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Design, Testing and Analysis of Journal Bearings for Construction Equipment Henrik Strand

Doctoral Thesis Department of Machine Design Royal Institute of Technology SE-100 44 Stockholm, Sweden

TRITA-MMK 2005:21 ISSN 1400-1179 ISRN/KTHMMK/R-05/21-SE ISBN 91-7178-142-0

TRITA-MMK 2005:21 ISSN 1400-1179 ISRN/KTHMMK/R-05/21-SE ISBN 91-7178-142-0 Design, Testing and Analysis of Journal Bearings for Construction Equipment Doctoral Thesis in Machine Design This is an academic thesis, which with approval of the Department of Machine Design, Royal Institute of Technology, will be presented for public review in fulfillment of the requirements for a Doctorate of Engineering in Machine Design. This public presentation will be made at the Roya l Institute of Technology, Room M3, Brinellvägen 64, Stockholm, Sweden, on October 7, 2005 at 14:15. © Henrik Strand 2005

HENRIK STRAND DESIGN , TESTING AND ANALYSIS OF JOURNAL BEARINGS FOR CONSTRUCTION EQUIPMENT

ABSTRACT Grease- lubricated journal bearings present a common challenge for construction equipment manufacturers in the world. The common design methodology is based on empirical data and has worked very well historically because the market and governments have accepted that bearings in construction equipment need frequent lubrication and exchange of worn parts. Legal and market requirements will soon demand lower environmental impact and increased machine efficiency. These requirements call for better methods of designing grease lubricated journal bearings. The goal of the outlined work was to develop better design methods for grease lubricated journal-bearing design used in heavy-duty construction equipment machines, in order to prolong life and lubrication intervals. The research approach of the project can roughly be divided into three phases: 1. Development of test apparatus and test methods for journal bearing studies. 2. Bench tests of grease lubricated journal bearing design. 3. Verification between bench tests and computer simulations. In the thesis the current state of the art in bearing design for construction equipment is discussed and summarized in the form of design guidelines. The suggested design steps are just a mean to get to the starting point of design. The simple guidelines do however serve a purpose when collected since most published bearing design guidelines are aimed at the bushing material or at continuously rotating bearings. The influence of housing, environment and load cases can not be ignored when designing a bearing. Long term field-testing and experience can not be replaced until better design criteria have been established. Paper A deals with the design of the bearing test apparatus that was built and evaluated. Comparisons between theoretical contact and contact elements in Finite Element program have been made and discussed in paper B. In paper C a replica technique for measuring wear of large field specimens was evaluated. A case study of bearing housing design was performed in paper D utilizing Finite Element program and then validated in paper E in the bearing test apparatus. The influence of grease groove design on bushing life was tested and evaluated in paper F. Wear simulation of a plain bushing has been performed with a Finite Element program and presented in paper G. Keywords; Contact Mechanics, Finite Element, Grease, Journal Bearing, Lubrication, Replica Technique, Test Apparatus, Tribology, Wear

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HENRIK STRAND DESIGN , TESTING AND ANALYSIS OF JOURNAL BEARINGS FOR CONSTRUCTION EQUIPMENT

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HENRIK STRAND DESIGN , TESTING AND ANALYSIS OF JOURNAL BEARINGS FOR CONSTRUCTION EQUIPMENT

SAMMANFATTNING (ABSTRACT IN SWEDISH) Fettsmorda glidlager är en gemensam utmaning för tillverkare av anläggningsmaskiner i hela världen. Det vanliga sättet att konstruera är baserad på empiriska data och har fungerat mycket väl fram till nu, då marknaden och regeringarna har accepterat att glidlagren i dessa maskiner behöver frekvent återsmörjning och utbyte av slitdelar. Striktare marknads- och lagkrav gällande miljöeffekter och ökad produktivitet gör att bättre metoder för konstruktion av fettsmorda glidlager behövs. Målet med det här arbetet var att utveckla bättre metoder för konstruktion av fettsmorda glidlager i tungt belastade entreprenadmaskiner för att öka livslängden och förlänga serviceintervallen. Tillvägagångssättet för forskningen i projektet är grovt indelat i tre faser: 1. Utvecklande av provapparat och provmetoder för studier av glidlager. 2. Bänk och fälttest av fettsmort glidlager konstruktion. 3. Verifiering mellan fält och bänktester med data simuleringar. I avhandlingen är den nuvarande metodiken för dimensionering av lagringar för anläggningsmaskiner diskuterad och sammanfattad i form av dimensioneringsregler. De föreslagna konstruktionsanvisningarna är en hjälp att komma till en startpunkt för fortsatt konstruktion. Dessa enkla anvisningar fyller dock sin funktion när de är ihopsamlade, då de flesta publicerade anvisningar för lagringskonstruktion är enbart avsedda för val av bussningsmaterial eller roterande lagringar. Inverkan av lagerhus, miljö och lastfall kan dock inte ignoreras när en lagring ska dimensioneras. Långtidsprov i fält och erfarenhet kan inte bli ersatta förrän bättre konstruktionskriteria är etablerade. Artikel A behandlar konstruktion av lagerprovapparaten som är byggd och evaluerad. Jämförelser mellan teoretisk kontakt och kontaktelement i finita element program är gjorda och diskuteras i artikel B. I artikel C är en replikeringsmetod för att mäta slitage av stora fältprover utvärderad. En fallstudie av lagerhusets inverkan på yttrycksfördelningen presenterad i artikel D och genomförd med hjälp av finit element program samt validerad i artikel E med hjälp av lagerprovapparaten. Inverkan av fettsmörjspårens utformning på bussningsslitage är studerat i artikel F. Nötningssimulering av en slät bussning är gjord med finit element program och presenteras i artikel G. Nyckelord; Avgjutningsteknik, Fett, Finita Element, Glidlager, Kontaktmekanik, Nötning, Provapparat, Smörjning, Tribologi

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HENRIK STRAND DESIGN , TESTING AND ANALYSIS OF JOURNAL BEARINGS FOR CONSTRUCTION EQUIPMENT

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HENRIK STRAND DESIGN , TESTING AND ANALYSIS OF JOURNAL BEARINGS FOR CONSTRUCTION EQUIPMENT

ACKNOWLEDGEMENTS The work presented in this thesis was carried out during 1998-2005 at the Department of Machine Design, Division of Machine Elements at the Royal Institute of Technology (KTH) in Stockholm and at Volvo Wheel Loaders AB in Eskilstuna. The financial support from the Swedish Research Council, the research program HiMeC and Volvo Construction Equipment is gratefully acknowledged. I would like to thank my supervisor Professor Sören Andersson and co-supervisor Dr. Stefan Björklund for all the help, support and guidance during these past years. My thanks go also to the colleagues at the division of Machine Elements for stimulating company and support. I would also like to thank my colleagues at department UM of Volvo Wheel Loaders AB for support and enduring my many questions. Special thanks to my manager M.Sc. Per Olson for his support and to Lic.Eng. Magnus Byggnevi for enduring my unending questions about solid mechanics. Last but not least, I would like to thank my wife Karin and the rest of my family for making me explain in various languages what I am researching and for proofing this thesis.

Eskilstuna, August 2005

Henrik Strand

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HENRIK STRAND DESIGN , TESTING AND ANALYSIS OF JOURNAL BEARINGS FOR CONSTRUCTION EQUIPMENT

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HENRIK STRAND DESIGN , TESTING AND ANALYSIS OF JOURNAL BEARINGS FOR CONSTRUCTION EQUIPMENT

LIST OF APPENDED PAPERS Slight adjustments in formatting for the appended papers have been made from the original. Henrik Strand is sole author of all papers. Papers E, F and G are submitted for publication. Paper A Strand, Henrik, “Apparatus for simulation of wear in heavy-duty oscillating journal bearings”, Proceedings OST – 99 Symposium on Machine Design, KTH, Stockholm, 1999 Paper A was orally presented at OST-99 in Stockholm, Sweden. Paper B Strand, Henrik, “Boundary conditions in finite element calculations utilizing conformal contact of cylindrical bodies”, 9th Symposium on tribology – Nordtrib 2000, VTT Technical Research Centre of Finland, Porvoo, 2000 Paper B was presented as a poster at Nordtrib 2000 in Porvoo, Finland. Paper C Strand, Henrik, “Wear measurement of plain journal bearings”, 10th Symposium on tribology – Nordtrib 2002, Royal Institute of Technology, Stockholm, Sweden, 2002. Paper C was presented orally at Nordtrib 2002 in Stockholm, Sweden. Paper D Strand, Henrik, “Journal Bearing Housing Design - A Statistical Study with FEM”, 11th Symposium on tribology – Nordtrib 2004, Tromsö, Norway, 2004. Paper D was presented orally at Nordtrib 2004 in Tromsö, Norway. To be published in Tribology International. Paper E Strand Henrik, “Influence of journal bearing housing stiffness on bushing wear” Paper F Strand, Henrik, “Influence of grease groove design on bushing wear” Paper G Strand, Henrik, “Simulation of bushing wear”

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HENRIK STRAND DESIGN , TESTING AND ANALYSIS OF JOURNAL BEARINGS FOR CONSTRUCTION EQUIPMENT

NOMENCLATURE a A α α

[m] [m2 ] [°] [1/K]

β c d

[°] [m] [m]

effective throat measure area half contact angle coefficient of heat expansion half oscillation angle eyebar radial width diameter

∆d

[m]

diametric change

D δ ∆

[m] [m] [m]

E E'

[N/m2 ] [N/m2 ]

diameter displacement diametric difference, ∆ > 0 ⇒ play, ∆ < 0 ⇒ interference, modulus of elasticity equivalent modulus of elasticity 1 1 − ?12 1 − ?22 = + E' E1 E2 force or load frictional force height hardness safety factor moment of inertia coefficient of wear coefficient of wear stress concentration factor length length difference

F Ff h H η I k K Kt

[N] [N] [m] [N/ m2 ] [-] [m4 ] [m2 /N] [-] [-]

L ∆L

[m] [m]

λ ν µ p q Q R R'

s S σ σy σUTS τ τy t

[W/mK] heat transfer coefficient [-] Poissons coefficient [-] coefficient of friction [N/m2 ] pressure [N/m] line load [W] heat effect [m] radius [m] equivalent radius 1 1 1 = − R' R1 R2 [m] seat width or distance [Nm] torque or torsion [N/m2 ] stress [N/m2 ] yield stress [N/m2 ] ultimate tensile stress [N/m2 ] shear stress [N/m2 ] yield shear stress [m] thickness

t

[s]

time

T ∆Τ v V w W

[K] [K] [m/s] [m3 ] [m] [m3 ]

ξ ψ z

[-] [-] [m]

temperature temperature difference sliding velocity volume bushing width bending or torsion resistance heat distribution relative bearing play height coordinate

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HENRIK STRAND DESIGN , TESTING AND ANALYSIS OF JOURNAL BEARINGS FOR CONSTRUCTION EQUIPMENT

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HENRIK STRAND DESIGN , TESTING AND ANALYSIS OF JOURNAL BEARINGS FOR CONSTRUCTION EQUIPMENT

TABLE OF CONTENTS ABSTRACT...................................................................................................................................................................III S AMMANFATTNING (ABSTRACT IN SWEDISH)......................................................................................................V ACKNOWLEDGEMENTS ...........................................................................................................................................VII LIST O F APPENDED PAPERS.................................................................................................................................... IX NOMENCLATURE ....................................................................................................................................................... XI TABLE O F CONTENTS ............................................................................................................................................XIII 1.

INTRODUCTION...................................................................................................................................................1

1.1. BACKGROUND ...........................................................................................................................................1 1.2. HISTORICAL BACKGROUND ....................................................................................................................4 1.3. INDUSTRIAL A ND A CADEMIC RELEVANCE...........................................................................................4 1.4. RESEARCH A PPROACH.............................................................................................................................5 1.4.1. Subtasks...........................................................................................................................................5 1.5. OUTLINE OF THESIS .................................................................................................................................6 2. BEARING DESIGN...............................................................................................................................................7 2.1. STRUCTURAL DESIGN ..............................................................................................................................7 2.1.1. Pin Diameter ..................................................................................................................................7 2.1.2. Flexural Pin Design......................................................................................................................7 2.1.3. Surface Pressure Distribution.....................................................................................................9 2.1.4. Bushing Width............................................................................................................................. 11 2.1.5. Pin Seat Width............................................................................................................................. 12 2.1.6. Housing........................................................................................................................................ 13 2.1.7. Lockplate Screw Joint................................................................................................................ 14 2.1.8. Lockplate Weld Size................................................................................................................... 14 2.1.9. Lockplate Lock Size.................................................................................................................... 15 2.1.10. Lateral Pin Lock ......................................................................................................................... 17 2.1.11. Bearing Play................................................................................................................................ 18 2.1.12. Velocity......................................................................................................................................... 19 2.1.13. Pressure Velocity........................................................................................................................ 20 2.1.14. Friction......................................................................................................................................... 21 2.1.15. Temperature ................................................................................................................................ 21 2.1.16. Surface Roughness..................................................................................................................... 22 2.1.17. Wear.............................................................................................................................................. 23 2.2. BEARING M ATERIAL SELECTION .........................................................................................................24 2.2.1. Pins ............................................................................................................................................... 24 2.2.2. Bushings....................................................................................................................................... 25 2.2.3. Seals.............................................................................................................................................. 27 2.2.4. Grease .......................................................................................................................................... 28 2.3. FINITE ELEMENT ANALYSIS..................................................................................................................29 3. S UMMARY OF APPENDED PAPERS .............................................................................................................. 30 3.1. PAPER A...................................................................................................................................................30 3.2. PAPER B ...................................................................................................................................................32 3.3. PAPER C ...................................................................................................................................................33 3.4. PAPER D...................................................................................................................................................34 3.5. PAPER E ...................................................................................................................................................35 3.6. PAPER F....................................................................................................................................................36 3.7. PAPER G...................................................................................................................................................37 4. DISCUSSION AND CONCLUSIONS ................................................................................................................. 38 5.

FUTURE R ESEARCH........................................................................................................................................ 39

6.

R EFERENCES .................................................................................................................................................... 40

7.

APPENDIX ......................................................................................................................................................... 41

APPENDED PAPERS A-G

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HENRIK STRAND DESIGN , TESTING AND ANALYSIS OF JOURNAL BEARINGS FOR CONSTRUCTION EQUIPMENT

1. INTRODUCTION Grease- lubricated journal bearings present a common challenge for construction equipment manufacturers in the world. The common design methodology is based on empirical data and has worked very well historically because the market and governments have accepted that bearings in construction equipment need frequent lubrication and exchange of worn parts. Legal and market requirements will soon demand lower environmental impact and increased machine efficiency. These requirements call for better methods of designing grease lubricated journal bearings. The goal of the outlined work was to develop better design methods for grease lubricated journal-bearing design used in heavy-duty construction equipment machines, in order to prolong life and lubrication intervals. Bearings for construction equipment are ge nerally subject to very high loads together with low sliding speeds. Typical values of pressure are 100-200 MPa and sliding speeds are in the range of 0.01-0.1 m/s. An example of a construction equipment machine with many oscillating journal bearings is a wheel loader shown in figure 1 below.

Figure 1. A Volvo wheel loader model L220E.

1.1. BACKGROUND Grease- lubricated journal bearings have followed mankind since the earliest civilizations. The principal design of a rotating or oscillating shaft in a grease- lubricated hole has not changed much since then [1]. Today these bearings consist of hardened steel axles sliding against plastic or metallic bushings lubricated with modern petrochemical grease, instead of wooden shafts sliding against wood lubricated by animal fat. 1

HENRIK STRAND DESIGN , TESTING AND ANALYSIS OF JOURNAL BEARINGS FOR CONSTRUCTION EQUIPMENT Knowledge of bearing design has been passed down mostly as rules of thumb. In today's heavy-duty industrial applications, problems develop due to increasing loads on the bearing while the size is reduced. This leads to high surface pressures and accelerated wear. The question how to optimize the bearing design of heavy-duty bearing applications while maintaining small size and low cost is becoming a very important issue. Substantial testing of different bushing materials has been done by a variety of different researchers with various often contradictory results [2]. These tests are only concerned with the material selection process, i.e. how to select the best combination of material for the pin and bushing. Still for the overall bearing design, a few direct conclusions can be drawn from these tests. Current design guidelines found in literature for journal bearings are mostly based on empirically derived correlations for grease lubricated journal bearings subject to static loads and continuous rotation. However this is not a very fitting description of the actual working condition for bearings in construction equipment. The loading unit of a Volvo wheel loader has 16 individual grease lubricated journal bearings as can be seen in figure 2 below.

Figure 2. A loading unit of a wheel loader. These journal bearings operate in very harsh environments with highly varying dynamic impact loads and oscillations under low sliding speeds and a wide range of ambient temperatures. This implies that the bearings are operating in the boundary- lubricated regime, which means insufficient lubrication to completely separate the axle pin from the bushing. The resulting pin-bushing contact results in wear that shortens the life of the bearing. High friction from insufficient lubrication is a major cause of energy loss in the load unit.

2

HENRIK STRAND DESIGN , TESTING AND ANALYSIS OF JOURNAL BEARINGS FOR CONSTRUCTION EQUIPMENT Another problem is how to keep grease inside the bearing until all of it is consumed and then replaced with new grease. Today grease lubricated journal bearings operate on a total loss system, which means that all inserted grease is released directly out into the environment after its lubrication properties are diminished. For a machine that is used very hard, replenishment of grease is needed every day. Every time a wheel loader is greased, a few kilograms of grease are released into the environment. Seals are used to keep the grease between the bushing and the pin while protecting against external contaminants. They need to be stiff enough to seal against dirt while keeping the grease inside, but resilient enough to allow replenishment. During heavy loads the pin is deflected during operation, thus allowing the grease to be pushed out through the deformed seals. To minimize deflection different designs of housing can be used. A typical journal bearing for construction equipment is shown in figure 3 below.

Figure 3. Cross section of a typical journal bearing housing for construction equipment. To understand which factors and interactions govern the performance of the bearing, both simulations and full-scale field tests need to be performed. A quick way to evaluate different housing designs is to use finite element programs with capability of contact simulation. Use of finite element analysis places great demand on accuracy and speed. The most critical aspect of every finite element model is how to apply the boundary conditions and loads that most accurately correspond to reality. To fine-tune the finite 3

HENRIK STRAND DESIGN , TESTING AND ANALYSIS OF JOURNAL BEARINGS FOR CONSTRUCTION EQUIPMENT element model, it is necessary to compare results from actual physical tests and theoretical contact mechanic calculations.

1.2. HISTORICAL BACKGROUND Man started using metallic bushings as early as 400 BC. It is documented that conical journal bearings were used in China around 300 BC. During the Roman Empire the use of metal bushings spread throughout Europe. After the fall of the empire the usage diminished rapidly. The continuous use of metallic bushings until 1000 AD has been documented in China. In Europe the use of metallic bushings did not become widely spread until the middle ages. Most common bearings were still made of wood since metal bearings were expensive and difficult to make. In the middle of the 16th century there was evidence of journal bearings of wood, iron and steel. Around the year 1600 in Italy, recommendations for design and dimensioning of bearings with bronze bushings together with steel pins were recorded. From a hundred years later there are references to softer bushing materials with split halves being used with the possibility of adjusting the play. From 1734 there is evidence of the usage of cast iron, brass and bell metal for bushing materials. In 1839, Isaac Babbitt obtained a patent on a tin-based alloy with antimony and copper for bushings, called Babbitt. In 1921 copper- lead based alloys were introduced. During the middle part of the 20th century no significant improvements to bearing design were contributed. Since the introduction of plastic materials, tremendous progress on the plastic bearing market has been achieved. Within the last decades, porous oil- filled materials, as well as composite bearings, have also been applied as bushing material with success. In the course of this progress of bearing materials it has proved important to understand which physical and tribological properties are needed for different applications [1].

1.3. INDUSTRIAL AND ACADEMIC RELEVANCE The bearings in the lifting frame unit of a wheel loader are one of the most crucial components for its operation. If the bearings do not work properly the machine will not work well at all. A comparison can easily be made between a machine with bad lubrication and a human with rheumatism; neither can do much heavy intensive work without serious injury. The bearing design problem described above is present in many construction equipment applications. Very little open research has been done in this field in the last few years. Research with grease as a lubricant has been aimed at rolling element bearings that operate under much different tribological conditions. This is largely because roller bearings have been regarded as state of the art during the 20th century. However for slow moving applications like lifting frame bearings in wheel loaders, roller bearings just do not work well eno ugh without being substantially oversized. The work presented in this thesis can be used as reference for other similar applications and the guidelines presented in chapter 2 will simplify the design process. 4

HENRIK STRAND DESIGN , TESTING AND ANALYSIS OF JOURNAL BEARINGS FOR CONSTRUCTION EQUIPMENT

1.4. RESEARCH APPROACH The research approach of this work can roughly be divided into three phases: 1. Development of test apparatus and test methods for journal bearing studies The first phase was the foundation for further studies. In this phase the test apparatus for oscillating bearing testing was built and evaluated. Methods of measuring results from the test apparatus were also important. Papers A, C, E are related to this phase. 2. Bench tests of grease lubricated journal bearing design This phase involves testing of different designs and materials. The measured result was how the design and material combinations affect wear, friction and lubricity of the sliding bearing surface. Papers E and F mostly cover this phase. 3. Verification between bench tests and computer simulations The final phase was meant to correlate field and bench test results with simulated results from finite element analysis and numerical wear modeling. The goal was to be able to predict wear life as a function of material, load, speed and design. Another goal was to formulate a wear model for grease lubricated journal bearings to predict when a bearing needs to be replaced, and how much a bearing is damaged when subject to loss of lubrication. In papers D and E this verification is discussed. Some wear simulations have been performed using a finite element program and the results are presented in paper G. 1.4.1. SUBTASKS To achieve the above stated goals a few subtasks need solving: •

Bearing design parameters

The object was to theoretically find the parameters that govern the bearing function. To help sort through the parameters of the testing process, it is important to try to use existing empirical formulas and make simplified calculations to select the most important parameters. The result from this work is compiled in chapter 2 on bearing design. •

Faster finite element contact analysis

In order to solve 3-dimensional solid mechanics problems with the finite element method, it is necessary to have correct boundary conditions to get accurate results. It is of great importance to use an appropriate contact element model that corresponds well to reality and theory to produce these boundary conditions for contacting surfaces. Increased use of contact elements in finite element programs has helped the engineer to get more accurate boundary conditions at the expense of heavily increased calculation times. The question of how to trim the contact elements to get accurate results and short calculation times is a major problem, especially for conformal contacts in journal bearings. If a finite element analysis could be used to predict wear, it is vital that the analysis is quickly solved with reasonable accuracy. This is discussed in paper B. 5

HENRIK STRAND DESIGN , TESTING AND ANALYSIS OF JOURNAL BEARINGS FOR CONSTRUCTION EQUIPMENT

1.5. OUTLINE OF THESIS In this chapter the background, purpose and research approach of the work is presented and explained. In chapter 2 the current state of the art in bearing design for construction equipme nt is compiled into a handbook of journal bearing design. The handbook is not complete in any way but will hopefully serve as a start of design. In chapter 3 a short overview of the accomplished work of the appended papers can be found. In chapter 4 the results presented in the thesis are discussed and concluded. And finally in chapter 5 some remarks about future research are made.

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HENRIK STRAND DESIGN , TESTING AND ANALYSIS OF JOURNAL BEARINGS FOR CONSTRUCTION EQUIPMENT

2. BEARING DESIGN Grease lubricated oscillating journal bearings are today mostly designed by empirical tests and traditional guidelines. Bearing manufacturers have good knowledge of bearing material interaction but the design engineers who buy them and build them into machines usually do not. Unfortunately bearing manufacturers have little knowledge of how the bearing will be used and they have to rely on in-house bench tests. There are in fact a lot of design parameters that rely on the surrounding environment and geometry that can not be tested in a general purpose bench test. In this chapter some design guidelines are outlined for grease lubricated oscillating journal bearings for construction equipment applications.

2.1. STRUCTURAL DESIGN The applied load F on the bearing together with the material yield stress σy serves as the basis for the bearing design process. A safety factor denoted by η is frequently appearing in the equations below but no guidelines about what factors to use are presented since it varies for different applications. 2.1.1. PIN DIAMETER The pin material yield stress in shearing gives a minimum value of the diameter that may serve as a starting point.

F 4⋅F 2⋅ F  = = 2 2 ⋅ A 2⋅ π ⋅ d π ⋅ d 2  ⇒d= σy 2 2  σ e,VonMises = σ + 3 ⋅τ = η  τ=

2 3 F ⋅η π σy

(1)

This equation yields a diameter that is much smaller than actually needed. A large safety factor is recommended against shear break of the pin. 2.1.2. FLEXURAL PIN DESIGN Blake [4] states that the bending stress alone makes the decisive stress if the pin diameter is smaller or equal to its free length. Since this is usually the case, it becomes easy to account for the maximum stress of the outer fiber by approximating the pin as a constrained beam loaded in the middle. The beam is considered constrained since the contact between pin and pin seat is rather tight. From beam theory the minimum diameter to avoid plastic deformation for the described load case is:

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HENRIK STRAND DESIGN , TESTING AND ANALYSIS OF JOURNAL BEARINGS FOR CONSTRUCTION EQUIPMENT

F⋅L 8 π ⋅d3 W = 32 S σ= W S=

σ e,VonMises

     4 F ⋅ L ⋅η ⇒ d = 3 π σy   σ  = σ 2 + 3 ⋅τ 2 = y  η 

(2)

The free length L is defined by Blake to be the length between the pin seats i.e. the bushing width plus the lateral play.

Figure 4. Model of a constrained beam and model for bending with a transversal hole. When there is a lubricating hole in the pin, the stress is increased due to stress concentration. The effect of a transversal hole in a round pin subject to bending can be read from stress concentration factor charts. The result is shown below with Kt calculated from empirical data presented by Pilkey [5].     S σ nom =  W  I pin I hole d 3  π 1  dhole    W= − = ⋅  − ⋅    ⇒ z max zmax 4  8 3  d    F ⋅L (4) S=  8  2 3 4 d d d d d           K t  hole ≤ 0.2  = 3 −11.3 ⋅  hole  + 51.6 ⋅  hole  −114.6 ⋅  hole  + 76.4 ⋅  hole    d   d   d   d   d   σ = K t ⋅ σ nom =

σy η

2

3

d  d  d  d  3 − 11.3 ⋅  hole  + 51.6 ⋅  hole  − 114.6 ⋅  hole  + 76.4 ⋅  hole  4 F ⋅ L ⋅η  d   d   d   d  ⇒d=3 ⋅ ⋅ 8 d π σy   1− ⋅  hole  3⋅π  d 

4

As shown in equations 4 above, a lubrication hole in the maximum loaded region will increase the minimum required pin diameter by 34 - 44%. The maximum stress in the transversal hole does not occur on the surface, but a small distance below the pin 8

HENRIK STRAND DESIGN , TESTING AND ANALYSIS OF JOURNAL BEARINGS FOR CONSTRUCTION EQUIPMENT surface inside the lubrication hole [5]. This fact makes it very important to make sure that the lubrication hole has smooth surfaces to avo id additional stress concentration. The transition zone between the hardened surface and the core material is also in the same region. Another way to deal with this stress concentration is to place the transversal lubrication hole in the neutral layer for bending, which can be difficult since the load direction is often not fixed relative to the pin. 2.1.3. SURFACE PRESSURE DISTRIBUTION The surface pressure by projected bearing area is the most common distribution seen in bearing catalogues. As a rule of thumb, the projected surface pressure should be kept below 50% of the softer material yield stress to avoid seizure if the velocity is kept below 0.02 m/s [6]. The projected surface pressure is: p=

σy F 2⋅F ≤ ⇒ d ⋅ w = Aprojected ≥ d ⋅w 2 σy

(5)

If the surface pressure exceeds the material hardness it will be plasticized resulting in increased play, wear and extrusion [7]. The projected pressure does not take into account the influence of misalignment and bearing play and their addition to the maximum pressure in the bearing.

Figure 5. Projected area for surface pressure calculations. The projected surface approach tells nothing about the maximum edge pressure that deforms the materials in the bearing. A simplified model of load distribution of a bearing is shown in the figure below. The real parabolic load distribution shape of the bearing is difficult to model without using finite element tools to analyze the housing stiffness. A simplified triangular load distribution usually provides a slightly more accurate model than the projected pressure distribution but conservative model. Using this load distribution doubles the maximum pressure compared to the projected surface pressure.

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HENRIK STRAND DESIGN , TESTING AND ANALYSIS OF JOURNAL BEARINGS FOR CONSTRUCTION EQUIPMENT

Figure 6. Left; bearing with loads. Center; pin with parabolic pressure distribution. Right; pin with simplified pressure distribution. In figure 6 above the load is only varied axially and no information about the circumferential distribution is given. This distribution can be more or less simplified. Different approaches to circumferential pressure distribution modeling are shown in figure 7 below.

Figure 7a-d. Examples of four different circumferential pressure distribution models subject to a load per unit length q [N/m]. Figure 7a shows projected pressure distribution. Figure 7b shows sinusoidal pressure distribution. Figure 7c shows Hertz pressure distribution. Figure 7d shows pressure distribution according to equation 9. Figure 7a shows the projected pressure distribution, whic h gives the average pressure over the bearing diameter. The maximum pressure in this case is: p MAX =

q d

(6)

where q is the normal load per unit length. Figure 7b shows a common pressure distribution model, a sinusoidal distribution that is rather accurate for a press- fitted loaded axle in a hole. The maximum pressure in this case is: p MAX =

10

4⋅q q ≈ 1.273 p ⋅d d

(7)

HENRIK STRAND DESIGN , TESTING AND ANALYSIS OF JOURNAL BEARINGS FOR CONSTRUCTION EQUIPMENT Figure 7c shows the Hertzian pressure distribution that is valid when the radial differences are large enough to get a contact angle α below 20°. The maximum pressure in this case is:

p MAX =

q ⋅ E' p ⋅ R'

(8)

Figure 7d shows the pressure distribution that was analytically derived by Persson [8], which is valid for plain stress solutions. The maximum pressure is calculated by:

p MAX

2      tan  a  + 1 + tan  a    a ln  2 ⋅ tan    2  2    2 ⋅ q 2   = + 2 2 2     d a a     a     p ⋅ tan   + 1 p ⋅ tan   1 + tan       2    2    2  

(9)

The advantage of the two first models is that no knowledge about modulus of elasticity and contact angles is needed to get an estimate of the pressure distribution. The disadvantage is, of course, that they are inaccurate and in most cases far from the real distribution. The Hertz pressure is very easily calculated and takes into account different material parameters and bearing play, but only works for contact angles less than 20° which is not good for tight fitting journal bearing designs. The distribution in equation 6 is not easily calculated and requires knowledge about the contact angle that can be found by forming the quota E⋅∆/2⋅q and reading the value from a previously calculated table found in the appendix. It should be noted that equation 9 only works for materials with the same modulus of elasticity. In the work by Persson [8] there are results for material combinations with different modulus of elasticity but they are not reproduced here. Typical grease lubricated journal bearings should have a contact angle 2α of about 80160 degrees between the pin and bushing. The contact angle is of course depending on the deflection. A stiffer bearing and pin can have a less play and thereby handle larger contact angles and a weaker bearing will deflect more and get a larger contact area with increased load. According to equation 9 the model in equation 7 is conservative for α = 72 degrees and materials with the same modulus of elasticity. Another simple approach is to use twice the projected pressure, which can be used for large play between axle and bushing. This model is then conservative for α = 38 degrees. 2.1.4. BUSHING WIDTH A way to get the minimum bearing width is to calculate the maximum edge pressure on the bushing and compare it to the bushing hardness measured in Brine ll (HB) or Vickers (HV). The bushing hardness is used since it is usually softer than the pin material. If the width of the bushing is smaller than calculated by equation 10 the bushing edges will be plastically deformed. In the formula below the simplified pressure distributions shown in figures 6 and 7b are used. 11

HENRIK STRAND DESIGN , TESTING AND ANALYSIS OF JOURNAL BEARINGS FOR CONSTRUCTION EQUIPMENT 2⋅ F   H 8 F 8 F ⋅η w = bushing ⇒ w =  ⇒ pMAX = 4 ⋅ qMAX  π d ⋅w η π d ⋅ H bushing =  π ⋅d 

qM AX = p MAX

(10)

If the bushing is too short, it is susceptible to large edge pressures due to pin misalignment. A long slender bushing is better at handling misalignment but has problems with bending if the axle is not stiff enough compared to the load, as can be seen in figure 8 below.

Figure 8. Edge loading due to axle misalignment (left) and deflection (right). Width to diameter ratio should be kept in the interval of 0.5 ≤

w ≤ 1 .5 d

(11)

to avoid large edge loads due to misalignment and bending. A w/d ratio of about 1.3 is recommended for grease lubricated journal bearings to minimize side leakage [6]. These recommendations do not take into account the influence of the bearing play, bushing elasticity and seals. 2.1.5. PIN SEAT WIDTH The minimum pin seat width can be calculated in a similar way as the bushing width above. Again, the triangular pressure distribution is used. F   4 F H 4 F ⋅η s = seat ⇒ s =  ⇒ pMAX = 4 ⋅ qMAX  π d ⋅s η π d ⋅ H seat =  π ⋅d 

q MAX = p MAX

(12)

If the hardness value is not known the following approximate relation between Brinell hardness and ultimate tensile strength can be used for hardness’s below 400 HBS [9]: H seat = 0.3 ⋅ σ uts 12

(13)

HENRIK STRAND DESIGN , TESTING AND ANALYSIS OF JOURNAL BEARINGS FOR CONSTRUCTION EQUIPMENT 2.1.6. HOUSING The bearing housing should be stiff to prevent the bearing from thermal and mechanical deformation leading to misalignment, but also weak enough to allow conformal deformation to prevent edge loading. The housing should be optimized for stiffness to avoid deflection while maintaining an even pressure distribution to avoid wear.

Figure 9. Cross section of a typical bearing for construction equipment A common practice in engineering design is to use an inner to outer diameter ratio between 2 and 4 [4]. It is also recommended that the width c is kept constant along the outer edge to avoid stress concentrations.

Figure 10. Eyebar design The presence of a loaded hole in an eyebar is compensated by a stress concentration factor [10]. Using equations 14 belo w provide a minimum width of the eyebar.

13

HENRIK STRAND DESIGN , TESTING AND ANALYSIS OF JOURNAL BEARINGS FOR CONSTRUCTION EQUIPMENT σy   σy η  ⋅ s⋅d ±  η F σ NOM =  ⇒ Dh = s ⋅ (Dh − d )   d Dh Kt = +  Dh d  σ = σ NOM ⋅ K t =

σy  η

2

σ  2 2  ⋅ s ⋅ d + 4 ⋅ y ⋅ w ⋅ d ⋅ F − 4 ⋅ F 2 η  σ F 2 y ⋅ s −  d η

(14)

2.1.7. LOCKPLATE SCREW JOINT The pin is prevented from rotation by screwing its lockplate to the pin seat. To dimension the screw for shear load the following equation can be used with figure 11. F ⋅ µ1 ⋅

d d d⋅F − 2 ⋅ F ⋅ µ 2 ⋅ − R ⋅ FPINLOCK = 0 ⇒ FPINLOCK = (µ1 − µ 2 ) 2 2 2⋅ R

(15)

The coefficient of friction µ1 is between pin and bushing and µ2 is between pin and pin seat in figure 11 below.

Figure 11. Calculation of weld size and lockplate screw force. The maximum shear load on the screw occurs when µ1 = 1 and µ2 = 0. In reality friction also occurs between the lockplate and pin seat side but that is neglected here due to the conservative use of the coefficients of friction. 2.1.8. LOCKPLATE WELD SIZE Most pins have a lockplate welded to one end of the pin. The function of the lockplate is to prevent the pin from rotating in the pin seat and to keep the pin from moving axially out of the pin seat. Generally there are three different weld designs in use; external, internal and vertic al weld as can be seen in figure 12 below.

14

HENRIK STRAND DESIGN , TESTING AND ANALYSIS OF JOURNAL BEARINGS FOR CONSTRUCTION EQUIPMENT

Figure 12. Different lockplate weld types with respective cross section below. Shown from left to right are an internal weld, external weld and vertical weld. To calculate the needed effective throat measure the maximum allowed shear stress must be known. The shear stress for the internal and external weld can be calculated using the equation below. The minimum effective throat measure is usually 2 - 3 mm.

 µ1 = 1  F ⋅d    ⇒ FPINLOCK =  2⋅ R   µ2 = 0 2 π ⋅a  W= Di + 2 ⋅ a  ⇒ Di = 2  σy  S FPINLOCK ⋅ R τ= = =  W W 3 ⋅ η 

(

)

3 F ⋅ d ⋅η − 2 ⋅a 2 π ⋅ a ⋅σ y

(16)

For internal welds there is a maximum diameter requirement for Di. It has to do with penetration of the weld and heat close to the induction hardened surface.

(

Di ≤ d − 2 ⋅ a ⋅ 1 + 2

)

(17)

There is also a minimum diameter requirement for internal welds related to the possibility of manufacturing, and it is about 20 - 25 mm depending on desired weld throat size and electrode used. This calculation does not take the fatigue life of the weld into account. The fatigue life is governed by the stress in the weld and the geometrical design crack in the root of the weld. This design crack serves as an initiation point of the crack. For further reading on fatigue life of welds see Byggnevi [11]. 2.1.9. LOCKPLATE LOCK SIZE There are several lockplate designs in use today. Two typical designs use a screw in a closed or open slot in the lockplate. It is important to have large radiuses in the slot to 15

HENRIK STRAND DESIGN , TESTING AND ANALYSIS OF JOURNAL BEARINGS FOR CONSTRUCTION EQUIPMENT reduce the effect of stress concentrations. The plate thickness t is usually between 5 - 15 mm and the material is usually low alloyed structural steel with low yield strength (σy ≤ 300 MPa). For calculation of bending stress the prongs of the lockplate are simplified as constrained beams. Since the prongs are quite short and thick the shear stress can not be neglected and have to be incorporated in the calculation. Open design

Figure 13. Open lockplate design.   S = F ⋅L  t ⋅ h2  W =  6  S 6⋅ F ⋅L  σ= = ⇒h= 2 W t⋅h  F  τ=  t⋅h  σy 2 2 σ e,VonMises = σ + 3 ⋅ τ = η 

 σ  6 ⋅ F ⋅  F + F 2 + 16 ⋅  y  η  σ 2 ⋅t ⋅ y η

Closed design

Figure 14. Closed lockplate design

16

2  2 2   ⋅ t ⋅ L    

(18)

HENRIK STRAND DESIGN , TESTING AND ANALYSIS OF JOURNAL BEARINGS FOR CONSTRUCTION EQUIPMENT

  δ1 = δ2  3  FE ⋅ (2 L ) 5 δ1 = F  ⇒ FE = 3⋅ E ⋅ I 32  2 3 F ⋅ (2 L)3   L   L   FE ⋅ (2 L)3  δ2 = 3⋅   −   −  6 ⋅ E ⋅ I   2 L   2 L   3 ⋅ E ⋅ I  11  ⋅ F ⋅ L 16  2 t ⋅h  W =  6  S 33 F ⋅ L  σ= = ⋅ ⇒ h= 2 W 8 t ⋅h  F  τ=  2⋅t⋅h  σy  2 2 σ e,VonMises = σ + 3 ⋅ τ = η 

(19)

S = F ⋅ L − FE ⋅ 2 ⋅ L =

 σ  6 ⋅ F ⋅  F + F 2 + 121 ⋅  y  η  σ 4 ⋅t ⋅ y η

2  2 2   ⋅ t ⋅ L    

(20)

Using equation 19 the necessary eyebar width c can then be calculated as: σ =

FE σ y 5 F ⋅η = ⇒c = ⋅ c ⋅t η 32 σ y ⋅ t

2.1.10.

(21)

LATERAL PIN LOCK

If the pin seat bores have a loose fit together with a long slender pin, lateral pin movement can occur. The bending of the pin will move the pin out of the bore furthest away from the lockplate due to deformation. To prevent this choose a tight fit between pin and seat and a pin length to diameter ratio less than 3. If lateral movement still occurs or if it is not possible to use the tighter fit or length/diameter ratio, a screwed on end plate might be necessary. The movement of the pin can be approximated by the following equation together with figure 15:

Figure 15. Lateral displacement 17

HENRIK STRAND DESIGN , TESTING AND ANALYSIS OF JOURNAL BEARINGS FOR CONSTRUCTION EQUIPMENT 2 2  Lb  L  2 + δ =       2 2  3 2 2  1 F⋅L ∆L 1  F ⋅L   δ = = 1 − 1 − 2  ⇒ 48 E ⋅ I L 24 E ⋅ I    ∆L = L − Lb   

(21)

In equation 21 above the total pin length (L=2⋅s +w) should be used. To prevent the pin end from moving a lateral force need to be applied. The necessary force can be calculated by equation 22 below if the friction between the pin and pin seat is neglected:   σ PIN = E ⋅ ε PIN  ∆L  ε PIN =  2  L 2  2  E ⋅π ⋅ d  64  F ⋅ L    FLATERAL = σ PIN ⋅ A ⇒ FLATERAL = ⋅ 1 − 1 −  4   4 9 E ⋅ π ⋅ d        π ⋅d 2  A= 4  4  π ⋅d  I= 64 

2.1.11.

(22)

BEARING PLAY

For grease lubricated heavily loaded bearings with metallic bushings, the recommended relative bearing play ψ for journal bearings is [6]: 0 .3 ‰ ≤ ψ ≤ 5 ‰  dbush − d pin  ψ =  d pin  

(23)

The bearing play depends on the type of load and application. The bearing play can be used to control the bearing sensitivity to bearing play increase due to wear and magnitude of edge load. For oscillating grease lubricated bearings with slow sliding speeds the play should be kept as small as possible. Since the bushing is often press fitted into the housing, there will be a deformation that influences the bearing play. To control the resulting play, four diametrical tolerances have to be known, namely the housing inner diameter, bushing outer diameter, bushing inner diameter and pin outer diameter, see figure 16 below. The deformation from the interference fit is in the same magnitude as the manufacturing tolerances, which makes it important to have control over the displacement of the inner diameter. In the case when the housing and bushing are of the same material the radial displacement of the inner surface diameter of the bushing can be approximated by the following equations adapted from Sundström [12]: 18

HENRIK STRAND DESIGN , TESTING AND ANALYSIS OF JOURNAL BEARINGS FOR CONSTRUCTION EQUIPMENT

( (

? d b D2 − Dh2 u=− ⋅ ⋅ 2 2 D db − Dh2

) )

 ? = Db − dh     D = d h ≈ Db 

(24)

Figure 16. Press fit of bushing into house. When the bushing and the housing material modulus of elasticity and Poisson coefficients are different, the interference fit equation becomes:  D 2 ⋅ db ⋅ (3 ⋅υ h − 5 )    D 2 − d b2 1 ∆   u=− ⋅ ⋅ 2 4 D ⋅ Eb    D     d b 2     1+      1+  1  D   1   D    h  + υ + ⋅ − υ  ⋅  h b 2 2    Eh   D   Eb   db    1−  D     1 −  D         h  

2.1.12.

(25)

VELOCITY

When calculating velocity it is important to use the actual velocity during movement for life calculations. The average velocity for pendular motion of journal bearings can be formulated by the equation below. v=

s 2⋅d ⋅β = t t

(26)

The equation is illustrated in figure 17 below. This is a simplification since an oscillating movement implies a lot of accelerations and decelerations. The influence of intermittent operation is not contained in the equation but influence the life of the bearing greatly. Journal bearings for construction equipment operate under very low 19

HENRIK STRAND DESIGN , TESTING AND ANALYSIS OF JOURNAL BEARINGS FOR CONSTRUCTION EQUIPMENT speeds and transient motion. This transient motion generates heat that has to be dissipated.

Figure 17. Sliding distance for calculation of average sliding velocity. 2.1.13.

PRESSURE VELOCITY

The maximum limit of the pressure p is correlated to the mechanical strength of the material and the limit of the sliding velocity v is regarding frictional heating of the materials. For a given material combination there is a maximum pv-product beyond which surfaces can no longer absorb the friction energy thus generating either localized seizure or melting of the materials. A typical pv- graph is shown in figure 18 below.

Figure 18. A typical pressure velocity graph with logarithmic axes. The pv- factor characterizes damage risks related to thermal and mechanical phenomena. It does not consider wear modes that might occur in industrial applications. In particular this factor does not describe the influence of phenomena such as abrasion and fretting corrosion. For journal bearings in construction equipment the velocity is so far away from the upper velocity limit of the pressure- velocity - graph that in reality only the pressure limit needs to be considered.

20

HENRIK STRAND DESIGN , TESTING AND ANALYSIS OF JOURNAL BEARINGS FOR CONSTRUCTION EQUIPMENT 2.1.14.

FRICTION

The friction force in a bearing is modeled as directly proportional to the normal load (Coulomb friction). Ff = µ ⋅ FN

(27)

This simplification only deals with the friction from a system point of view and states nothing about the local shear stresses at different locations with different pressures. For unevenly distributed normal force the friction force will vary, and this results in unfavorable stress gradients in the materials. For grease lubricated oscillating journal bearings, the coefficient of friction is typically in the range of 0.1 - 0.8. This large range is due to the material combination and amount of lubricant present between them. A newly greased lubricated bearing will have a coefficient of friction of about 0.1, and then it will gradually increase to reach typical values of the dry surfaces since the grease escapes from the sliding area due to heat or mechanical action. The upper limit is when seizure or excessive wear occurs between the bushing and the pin. This is not really sliding friction but rather shearing of the material at the contact interface. 2.1.15.

TEMPERATURE

The generated heat between pin and bushing is considered to be proportio nal to the frictional force and sliding speed. The partitioning of absorbed heat by the pin and bushing is depending on the heat conductivity and bulk temperatures. Due to the low sliding speeds both the pin and bushing can be regarded as stationary and the partitioning of heat entering each body can be calculated by the following equation adapted from Bhushan [13]: Q = F ⋅ µ ⋅ v = Qbush + Qpin Qbush = ξ ⋅ Q Qpin = (1 − ξ ) ⋅ Q ξ= 1+

(28)

1 λpin λbush

As can be seen the conductivities of the bodies govern the partitioning of the frictional heat and the most heat will enter the body with highest thermal conductivity. To get the temperature on the contact surface it is usually necessary to measure the temperature a short distance away. The following equation can be used to estimate the temperature in the contact interface if the temperature is measured on the backside of a bushing with thickness t [14]:

21

HENRIK STRAND DESIGN , TESTING AND ANALYSIS OF JOURNAL BEARINGS FOR CONSTRUCTION EQUIPMENT T surface − Tmeasured   d + 2 ⋅t   Qbush ⋅ ln  t    d  ⇒ T = T + π ⋅ w ⋅t  surface measured Am = λ ⋅ π ⋅ w bush   d + 2⋅t  ln     d  

Qbush = λbush ⋅ Am ⋅

(29)

The material combination governs the maximum temperature that can be allowed without resulting in thermal degradation. If the temperature gets too high in grease- lubricated bearings, the ultimate bearing resistance will be decreased with increased friction and wear as a result. The grease will separate and bleed out all oil, and only the soap will remain in the bearing. The soap will then be destroyed resulting in high friction and wear.

Figure 19. Heat dissipation. The pin and bushing does also shrink due to temperature differences. This is commonly used in production to fit bushings and pins. The diameter of the pin and bushing shrinks and expands according to the following equation:

∆d = d ⋅α ⋅ ∆T

(30)

Typical values of α are 12⋅10-6 [1/K] for steel, 18⋅10-6 [1/K] for bronze and 100⋅10-6 [1/K] for nylon plastics. It is important to have control over the temperature dependency since the magnitude of heat expansion is usually of the same range as manufacturing tolerances and bearing play. 2.1.16.

SURFACE ROUGHNESS

Surface roughness is an important factor to control wear and lubrication film formation. The surface roughness of both axle and bushing should be kept below 1µm Ra to control wear [5]. The requirement on the surface roughness properties is closely related to 22

HENRIK STRAND DESIGN , TESTING AND ANALYSIS OF JOURNAL BEARINGS FOR CONSTRUCTION EQUIPMENT material combination and surface treatment of the sliding surfaces. The roughness also influences the lubricant's ability to adhere to the surfaces. A very smooth surface may result in the lubricant not protecting the surface due to bad adhesion. A good way to characterize surface properties is to use the bearing area curve that is a cumulative height function. This curve is shown in figure 20 below.

Figure 20. Bearing area curve. The leftmost picture is a representation of a measured surface with exaggerated z scale. The graph in the middle is the height distribution and the graph to the right is the cumulative height distribution. 2.1.17.

WEAR

The Archard wear law is commonly used to model wear of the bushing and pin in a bearing. It is not certain that the equation models the wear in a bearing correctly, but it is used commonly anyway.

V F =K⋅ s H soft

(31)

For simulation purposes it is better to use the generalized Archard wear law that is produced by dividing equation 31 with the contact area A.  V K F  V K F = ⋅ ⇒ h = , k = , p = , s = v ⋅ t  ⇒ h = k ⋅ p ⋅ v ⋅t s ⋅ A H soft A A H soft A  

(32)

The difficulty in using these laws is the unknown coefficient of wear k which depends on many factors like pressure, velocity, temperature, surface roughness, materials, type and amount of lubricant. The coefficient has to be determined through testing. Wear mechanisms common to bearings are adhesive, oxidative and fatigue wear and they depend on the material combination and lubrication regime. Abrasive wear can be avoided by using good seals and sufficient lubrication.

23

HENRIK STRAND DESIGN , TESTING AND ANALYSIS OF JOURNAL BEARINGS FOR CONSTRUCTION EQUIPMENT

2.2. BEARING MATERIAL SELECTION It is very difficult to find material combinations that have ideal performance in every aspect of operation. Different bulk and surface properties have to be considered when selecting materials for journal bearing applications. 2.2.1. PINS Material properties: Suitable materials for pins are low alloyed carbon steels that can be hardened and tempered. The higher the carbon content the harder the pin will be when hardened. The problem with higher carbon content is that if a lockplate is welded to the pin there will be martensite remains if the pin is not welded at raised temperature. Also the fatigue life can be reduced by higher carbon content. To get as much performance as possible from the pin it should be hard on the surface and soft in the core. This is usually achieved by hardening processes. Through- hardened low temperature tempered pins are not recommended for dynamically loaded bearings due to brittleness. Thermal properties: Good heat transfer is required to lead away heat from the contact. Heat transfer is generally good for steel materials without nonmetallic coatings. Tribological and chemical properties: To achieve adequate surface finish, grinding of a hardened surface is generally required (Ra 0.2-1.0 µm). To protect from corrosive environment and decrease friction, surface coating or heat treatments can be used. Another reason for coated pins are fretting corrosion in the pin seats which can make disassembly difficult. Hard chromium coatings are common but have the disadvantage of a toxic process and cost. The advantage is a very hard surface of about 900 HV with very good corrosion resistance. To work properly hard chromium needs a hardened surface to adhere to. Other coatings used are low friction coatings like PTFE, graphite or molybdenum disulphide. These treatments can lower the coefficient of friction below 0.1 and work for short durations without lubricant but they are fairly expensive and relatively uncommon. The most common heat treatment is induction hardening with tempering which makes the surface hard while keeping the core soft. Induction hardening is commonly used for steels with carbon content less then 0.5%. Case hardening can be used if the material has carbon content below 0.3%. These steels can get a very hard surface but the yield strength is generally lower then for tempered steels. There are also nitriding processes like carbo nitriding, gas nitriding, plasma nitriding and nitro-carburizing that can be suitable for pins. These processes increase fatigue endurance, corrosion resistance and lowers the coefficient of friction. 24

HENRIK STRAND DESIGN , TESTING AND ANALYSIS OF JOURNAL BEARINGS FOR CONSTRUCTION EQUIPMENT Manufacturing and procurement properties: Pin prices generally consist of 50% material cost and 50% manufacturing cost. Steel prices and steel availability is a difficulty since not many steel mills provide tempered steels with low carbon content in large diameters. One important factor when buying round steel bars for pins is that the rate of reduction from slab to bar is high to avoid large slag inclusions and segregations. High rate of reduction also provides a good homogenization of the material. Another important manufacturing factor is hydrogen degassing that can influence life at static loads and also improves the purity of the steel. 2.2.2. BUSHINGS Many different bushing designs flood the world market. All claim superiority to the rest. The truth is that almost every bushing works for some type of application. Bushings could be divided into solid bushings and rolled strip bushings as illustrated in figure 21 below.

Figure 21. Solid bushing and rolled strip bushing with interlocking flanges. Solid bushings generally have thick walls in order to withstand deformation and provide long wear life. Solid bushings are mostly made of metals, but recently polymer composite bushings have been designed that are almost as good. There are also selflubricating ceramic bushings available, but they tend to break at high impact loads. Material properties: Steel alloy bushings are most commonly case hardened and can achieve hardness values of about 600 - 800 HV. This is in the same hardness region as induction hardened pins which implies that a chromium coating is needed to achieve a sufficient hardness difference of 100 HB if the bushing is regarded as the sacrificial part. One drawback of case hardening is geometric distortion since bushings are usually rather thin walled. Copper alloys used in construction equipment are copper-tin and copper-aluminum bronzes and some copper-zink brasses. In the past a lot of copper- lead alloys were used but since lead is not considered healthy, these are not so common anymore. Tin (Sn) bronze alloys are the most common bronze alloy with a tin content of about 8 14%. Hardness values of tin bronze are in the range of 90 - 130 HB. 25

HENRIK STRAND DESIGN , TESTING AND ANALYSIS OF JOURNAL BEARINGS FOR CONSTRUCTION EQUIPMENT Another tough bronze is aluminum bronze with an Al content of about 10% and some nickel (Ni) and iron (Fe). Typical hardness values are 140 - 230 HB. Manganese brass is also a used copper alloy for bushings. 25% Zink (Zn) and 10% aluminum (Al) and iron (Fe) and manganese (Mn) makes a very hard alloy with typical hardness values of 190 - 230 HB. Most copper alloys can not be successfully hardened. Some aluminum bronzes can be hardened but they turn hard and brittle. Recently some progress in hardening bronze with tin and nickel have been made and resulting in hardness values of about 275 - 375 HV without much embrittlement [15]. Thermal properties: Steel and copper alloys have very good thermal properties for leading away heat from the contact surface. Plastics are generally good insulators and can have difficulties with heat transport and their physical properties change rapidly with temperature. This can lead to problems with interference fit and deformation if the temperature dependence taken into account. Tribological and chemical properties: Steel sliding against steel is not a good tribological solution but is commonly used anyway. Steel bushings work well under heavy loads and slow sliding speeds with good lubrication. If good lubrication can not be provided different surface treatments can be added to prevent high friction and wear. Steel bushings have also fairly good resistance against abrasive wear. Uncoated or untreated steels have poor corrosion resistance. Bronze bushings are soft due to the high copper content. The copper content makes the alloy adhere to lubricants very well by forming grease like films with hydrocarbons that stick to the surface. Most coppers are fairly resistant to corrosion and work with most lubricants. To enhance dry rubbing performance sometimes solid graphite blocks are drilled into to the bronze bushing thereby forming a solid lubricant film. Manufacturing and procurement properties: Solid steel and bronze bushings are fairly easy to acquire, but the problem is to choose between all the different formulations that are offered. Some bronze alloys are difficult to turn due to the good lubricity of the material. Rolled bushings are generally thin walled and cheap to manufacture. They are often rolled of a metallic plate in steel or bronze lined with self- lubricated friction reducing material like polytetrafluoroethylene (PTFE), graphite or molybdenumdisulphide (MoS2 ). The disadvantage of rolled bushings is that they have a slot where the ends of the bushing meet. This slot has to be locked to avoid separation when the housing is loaded. From an assembly point of view they also present a problem since they need a special mounting tool to avoid bushing overlapping. Bushing overlapping is generally not a problem for small diameters.

26

HENRIK STRAND DESIGN , TESTING AND ANALYSIS OF JOURNAL BEARINGS FOR CONSTRUCTION EQUIPMENT 2.2.3. SEALS Seals are added to protect the bearing from dirt and contaminants. Usually only one set of seals are used per bearing but sometimes there are two sets. The outer seal is then to protect from dust, weather and operators cleaning with high pressure equipment. The inner seals function is to keep the grease from escaping the contact zone between the pin and bushing while preventing contaminations from entering. The most common seals used for oscillating bearings are o-rings, lip seals and scraper rings as shown in figure 22 below.

Figure 22. Example of seal cross sections. Material properties: Common materials for seals are different types of elastomers. Depending on design the elastomers should be stiffer or softer. A general requirement is to ensure the elastomer can handle the deflection in the bearing so there will not be a gap where grease can escape and contaminants can enter. On the other hand the elastomer needs to be stiff enough to ensure that the grease is only allowed to escape when it is being replaced by new grease. Thermal properties: The elastomers used in seals should be able to function in a temperature range of -40°C to +100°C. Tribological and chemical properties: The elastomers have to be compatible with oils and greases and also have little wear in dry contact with steels. The seals must also be resistant to a wide range of chemicals depending on the machine operating environment. Manufacturing and procurement properties: To avoid excessive wear it is important to have good surface finish on the axle. The design of the seal has to be adjusted for the application. Bearing deflection, contaminant level, grease consistency, frictional properties and period of relubrication all have influence on the sealing design.

27

HENRIK STRAND DESIGN , TESTING AND ANALYSIS OF JOURNAL BEARINGS FOR CONSTRUCTION EQUIPMENT 2.2.4. GREASE The recommended amount of grease required for adequate lubrication varies greatly in literature. Generally the stated amount is too large. Only a small amount of grease is really needed for sufficient lubrication of the bearing. The excess grease is mainly for pushing out contaminants and moisture. The frequent need for grease replenishment is often due to inadequate seals and bad bearing design. The trick of good bearing design is to keep grease from escaping the loaded area to the unloaded area of the bearing. The grease does not return to the contact surface when it is unloaded since grease only flows when it is loaded. To keep the surfaces lubricated at all times very sticky grease is needed. Very sticky grease cannot be replenished easily due to pumping resistance. Smooth and soft grease is easy to transport but does not adhere to the surfaces very well. The ideal grease is, of course, sticky grease that lubricates well and is easy to transport to the bearing. Since this is a contradiction other means have to be used. It is also very difficult to specify special grease for machines that are shipped worldwide since the grease properties vary greatly from continent to continent even for the same formulation and brand. Lubrication grooves and grease reservoirs are usually machined into the bushing. The grooves distribute the grease from the lubrication entry point along the sliding surface, while the reservoirs help maintain lubrication for longer periods of time. The edges of the grooves and reservoirs should be rounded to prevent the lubricant from being removed from the opposing surface. The grooves should also have a cross section large enough to ensure grease transport without having the grease bleed out any oil due to extrusion.

Figure 23. Examples of lubrication grooves and pockets. Stand-alone reservoirs in the bushing are a way to keep grease where it is needed. These reservo irs can have different shapes such as half spheres, holes and rhombs. The disadvantage of grooves is that they reduce the load bearing area if the grease is not pressurized and they are expensive to manufacture.

28

HENRIK STRAND DESIGN , TESTING AND ANALYSIS OF JOURNAL BEARINGS FOR CONSTRUCTION EQUIPMENT

2.3. FINITE ELEMENT ANALYSIS Finite element analysis can be used for contact studies in the bearing without regard to lubricant. Through computer simulations, the surface pressure distribution and contact area can be found for the given load and bearing geometry. The deflection, contact area, and pressure distribution of the bearing contact surface are important for wear modeling and prediction. With the help of finite element program a better approach to the axial and circumferential pressure distribution can be made. It is necessary to have good understanding of how this distribution of load is made since the bearing is often the force or displacement boundary condition for structural finite element calculations. A slight change in boundary condition can have very large impact on the stresses of the entire structure, as shown by Pettersson [16]. These contact calculations of the bearing surfaces are very time consuming to solve with high precision in finite element programs. If it is known how the bearing assembly distributes the load, a simplification of the bearing in the finite element model can be made to improve calculation times as is shown in the appended paper B. This simplification should only be used when the important results are some distance away from the bearing.

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HENRIK STRAND DESIGN , TESTING AND ANALYSIS OF JOURNAL BEARINGS FOR CONSTRUCTION EQUIPMENT

3. SUMMARY OF APPENDED PAPERS 3.1. PAPER A Paper A describes design of the test apparatus for heavy-duty grease-lubricated journal bearings. Characteristics of the apparatus were designed to simulate real operations of a journal bearing in the lifting frame of a wheel loader or excavator. The apparatus can vary normal force, oscillating frequency and angle of oscillation (i.e. sliding distance and velocity). The provided amount of grease and the geometry of the bushing and pin seat can also be varied. The collected output data include; normal force, friction torque, sliding velocity, oscillating angle, bearing temperature and structural stress. A photo of the bearing test apparatus is shown in figure 24. Two hydraulic cylinders induce both load and movement to the test sample that is positioned in the middle of the pivot arms. To provide the load, hydraulic pressure is built up on the positive side of both cylinders. The pressure can be varied up to a maximum of 300 bars, which results in a maximum force of 1.25 MN per cylinder. An open connection between the two lower sides of the cylinders creates an equal force in each. This reduces the need for constant replenishment of large volumes of pressurized hydraulic fluid for the cylinders.

Figure 24. Photo of the bearing test apparatus. 30

HENRIK STRAND DESIGN , TESTING AND ANALYSIS OF JOURNAL BEARINGS FOR CONSTRUCTION EQUIPMENT The two vertically standing hydraulic cylinders are connected by a yoke with the test object in the middle as can be seen in figure 25 below.

Figure 25. Detail of test apparatus with journal bearing test layout The pressure of lower sides of the cylinders manages the load on the bearing and the sliding speed and oscillation is regulated by alternatively pressurizing the upper sides of the hydraulic cylinders. The maximum applied load is 2.5 MN, maximum sliding speed is 10 degrees/s and the maximum oscillation angle is ±45 degrees. All of the bearing parts can be changed to test housing design, pin lengths and diameters etc. as can be seen in figure 26 below. The test apparatus can be seen in figure 4 above.

Figure 26. Detailed layout of grease-lubricated journal bearing test. Screw (1) Pin (2), pin holder with lock (3), scraper ring seals (4), bushing (5), bearing house (6) and pin seat (7).

31

HENRIK STRAND DESIGN , TESTING AND ANALYSIS OF JOURNAL BEARINGS FOR CONSTRUCTION EQUIPMENT

3.2. PAPER B Paper B is a study of the theoretical solution of the conformal contact of a pin in a hole compared to different finite element solutions. The goal of this study was to evaluate the possibility of speeding up contact calculations in finite element programs by using less calculation expensive elements than contact elements. This is mainly usable for large structures with many bearings. The idea was to replace the modeled pin by link and beam elements as can be seen in figure 27 below.

Figure 27. 3-dimensional setup with quarter models. This method provides fairly accurate estimates, at a low computational cost, especially at a distance from the loaded surface, if the correct stiffness values are used. These stiffness values can be found by trimming the stiffness values in 2-dimensional calculations with the theoretical solution or contact elements.

32

HENRIK STRAND DESIGN , TESTING AND ANALYSIS OF JOURNAL BEARINGS FOR CONSTRUCTION EQUIPMENT

3.3. PAPER C In Paper C a method of measuring bushing wear in situ without removing the bushing from the machine was evaluated. This method of measurement is made possible by using replica technique with cast plastic at room temperature. Some extra parts to prevent the plastic from escaping can be seen in figure 28 below.

Figure 28. Schematic procedure from mold assembly to finished cast replica. The resulting plastic replica can be seen in figure 29 below.

Figure 29. Cast replica of worn bushing. The results for the surface roughness were good but the form of the replica was distorted during curing. Unfortunately the form is the interesting part when investigating wear of large samples. Since this method can not accurately reproduce the surface without distortion, it is not a good way to get results if shape is interesting, but the results are good enough to get comparative answers to how the bearing is worn in field measurements. A great benefit from this method is that the worn surfaces are preserved in the cast for possible further analysis and comparison with other samples. 33

HENRIK STRAND DESIGN , TESTING AND ANALYSIS OF JOURNAL BEARINGS FOR CONSTRUCTION EQUIPMENT

3.4. PAPER D In Paper D the design of the journal bearing housing is studied. The idea was to find what parameters govern the pressure distribution between the pin and bushing utilizing a finite element model and applying statistical methods to evaluate the results. The parametric model can be seen in figure 31 belo w.

Figure 30. Left; Cross-section of the evaluated design with the designated parameters A (set ring thickness), B (set ring width) and C (effective throat measure). Right; a quarter of the complete bearing assembly used for FE-analysis.

Figure 31. Response surface of parameter A and B and their influence on the normalized contact pressure curve slope. The results show that the contact pressure distribution is most influenced by the height (A) and width (B) of the set ring, and that a fairly thin set ring provides the smoothest distribution. The influence of the weld size was shown not to be as important as believed initially. 34

HENRIK STRAND DESIGN , TESTING AND ANALYSIS OF JOURNAL BEARINGS FOR CONSTRUCTION EQUIPMENT

3.5. PAPER E In Paper E the results from paper D were validated in the test apparatus. Three housings with different set ring diameters were tested. The load cycle used for testing can be seen in figure 32 below.

Figure 32. Load cycle used for testing. A total of 9 bushings were tested in three different houses. The average wear results for the different houses are shown in figure 33 below.

Figure 33. Average wear curves for houses 1-3. From the wear curves in figure 33 it can be deduced that none of the housings have optimum design. Ho use 1 has a too compliant set ring and thereby inducing higher pressures in the center region. House 2 has low wear close to the center but a spike at the edge from a too stiff set ring. House 3 has high wear both at the center and on the edge. From these curves it can be deduced that the optimum housing should be somewhere between house 1 and 2 if the set ring cross section is kept rectangular.

Figure 34. Calculated pressure distributions. When comparing the pressure distributions from the finite element (FE) calculation shown in figure 34 with the accumulated wear curves, the likeness of the curves are quite good considering that the FE- model did not include a central grease groove. If a grease groove had been modeled, an increase of pressure close to the groove would be anticipated. The deviation in the edge region of the bushing is most likely due to coarse element mesh and how the load was applied in the FE- model.

35

HENRIK STRAND DESIGN , TESTING AND ANALYSIS OF JOURNAL BEARINGS FOR CONSTRUCTION EQUIPMENT

3.6. PAPER F In Paper F two different bushing grease groove designs and their influence on wear were studied and compared with a bushing without grooves. The bushings tested are shown in figure 35 below.

Figure 35. Bushing groove designs tested. No groove, X-groove and H-groove. The arrow shows the direction of the load and the dashed line shows the location of the measured profile. The wear results from testing of the different bushings are shown in figure 36 below.

Figure 36. Averaged wear curves for the different groove designs. From figure 36 can be observed that both the plain bushing and the H- groove bushing have less wear than the X-groove bushing. It was expected that the H-groove would be better then the plain bushing but not that the plain bushing would be better then the Xgroove bushing. Both the X-groove bushing and the plain bushing have a lot of wear particles that have deformed and stuck to the bushing surface. All bushings have transferred material to the pin in greater or lesser degree. The fact that the X-groove bushing have twice the wear in the region x = 4-16 mm can be from the fact that the grooves comes together and decrease the supporting area there. Another possibility is that the grease is dragged outside the groove in that region and no grease is available thereby generating more wear. The results from the tests show that the best groove for grease lubricated oscillating journal bearings are perpendicular to the sliding direction and that a groove with 45 degree angle to the sliding direction is no better than a bushing without a groove.

36

HENRIK STRAND DESIGN , TESTING AND ANALYSIS OF JOURNAL BEARINGS FOR CONSTRUCTION EQUIPMENT

3.7. PAPER G In Paper G a wear simulation of house 2 of paper E was made. A finite element model of the pin, bushing and housing was used to calculate pressure distributions for different loads and then used as input for wear calculation with the wear results from paper E. Then the model was updated and the pressure calculations were rerun in two steps. The results were then compared to the actual wear from the performed tests. The model used is shown in figure 37 below.

Figure 37. Left; Finite element model of the house, bushing and pin. Right; Detail of pin - bushing interface. The measured and calculated wear profiles are shown in figure 38 below.

Figure 38. Bushing wear curves. Measured wear compared to calculated wear in one step and wear calculated in three steps. The differences between the pressure distributions of the different worn bushings were not significant thereby validating the use of the linear wear equation. In figure 38 the difference between using a one step wear calculation and using three steps was minor except for the parts close to the central groove and the edge. The discrepancy from the calculated wear and the measured wear was probably due to neglecting the influence of initial lubrication of the bushing. Since the grease was introduced to the groove in the middle of the bushing before testing it is likely to assume that more grease was available in the center of the bushing thereby protecting the surface from wear. Assuming the edge region of the measured wear profile was unlubricated a larger coefficient of wear would be expected. 37

HENRIK STRAND DESIGN , TESTING AND ANALYSIS OF JOURNAL BEARINGS FOR CONSTRUCTION EQUIPMENT

4. DISCUSSION AND CONCLUSIONS The bearing design guide in chapter 2 should be used as a means to get to a starting point of design. The equations are compiled to be used as is. These guidelines were collected and presented, since most published bearing design guidelines are aimed at the bushing material or at continuously rotating bearings. The influence of housing, environment and load cases cannot be ignored when designing a bearing. Long term field-testing and experience cannot yet be replaced by theoretical reasoning until better formulations have been established. The vision when starting to compile these equations and guides was to serve as a tool for both original equipment manufacturer engineers and bearing suppliers to help them communicate and make better decisions about bearing design. Looking back at the research work performed a few conclusions can be made:

38

•

Finite element programs are a necessary tool to calculate accurate pressure distributions in the bearing.

•

Contact elements in finite programs can only be replaced by simpler elements if the region of interest is not too close to the bearing.

•

Replacing pins with beam and link elements in finite element analysis is a very fast method to get results for large structures with a lot of bearings but can be inaccurate if the element properties are not carefully chosen.

•

When constructing finite element models for calculation of pressure distributions and wear it is important to use small enough elements to get accurate results.

•

Roughness can be measured very accurately by plastic replicas.

•

Form of large objects is very difficult to measure using replica techniques since the replica is often deformed during curing.

•

Form is the most important factor when measuring wear over large specimens.

•

Plastic replicas of test surfaces are a good way to store worn surfaces for future reference.

•

Frictional heating can be a problem for the lubricants in the bearing even at slow sliding speeds if the intensity of oscillation and the pressure is high enough.

•

Grease grooves should be perpendicular to the sliding direction for best result.

HENRIK STRAND DESIGN , TESTING AND ANALYSIS OF JOURNAL BEARINGS FOR CONSTRUCTION EQUIPMENT

5. FUTURE RESEARCH Since this thesis has not resolved all issues about journal bearing design for construction equipment, there are a few important tasks to consider for the future: •

Some of the guidelines in chapter 2 need to be verified by simulations and testing before they can be fully trusted.

•

Find the correlation between field tests and test apparatus to be able to trust the results from bench tests and know what they mean in reality.

•

Influence of grease lubrication is only briefly touched in this thesis. The questions of ultimate lubrication interval, amount and type of lubricant to keep down wear and friction is a very important issue. Also to be able to calculate grease lubrication films is also a challenge.

•

Environmentally friendlier lubricants are also coming up as a market and government demand. The question of how environmentally friendlier lubricants can be used in construction equipment without degrading performance and becoming hazardous when they are used is a difficult question to solve.

•

Legislation about chromium plating is about to happen and which surface treatments to replace chromium with are something to consider. Hard chromium is not affected by the legislation yet, but since the market for ordinary chromium treatments will be reduced, hard chromium plating will be affected. To find out what surface treatme nts for pins that can be used instead is a question that needs to be solved to be able to use hard materials for bushings in the future.

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HENRIK STRAND DESIGN , TESTING AND ANALYSIS OF JOURNAL BEARINGS FOR CONSTRUCTION EQUIPMENT

6. REFERENCES [1]

Dowson, D., “A History of Tribology”, Professional Engineering Publishing, London, 1998.

[2]

Neale, M.J., “The Tribology Handbook”, Butterworth-Heinemann, Oxford, 1995.

[3]

Strand, H., “Apparatus for simulation of wear in heavy-duty oscillating journal bearings”, Proceedings OST – 99 Symposium on Machine Design, KTH, Stockholm, 1999.

[4]

Blake, A., “Practical Stress Analysis in Engineering Design”, Marcel Dekker Inc, New York, 1982.

[5]

Pilkey, W.D., “Peterson’s stress concentration factors”, 2nd-edition, John Wiley & Sons, New York, 1997.

[6]

Essinger, I., “Glidlager, Fettsmorda och Självsmörjande”, Johnson Metall AB, Örebro, 1989 (In Swedish).

[7]

Hogmark, S., Jacobson, S., “Tribologi – Friktion, smörjning och nötning”, Liber Utbildning AB, Stockholm, 1996 (In Swedish).

[8]

Persson, B.G.A., “On the stress distribution of cylindrical elastic bodies in contact”, Doctoral Dissertation, Chalmers University, Göteborg, 1964.

[9]

Beckman, S., “Approximate relationship between Vickers, Brinell and Rockwell hardness”, Volvo internal standard 1002,432 issue 4; http://www.tech.volvo.se/standard/docs/1002432.pdf

[10] SSAB Svenskt Stål, “Konstruera med Weldox- och Hardox-plåt”, SSAB Oxelösund AB, Oxelösund, 1991 (In Swedish). [11] Byggnevi, M., “LEFM Analysis and Fatigue Testing of Welded Structures”, Licentiate Thesis, KTH, Stockholm, 2005. [12] Sundström, B., “Allmänna tillstånd och dimensioneringskriteria, 6:e upplagan”, Kungliga Tekniska Högskolan, Stockholm, 1993 (In Swedish) [13] Bhushan, B., “Modern Tribology Handbook”, CRC Press, Boca Raton, 2001 [14] Ekroth, I., “Tillämpad Termodynamik”, Kungliga Tekniska Högskolan, Stockholm, 1994 (In Swedis h). [15] http://www.brushwellman.com/alloy/products/copper_nickel_tin/copper_nickel_ti n.asp [16] Pettersson, G., “Fatigue assessment of welded structures with non-linear boundary conditions”, Licentiate Thesis, KTH, Stockholm, 2005.

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HENRIK STRAND DESIGN , TESTING AND ANALYSIS OF JOURNAL BEARINGS FOR CONSTRUCTION EQUIPMENT

7. APPENDIX Tabulated values for equation 9 in chapter 2.1.3: α  tan   2

α

E ⋅∆ 2⋅q

d ⋅ pMAX 2⋅ q

α  tan   2

α

E ⋅∆ 2⋅q

d ⋅ pMAX 2⋅ q

0,020 0,040 0,060 0,080 0,10 0,12 0,14 0,16 0,18 0,20 0,22 0,24 0,26 0,28 0,30 0,32 0,34 0,36 0,38 0,40 0,42 0,44 0,46 0,48 0,50 0,52 0,54 0,56 0,58 0,60 0,62 0,64 0,66 0,68 0,70 0,72 0,74 0,76 0,78 0,80 0,82 0,84 0,86 0,88 0,90 0,92 0,94 0,96 0,98 1,0 1,1 1,2 1,3

2,29 4,58 6,87 9,15 11,42 13,69 15,94 18,18 20,41 22,62 24,81 26,99 29,15 31,28 33,40 35,49 37,56 39,60 41,61 43,60 45,56 47,50 49,40 51,28 53,13 54,95 56,74 58,50 60,23 61,93 63,60 65,24 66,85 68,43 69,98 71,51 73,00 74,47 75,91 77,32 78,70 80,06 81,39 82,70 83,97 85,23 86,46 87,66 88,84 90,00 95,45 100,4 104,9

1591 396,9 175,9 98,52 62,71 43,26 31,53 23,92 18,71 14,98 12,22 10,12 8,492 7,199 6,157 5,305 4,600 4,011 3,512 3,088 2,723 2,408 2,134 1,894 1,683 1,496 1,330 1,183 1,051 0,9319 0,8250 0,7284 0,6408 0,5612 0,4887 0,4225 0,3619 0,3063 0,2552 0,2081 0,1646 0,1244 0,0872 0,0526 0,0205 -0,0094 -0,0372 -0,0632 -0,0876 -0,1103 -0,2049 -0,2753 -0,3292

15,92 7,968 5,321 4,000 3,210 2,685 2,311 2,032 1,817 1,645 1,506 1,391 1,294 1,213 1,142 1,082 1,029 0,9824 0,9415 0,9054 0,8732 0,8445 0,8187 0,7956 0,7749 0,7561 0,7392 0,7238 0,7100 0,6974 0,6859 0,6755 0,6661 0,6575 0,6496 0,6425 0,6360 0,6301 0,6248 0,6199 0,6155 0,6114 0,6078 0,6045 0,6015 0,5988 0,5964 0,5942 0,5922 0,5904 0,5842 0,5811 0,5801

1,4 1,5 1,6 1,7 1,8 1,9 2,0 2,1 2,2 2,3 2,4 2,5 2,6 2,7 2,8 2,9 3,0 3,1 3,2 3,3 3,4 3,5 3,6 3,7 3,8 3,9 4,0 4,1 4,2 4,3 4,4 4,5 4,6 4,7 4,8 4,9 5,0 6,0 7,0 8,0 9,0 10 11 12 13 14 15 20 25 30 40 50 8

108,9 112,6 116,0 119,1 121,9 124,5 126,9 129,1 131,1 133,0 134,8 136,4 137,9 139,4 140,7 141,9 143,1 144,2 145,3 146,3 147,2 148,1 149,0 149,8 150,5 151,2 151,9 152,6 153,2 153,8 154,4 154,9 155,5 156,0 156,5 156,9 157,4 161,1 163,7 165,7 167,3 168,6 169,6 170,5 171,2 171,8 172,4 174,3 175,4 176,2 177,1 177,7 180,0

-0,3714 -0,4050 -0,4323 -0,4548 -0,4736 -0,4895 -0,5031 -0,5148 -0,5250 -0,5339 -0,5417 -0,5487 -0,5549 -0,5604 -0,5654 -0,5699 -0,5740 -0,5777 -0,5811 -0,5842 -0,5871 -0,5897 -0,5921 -0,5944 -0,5964 -0,5984 -0,6001 -0,6018 -0,6034 -0,6048 -0,6062 -0,6074 -0,6086 -0,6098 -0,6108 -0,6118 -0,6128 -0,6198 -0,6241 -0,6270 -0,6290 -0,6304 -0,6315 -0,6323 -0,6329 -0,6334 -0,6338 -0,6350 -0,6356 -0,6359 -0,6362 -0,6364 -2/π

0,5805 0,5817 0,5835 0,5856 0,5878 0,5901 0,5924 0,5946 0,5968 0,5988 0,6008 0,6027 0,6044 0,6060 0,6076 0,6090 0,6104 0,6117 0,6128 0,6140 0,6150 0,6160 0,6169 0,6178 0,6186 0,6193 0,6201 0,6207 0,6214 0,6220 0,6226 0,6231 0,6236 0,6241 0,6245 0,6250 0,6254 0,6286 0,6306 0,6319 0,6329 0,6336 0,6341 0,6345 0,6348 0,6350 0,6352 0,6358 0,6361 0,6363 0,6364 0,6365 -2/π

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HENRIK STRAND DESIGN , TESTING AND ANALYSIS OF JOURNAL BEARINGS FOR CONSTRUCTION EQUIPMENT

42