear is done by a spirit level, the accuracy is not sufficient to ensure the exact same measurement angle; an example of the magnitude of the problem can be seen in Figure 3.5. Also, the starting position of the stylus instrument will vary by some tens of points, corresponding to roughly ± 10µm.
Figure 3.5: A sample of measurements illustrating the effect of the angular position error on the shape and position of the measurements [35]
In order to directly compare the measured profiles, the profiles need to be transformed to a common reference. The theoretical involute profile is suitable in this regard. The theoretical profile was generated using the gear parameters
CHAPTER 3. GEAR PERFORMANCE EVALUATION METHODOLOGY
listed in Table 3.1. An involute profile is the path the end of a straight line follows when the line is rolled over a circle. To generate the involute profile first a coordinate system O (x, y) is created with origin in the centre of the gear.
Figure 3.6 shows the generation of an involute profile, where the start of the involute profile P0 is on the vertical axis and lies on the base circle rb. At any point P on the involute profile, the normal N is tangent to the base circle rB
in point B. The involute radius of curvature in point P is given by the distance PB, which is also equal to the length of the arc segment betweenP_0B.
N
O P
0y
x
Base Circle
T
B Generating line Gear tooth profile
P r
r
bϑ
Figure 3.6: Generation of an involute curve [35]
The position of any point, P, along the involute profile can thus be described using a position vector rϑ. The function for calculating the position vector rϑ [36] is given by:
r (ϑ) = x (θ) i + y (ϑ) j = rb[(sinϑ − ϑcosϑ) i + (cosϑ + ϑsinϑ) j] (3.9) Where i and j are the unit vectors of the x and y axes, and the parameter ϑ varies in the interval [ϑF, ϑa].
The tooth profile was measured using a stylus instrument, initially and after each performed test; the measured tooth profile rm contains the coordinates
3.4. CALCULATIONS
for each measured point n as can be seen in Equation 3.10.
rmi = {xmi, ymi}ni=1 (3.10) As each measured profile n is located in its own local coordinate system, XY Zn, the theoretical involute profile is generated in a global machine coor- dinate system, xyz.
X (X, Y, Z)T, x (x, y, z)T
The coordinate systems can be related to one another using Equation 3.11, where R is a rotational matrix and X0 is the origin of the model coordinate frame xyz, referenced to the machine coordinate frame XY Z.
x = R (X − X0) (3.11)
The end goal is to find the solution X0 that has the smallest geometric distance to each point of the measured profile X. The geometric distance is a suitable measurement for the error as it is invariant to coordinate transfor- mation, i.e. rotation and translation. In order to find the best solution for the problem described, the Orthogonal Distance Fitting (ODF) model can be used. Several fitting parameters a need to be optimised: ag form parameters, ar rotation parameters, and ap position parameters. As the form of the theo- retical involute is fixed, the complexity of the problem can be reduced by using template matching. Template matching is a special case of ODF where the shape and size of the object is known; the form parameter ag can therefore be ignored. To solve the ODF two cost functions are used as performance indices σ20, and the goal is to minimise both of them. Where Equation 3.12 is the square sum, and Equation 3.13 is the distance between the measured points and the corresponding points on the modelled involute profile.
σ02I = kX − X0kTPTP kX − X0k (3.12)
σ02
II = (X − X0)TPTP (X − X0) (3.13) Here PTP is the weighting matrix, for most ODF applications the weighting matrix can be replaced by the identity matrix I [37], a n × n zero matrix with ones in the diagonal.
PTP = I =
1 0 0 0 1 0 0 0 1
By using the variable-separation method [37] the optimisation problem can
CHAPTER 3. GEAR PERFORMANCE EVALUATION METHODOLOGY
be solved using a nested iteration scheme, Equation 3.14. The model parame- ters a and the minimum distance points X0 are solved.
min
a={ap,ar} min
{X0i}mi=1σ02 {Xi0(a)}mi=1 (3.14) The inner loop of the optimisation is performed every iteration cycle and calculates the minimum distance points for the current set of parameters. The outer loop updates the parameter set. The optimisation is terminated when no more improvement to the performance indices σ20I,II can be achieved, Equa- tions 3.12 and 3.13. The optimisation problem can then be solved using a numerical solving method of choice, such as the Newton method, the Gauss- Newton method, or the Gradient Descent method.
By substituting the notations from our measurements into the general Equa- tions 3.12 and 3.13, we obtain two performance indices that should be min- imised:
σ02
I = krm− r0kTPTP krm− r0k σ02II = (rm− r0)TPTP (rm− r0) Where r’ can be obtained from rearranging Equation 3.11:
r’ = rR−1+r0
The minimum distance dminbetween the measured profile and the optimised theoretical profile can thus be calculated:
dmin=rm−r´ (3.15)
As the profiles are worn and damaged, the deviation from the theoretical profile is large and the fitting is troublesome. To mitigate this, the measured profiles were filtered using a spline high-pass filter [38] with a cut-off length of λc = 0.08mm. The mean deviation to the profile measured before run- ning in was calculated, as well as the standard deviation. Line segments that deviated more than one standard deviation from the initial measurement were omitted while aligning the profiles. The profiles were finally aligned using a cross correlation algorithm on the undamaged parts of the profiles.
After the fitting and alignment procedure is complete, the measured profile show a nearly perfect match, as can be seen in Figure 3.7.
In order to evaluate the form changes, one option is to look at the cumulative wear, Equation 3.16, the difference between each measured profile compared to the initial measurement. The cumulative wear gives a representation of the full extent of the damage over time; the damage progression can therefore be followed. An example is presented in Figure 3.8.
Wi= dTi− dT1, i ∈ {2, · · · , n} (3.16)
