Dimensioning of a cutter wheel bearings

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Dimensioning of a cutter wheel bearings

Dimensionering av lagring till cutterhjul

Kebin Xie

Faculty of Health, Science and Technology

Degree Project for Master of Science in Engineering, Mechanical Engineering 30HP

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Abstract

Mobile Miner 40V is a machine used for rock excavation and developed by Epiroc. This machine is equipped with a large cutter wheel to perform the excavation. After a test run, some surfaces associated with bearings within the cutter wheel were found to be damaged due to scuffing - severe sliding wear. There is a static load applied to the surfaces due to gravity. However, the reason for this damaged issue was believed that there is a large dynamic load applied to the surfaces during the excavation. This dynamic load was not found in a previous FE model used to verify safety issues. Therefore, a new FE model that is more in line with reality, and a failure analysis were required. Additionally, a feasibility study for a cutter wheel with a larger dimension was also needed since a larger cutter wheel is desirable.

Firstly, wear mechanisms were reviewed, and some theories were chosen to analyze the damaged issue. Since it was unknown whether the surfaces were well-lubricated or not, both cases were investigated. The Archard wear equation was used to analyze the poor-lubricated situation, while the lubrication number and the Reynolds equation were used to analyze the well-lubricated case. Secondly, contact mechanisms between the surfaces were also investigated. The investigation of the contact mechanisms involved several theories, such as the Hertzian contact theory and the impact load factor. Besides these theoretical analyses, a numerical analysis was performed. Lastly, a new FE model was established in Ansys. Both the cutter wheel which was subjected to scuffing(existing cutter wheel), and the cutter wheel with a larger dimension(larger cutter wheel) were analyzed by the use of the new FE model.

The maximum and minimum wear rates obtained by the Archard wear equation are approximately 1.9・10-2_mm3_{/m and 4.8・10}-3_mm3_{/m, which are considered as a completely}

unacceptable level in engineering applications. The maximum and minimum critical loads obtained by the Reynold equation are approximately 1.8kN and 24.8kN, which both are larger than the static load applied to the surfaces. The maximum and minimum critical mean contact pressures obtained by the lubrication number are approximately 65MPa and 240MPa, which both are larger than the mean contact pressure generated by the static load. No evidence shows that there is a large dynamic load applied to the surfaces during the excavation. The largest possible contact pressure on the bearings in the existing cutter wheel is very close to the limit of severely damaged. The largest possible contact pressure on the bearings in the larger cutter wheel is believed to exceed the limit of severely damaged.

The previous assumption that the surfaces were damaged due to a large dynamic load was wrong. The obtained results support that the surfaces were only subjected to a static load and were damaged due to inadequate lubrication. The existing cutter wheel is operated safely with the current load cases. However, the forward thrust force is suggested to decrease when the cutting angle is large. There is a high risk if the larger cutter wheel is operated with the current load cases.

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Sammanfattning

Mobile Miner 40V är en maskin som används för bergavverkning och är utvecklad av Epiroc. Denna maskin är utrustad med ett stort cutterhjul för att utföra bergavverkningen. Efter en testkörning upptäcktes det att vissa ytor associerade med lagring i cutterhjulet skadats genom scuffing - allvarlig glidande slitage. Det finns en statisk belastning som verkar på ytorna på grund av tyngdkraften. Anledningen till det här haveriet ansågs dock bero på en stor dynamisk belastning som verkar på ytorna under bergavverkningen. Denna dynamiska belastning har inte påvisats i en tidigare FE modell som används för att verifiera säkerhetsproblem. Därför krävs en ny FE-modell som bättre simulerar verkligheten, såväl som en felanalys. Dessutom krävs det en genomförbarhetsstudie för ett cutterhjul som har en större dimension med hjälp av den nya FE modellen eftersom ett större cutterhjul är önskvärt.

För det första var slitmekanismer studerade, och några teorier valdes för att analysera de skadade ytorna. Eftersom det var okänt om ytorna var välsmorda eller inte, analyserades båda fallen. Archards nötningsekvation användes för att analysera det dåligt smorda fallet, medan ett “lubrication number” och Reynolds ekvation användes för att analysera det välsmorda fallet. Även kontaktmekanismer mellan de skadade ytorna analyserades. Detta involverade flera teorier såsom Hertzians kontaktteori och slagbelastningsfaktorn. Förutom dessa teoretiska analyser har en numerisk analys utförts. Slutligen etablerades en ny FE modell i Ansys. Både det cutterhjulet som utsatts för scuffing(befintliga cutterhjulet), och det cutterhjulet som har en större dimension(större cutterhjulet) har analyserats med hjälp av den nya FE modellen. De maximala och minimala nötningsgraderna som har erhållits med hjälp av Archards nötningsekvation är cirka 1,9・10-2_mm3_{/m och 4,8・10}-3_mm3_{/m, vilket anses vara en helt}

oacceptabel nivå i tekniska applikationer. De maximala och minimala kritiska belastningarna som erhållits med hjälp av Reynolds ekvation är cirka 1,8kN och 24,8kN, vilket i båda fallen är större än den statiska belastningen som verkar på ytorna. De maximala och minimala kritiska medelkontakttrycken som erhållits med hjälp av ett “lubrication number” är cirka 65MPa och 240MPa, båda är större än det kontakttrycket som genereras av den statiska belastningen. Det finns inga bevis som visar att det finns en stor dynamisk belastning som verkar på ytorna under bergavverkningen. Det största möjliga kontakttrycket på lagringen i det befintliga cutterhjulet är mycket nära gränsen för allvarlig skada. Det största möjliga kontakttrycket på lagringen i det större cutterhjulet tros överskrida gränsen för allvarlig skada.

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Acknowledgments

First, I would like to express sincere gratitude to Camatec Industriteknik AB and Epiroc

Rock Drills AB for offering this exciting project and allowing me to be a part of it.

Secondly, I want to thank Camatec Industriteknik AB again for letting me do my master thesis in their facility, as well as Epiroc Rock Drills AB, for showing me their mine in Kvarntorp, and to all the helpful staff members of the companies for showing kindness. Additionally, I would also like to take the opportunity to thank some people individually that have helped me throughout this project. Thank you, Joakim Bengtsson and Göran Karlsson, for providing the opportunity for me to do my master thesis work at Camatec

Industriteknik AB. I profoundly thank Peter Wigarthsson for continuous guidance from

the initial start to the end and patience of helping me. I also like to show my appreciation to Michael Olofsson and Jonas Andersson for providing support, exciting discussions, and advice. I also sincerely thank Fredrik Saf, Anita Sandström, and Robert for their data, knowledge, and genuine assistance.

Lastly, I want to say thanks to Anders Gåård and Jens Bergström from Karlstad University for valuable supervision and support during the project.

At the end of all, I want to give a special thanks to my co-worker William Adam Fagrell for working together with me in this project, and all his treasured wisdom and discussions.

Kebin Xie May 8, 2020 Karlstad, Sweden

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Nomenclature

P Contact pressure F Normal load

w Length of contact body or journal bearing d Diameter of contact body or journal bearing E Elastic modulus

ν Poisson's ratio

E’ Equivalent elastic modulus

R Radius of contact body or journal bearing R’ Equivalent radius

α Contact angle

Δ Diametric difference Δ > 0 ⇒ play

Δ < 0 ⇒ interference n Impact load factor

m Mass of object or contact body g Earth gravity

g = 9.81 [m/s2_{] in this paper}

h Falling height

δ Deflection of object or contact body Q Volume worn per unit sliding distance K Wear coefficient

H Hardness of material λ Lambda ratio

hl Fluid film thickness or radial clearances Ra Combined surface roughness

L Lubrication number U Sum velocity

U = U1 + U2

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ε Eccentricity ratio ε = e/c

e Eccentricity e = (hlmax - hlmin)/2

c Mean film thickness c = (hlmax + hlmin)/2

Fr Forward thrust force Fa Side load

Fmg Static weight of cutter head and support ring β Cutting angle

γ Disc angle

Qz Force parallel with disc cutter Qx Force normal to disc cutter M Bending moment of disc cutter

MM Mobile Miner

TBM Tunnel Boring Machine

BCC Body-Centered Cubic Structure

FCC Face Centered Cubic Structure

(E)HL (Elasto)hydrodynamic lubrication

ML Mixed lubrication

BL Boundary lubrication

LCC Lubricated concentrated contacts

IRG-OECD International Research Group on Wear of Engineering Materials

LH Left hand

RH Right hand

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1 Introduction

1.1 Background

Camatec Industriteknik AB is an engineering consulting firm with expertise in several

technical areas. One of their customers is the company Epiroc Rock Drills AB, which is one of the global leaders in the mining and infrastructure industries. They produce equipment, consumables, and services for use in applications such as mining, well drilling, infrastructure, and civil works. These companies have together identified two specific projects that are adapted to be master thesis projects for two students. These projects have their own goal and purpose, but a sizable proportion of them are cooperative, which means that the students need to perform some project works jointly. Both projects are centered on an existing machine within the field of mechanical mining equipment.

Rock excavation is a process that removes diverse types of rocks from boreholes or tunnels by four basic mechanisms - thermal spalling, fusion and vaporization, mechanical stresses, and chemical reactions [1]. Rock excavation is performed for two purposes. The first purpose is to create tunnels or networks of tunnels in which people and vehicles can be and travel through. The other purpose is to extract substances, minerals, or other objects from the rock being excavated. Before the industrialization age, rocks were excavated by either hand with primitive tools or use of black powder. With new inventions from the industrialization age, machine-driven devices were introduced, and black powder was replaced with dynamite [2].

The use of explosives causes vibrations and residual stresses in the rock around tunnels, which in turn increases the need for rock reinforcement to avoid the collapse of the tunnels. Because of this, there is a high market demand for mechanical mining equipment that is at least as effective as the use of explosives. To meet this need, Epiroc has developed a series of machines for mechanical rock excavation, in which the series is called Mobile Miner (MM). MM can be used in both tunneling and mining. In the application of tunneling, they have better flexibility compared to a Tunnel Boring

Machine, TBM, due to a reduced turning radius. In the application of mining, MM is a

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MM is equipped with a large cutter wheel that performs rock excavation. The cutter wheel is oriented either vertically or horizontally, which leads to different tunnel profiles. For example,MM 22H has a horizontally oriented cutter wheel that excavates a tunnel profile with a ceiling height of 2.2 meters. In comparison, MM 40V has a vertically oriented cutter wheel that excavates a tunnel profile with a ceiling height of 4.0 meters. However, the tunnel profile excavated byMM 40V has a relatively narrow width. Figure 1.1 shows the mentioned machines above.

Figure 1.1 The machines of the MM series. a) Mobile Miner 22H. b) Mobile Miner 40V. [3]

The cutter wheel used in MM 22H is the same as used in MM 40V, in that it aims to simplify the development.MM 22H was the first developed machine and got remarkable success in South Africa. MM 40V is the machine under development, and some failures within the cutter wheel are found after a test run. The failures are found at the same cutter wheel in MM 22H as well, but the damage is less severe.

1.2 Problem description

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Figure 1.2 These pictures were taken when the cutter wheel was demounted after the

test run. a) The surface damaged due to scuffing. b) The surface damaged by the metal chips. c) A pile of metal chips was found inside the cutter wheel after the test run. [4] Due to gravity, the damaged surfaces are subjected to a static load. However, the static load wasn’t believed to be large enough to cause scuffing. Meanwhile, a large dynamic load that occurs during the excavation was considered as the cause of scuffing. During the development of MM 22H, an FE model with certain assumed load cases was established to verify safety issues of the cutter wheel during rock excavation. The previous FE model didn’t expose the scuffing problem and the assumed dynamic load. Therefore, this verification is considered incorrect. It is still uncertain whether the previous FE model is incorrect, or the assumed load cases are incorrect. Thereby, a new FE model is desirable as well as a failure analysis based on the newly developed FE model to the existing cutter wheel. Besides, the assumed load cases need also to reconsider, in which this work is cooperatively performed with Fagrell, who is the co-worker of this cooperative project [5]. Moreover, Epiroc wants to know if the same bearings can still be used in a larger cutter wheel that is applied to the same load cases. A feasibility study of a larger cutter wheel is desirable because of this reason.

1.3 Purpose

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1.4 Goals

The project consists of one main goal and two subgoals. The achievement of the main goal requires that the subgoals are achieved first.

● The main goal of this project is to estimate if it is still safe to use the same bearings in a larger cutter wheel.

1. One of the subgoals is to determine the failure mechanism.

2. Another subgoal is to use the software Ansys to develop a new FE model that can analyze the stresses and deformations on the bearings. This FE model should have the ability to be adapted with different load cases and different dimensions of the cutter wheel.

1.5 Declaration of confidentiality

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2 Theory

2.1 Contact mechanism

Pressure is defined as force per unit area. If a load is applied to an infinitely small area, surface pressure would be infinitely large at this point. When two non-conforming surfaces meet each other, contact will initially occur at such an exceedingly small area. This small area could be a point or line depending on the geometry of contact bodies. It implies that this small area could be a location where has extremely high contact pressure, even though the load is moderate. However, the contact pressure is impossible to be infinite since the contact bodies will deform, initially elastically, to provide a large contact area to balance out the load.

When a cylinder meets a surface, the first contact area will be a strip. This contact type is also termed as line contact or non-conforming surfaces in contact. As mentioned in the previous section, the contact area will be increased due to deformation. Therefore, the contact pressure doesn’t only depend on the initial contact area and the load normal to it but also the elastic modulus of the contact bodies since they determine how much the contact area can be increased. Strand has summarized and presented four methods to calculate the contact pressure of such a case in his doctoral thesis [6]. The first method uses the projected area of the cylinder to calculate contact pressure. The maximum contact pressure obtained by this method is:

WhereP_max is the maximum contact pressure, F is the normal load, w is the length, and d is the diameter. The first method gives only an average pressure and doesn’t consider the high pressure in the middle of the deformed area. However, this method is still used commonly as a rule of thumb for calculation of bearings due to simplicity [6]. The second method uses a sinusoidal distribution to calculate contact pressure. The maximum contact pressure obtained by this method is:

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Where E’ represents an equivalent elastic modulus, and R’ represents an equivalent radius. The Hertzian contact theory is a classical theory of contact mechanics and is an especially useful tool for engineers and researchers, even though it has some limitations. One of them is that it is only valid when the contact angle is less than 20° [6]. Figure 2.1 illustrates the Hertzian contact pressure distribution and the contact angle.

Figure 2.1 Hertzian contact pressure distribution with a contact angle of 20°. [6]

The contact angle describes the size of the contact area in line contact and depends on load, geometry, and elastic modulus. These factors together form a value that is termed

quota. Each contact angle in a given system corresponds to a quota. Thereby, the contact

angle can be obtained by calculating the quota. The corresponding values of different contact angles have tabulated in the appendix. The quota is defined as:

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Where α represents the contact angle. Unlike the Hertzian contact theory, the method above is valid for all contact angles. But the material of contact bodies must have the same elastic modulus [6].

2.1.1 Impact load factor

When an object falls freely down from a certain height and hits another fixed object, meanwhile, the time of collision is short, the load applied to the fixed object will be much larger than the static load of the falling object. Therefore, engineers should not neglect this short-time applied load since it may lead to failures in engineering applications. This short-time applied load is also termed impact load.

Rather than perform complex dynamic analysis, an equivalent static analysis is preferred since it is much easier. Dattakumar and Ganeshan [7] have presented a method of such a simple equivalent static analysis in their master’s thesis. In this equivalent static analysis, the static load is multiplied by an impact load factor to obtain a reasonable estimate of a corresponding dynamic load. Further, this approximate dynamic load can be used to calculate the corresponding maximum dynamic deflection and stress.

The impact load factor depends on the collision speed and the elasticity of the impacted object. However, for the case of a free-fall, the relation can be extended and expressed as that it depends only on the falling height and the static deflection. Figure 2.2 illustrates such a case of free-fall.

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The equivalent dynamic load and the impact load factor could be obtained by:

WhereFdynamic is the dynamic load, n is the impact load factor, m is the mass of the falling object, g is the earth gravitational acceleration, h is the falling height, and δ is the static deflection. Equation 7 is a simplified version of the dynamic impact load factor in which it is derived just for the free-fall case and doesn’t consider modal vibrations [7].

2.2 Sliding wear

When there is a relative motion between two solid surfaces, substance loss may occur progressively on the surfaces, even though the surfaces are lubricated. If this process doesn’t involve the presence of hard particles, it is considered as sliding wear [8]. Otherwise, it is considered as abrasive wear. The hard particle could be a separate component between the surfaces or an element of the structure of one or both surfaces. Scuffing is defined as severe sliding wear and occurred in inadequately lubricated metallic surfaces. The scuffing can cause observable changes in the surface texture with features related to the direction of relative motion.

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2.2.1 Asperities junction & Adhesion

A solid surface that is truly smooth doesn’t exist. Even in the highly polished surface, unevenness is found on the microscopic level. This unevenness consists of many rugged projections that are termed as asperities in material science. When two solid surfaces are brought close together gradually, the contact between them will initially occur on some of the asperities that are relatively higher. At that moment, the real contact area covers only an exceedingly small portion of the nominal area [8].

Since the real contact area is so small, a small load is already large enough to deform the asperities elastically [8]. Once a load applies to the contact surfaces, the asperities become flattened due to the deformation and formed so-called asperities junctions. When the load increases, the asperities junctions will keep growing until they can support the applied load. This growth implies that the real contact area doesn’t depend on the nominal area, but the applied load and the surface hardness. Figure 2.3 illustrates the asperities and asperities junctions.

Figure 2.3 a) Illustration of a solid surface that has four asperities. b) Illustration of the

forming of asperities junctions.

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When a tangential load is applied to the contact surfaces, the asperities junctions experience additional shear stress [8]. In order to support this additional load, the asperities junctions must grow tangentially to obtain a larger contact area. However, the growth of the asperities junctions has a limit in reality. Once the tangential load overcomes the shear strength of materials, the asperities junctions will be broken and result in a fracture. During the shearing, some debris particles are formed and taken away by the counter surface. This process is the mechanism of adhesive wear. The fracture occurs more likely on the surface with lower hardness. Figure 2.4 illustrates a representative example.

Figure 2.4 Cracks are developed at the red lines and formed debris particles at the peak

of the asperities.

There are several models regarding the sliding wear mechanism, which differ in the detailed process of how the debris is formed [8]. The debris may not be formed as lumps or fragments from asperity peaks, like the example shown in Figure 2.4. However, it is no doubt that asperity contacts can lead directly to the formation of wear debris.

2.2.2 Archard equation

In order to describe the unlubricated sliding wear, Archard and Holm have introduced a simple theoretical analysis model [8]. The analytical model is based on the asperities contact theory, which was introduced briefly in Chapter 2.2.1. The assumption of this model is that the local deformation of the asperities is plastic. This model associates the quantity of wear with the real contact area, in which the real contact area is, in turn, associated with the normal load and the surface hardness. Eventually, Archard wear

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Where Q represents volume worn per unit sliding distance, K represents a constant, F represents the normal load, and H represents the lowest surface hardness of the contact surfaces. Since F/H represents the real contact area, and K is a constant, the volume worn per unit sliding distance is actually a function of the real contact area. A large real contact area implies more severe wear occurs in a unit sliding distance since the debris particles are formed only on the surfaces where they are truly in contact. In other words, the volume worn per unit sliding can’t exceed the real contact area. More specifically, K isn’t allowed to be larger than 1. Therefore, K could be interpreted as the probability in which debris particles are formed when an asperities junction is broken [8].

K is termed as wear coefficient and is a property of a specific sliding system, not a

property of a specific material [8]. In other words, the wear coefficient depends on the operating conditions and could be changed if the operating conditions are changed. Many parameters are able to, more or less, direct or indirect, affect the operating conditions. For example, when the normal load is over a critical limit, the wear coefficient will exhibit an increase [8]. Steels with higher hardness are intended to have a lower wear coefficient [9]. A high temperature can change the wear coefficient indirectly by changing the generation rate of the oxide layer and can change even the hardness when the temperature is sufficiently high [8,10]. High sliding speed increases the local temperature, thereby changing the wear coefficient indirectly [8,10]. Surface roughness could have a strong influence on the dry contact [11]. Vibration may have either a positive or a negative effect on the wear coefficient, depending on the material of the sliding surfaces [12,13]. All the above mentioned are the examples of how the operating conditions affect the wear coefficient.

As a summary, the wear coefficient isn’t a constant value but a possible value in the range of adhesive wear rate for a given sliding system. Since many parameters can affect the wear coefficient, most of which are interrelated, sliding systems with the same material combination may have a different wear coefficient. There is a theoretical approximation model that is established to predict the wear coefficient for a given sliding system. Still, this model has shown significant discrepancies compared with published experimental values [14]. This failure implies a great difficulty in predicting an accuracy wear coefficient for a given sliding system rather than using experimental values. However, the Archard wear equation is still a passable tool to get a good-enough approximation when the operating conditions in a given sliding system are nearly unchanged. Usually, it is so high as to be completely unacceptable in engineering applications when the wear rate is up to 10-2- 10-3mm3_{/m [8,10,15]. Even for}

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2.2.3 Lubrication regimes

Lubrication is probably the most effective way to reduce friction and wear for a sliding system. Lubrication reduces the sliding wear by introducing a layer of material to prevent the asperities contact completely or partially [8]. Fluid lubrication is one of the lubrication types that is widely used today. In some circumstances, the fluid lubrication creates a lubricant film to prevent the asperities contact. Depending on the circumstances, the lubricant film is divided into two types. When there is a relative motion on the surfaces of fluid lubrication, a hydrodynamic pressure within the lubrication is generated, and then this pressure further withstands the normal load. At that moment, a fluid film is created to prevent the asperities contact. When the fluid lubrication contains a suitable additive or lubricant, the molecules of them may lay down and get absorbed in the asperities. These absorbed molecules create an adsorbed

molecular film that can prevent the asperities contact when the fluid film doesn’t

present. The two types of lubricant film can exist at the same time.

Depending on how the normal load is carried, three lubrication regiments can be distinguished - full film lubrication which is also known as (elasto)hydrodynamic

lubrication, mixed lubrication, and boundary lubrication [8]. In (E)HL regime, the normal

load is entirely carried by the fluid film, which leads to a complete separation between the surfaces. Thus, the asperities contact could be neglected since the thickness of the fluid film is greater than the asperity heights. In ML regime, the normal load is partially carried by the fluid film and partially carried by the asperities, which leads to a partial asperities contact. In BL regime, the normal load is entirely carried by the adsorbed molecular film.

As a summary, (E)HL regime is the ideal regime since the wear rate of the sliding surfaces could be neglected before (E)HL begins to break down. In order to determine if (E)HL is beginning to break down, a lambda ratio is used for determination [8]. The lambda ratio is:

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The asperities contact occurs when λ < 3, but severe wear doesn’t need to occur directly. The adsorbed molecular film can protect the asperities if this film exists and isn’t penetrated. Otherwise, unlubricated sliding is dominant at the contact surfaces. As mentioned in Chapter 2.2.2, for an unlubricated sliding, wear is dependent on the wear coefficient and the real contact area. As a summary, the breakdown of (E)HL regime is the necessary condition for the occurrence of scuffing, not the sufficient condition. Simply put, scuffing isn’t 100% sure will occur when (E)HL regime is broken down [16]. Contact occurs between two lubricated non-conforming surfaces is termed as

Lubricated concentrated contacts, LCC. The lubricant film is probably not sufficient to

support the large pressure that occurred under LCC, thereby scuffing is more likely to occur under LCC. International Research Group on Wear of Engineering Materials, IRG-OECD has studied the failure by scuffing of LCC extensively, and one of their results is IRG transition diagram [17]. This diagram plots the normal load against the sliding velocity and distinguishes three regions in it. Figure 2.5 illustrates the mentioned diagram.

Figure 2.5 IRG transition diagram is roughly divided into three regions, I, II, and III. [17]

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decreased over time, and the contact is restored from adhesive wear to mild wear. However, mild wear may already be unacceptable under certain circumstances. Thus, Region II should also be avoided for any engineering applications. Region I is considered to be a safety region where the wear could be neglected [8, 17]. The effect of fluid film in this region isn’t neglectable, and the dominated lubrication regime is therefore defined as partial (E)HL.

However, more recent researches show that the scuffing may occur in Region I under certain circumstances [16, 17]. This research means that Region I is no longer considered to be a completely "safe" region. Moreover, this research shows also that the lubrication regime of when scuffing occurred in Region I is ML rather than (E)HL or BL [17]. Therefore, the assumption that the failure occurs when the normal load is entirely carried by the asperities contact is doubtful and probably wrong. Based on these mentioned results above, this research suggests that Region I can be subdivided into three subregions depend on the lubrication regimes, i.e. (E)HL, ML, and BL [16, 17].

2.2.4 Lubrication number & Reynolds equation

Lubrication number is a parameter used to describe the condition of lubrication in a

sliding system. Schipper and de Gee relate the lubrication number with the friction coefficient in their article [17]. This research implies that the lubrication number can be used to determine the lubrication regime since the friction coefficient usually increases when the fluid film is broken down. The relation between the lubrication number and the friction coefficient is presented in a Stribeck-type diagram shown in Figure 2.6.

Figure 2.6 Generalised Stribeck curve. [17] Reprinted with permission of John Wiley &

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In this diagram, the friction coefficient is considered to be a function of the lubrication number. The corresponding friction coefficients in this diagram were obtained by experiments based on lubricated AISI-52100 steels. The setting of experiments means that the friction coefficients in the diagram can't be used as general data to all material. The curve in this diagram is a monotonically decreasing function. In general, this curve includes three slopes, and thereby, three intervals are identified and corporated with the three lubrication regimes, i.e. (E)HL, ML, and BL. The lubrication number is determined by lubricant viscosity, sliding velocity, contact pressure, and surface roughness. For specifically, the lubrication number is defined as:

Where η is the inlet lubricant viscosity, U is the sum of sliding velocity, Pmean is the mean pressure of the contact surfaces, and R_a is the surface roughness that combined with the contact surfaces. For a line contact, Pmean is equal to:

WhereP_max is the maximum contact pressure. The transition between (E)HL regime and ML regime is of practical interest since the scuffing may occur already in Region I. But, this transition isn’t sharp. There isn’t a significant critical limit to distinguish between the two regimes. This phenomenon is actually consistent with physics since the fluid film will not collapse suddenly but collapse gradually. However, it is still possible to assume a critical limit as an approximation.

Reynolds equation is a partial differential equation used to calculate the pressure

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Figure 2.7 A plain journal bearing is steadily loaded by a normal force during rotating.

[8]

The journal is rotated eccentrically within the bearing due to the normal load and causes differences to the thickness of the film. This change in thickness generates pressure within the film. The normal load carried by this pressure is:

Where F is the normal load, S is a dimensionless number which is termed Sommerfeld

number, η is the viscosity of the film, U is the peripheral speed of the journal, R is the journal radius, c is the mean film thickness, and w is the bearing length. The mean film thickness is equal to half of the sum of the maximum film thickness hl_max and the minimum film thickness hlmin. The Sommerfeld number depends on if the bearing is short or long. For a short bearing with w/D < 1/3, the Sommerfeld number is:

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2.3 Rolling wear

Rolling contact becomes a major contact type in nowadays engineering applications due to the feature of low friction. Therefore, the wear of rolling becomes also a major concern in engineering design. Rolling wear is actually not a wear mechanism but a phenomenon that results from several material removal mechanisms. The major wear mechanisms of the rolling wear are known as abrasive wear, corrosive wear, and adhesive wear [18]. For a rolling contact, the contact area is divided into two regions

-Stick and Slip [19]. Stick is the region where no relative motion between the contact

surfaces occurs. Pure rolling has only the stick region. Slip, on the contrary, is the region that has relative motion between the contact surfaces. Slip ratio is usually used to denote the proportion of the two regions during a rolling contact. Figure 2.8 illustrates the difference between the two regions.

Figure 2.8 The contact area of a ball that is rolling toward the left. The stick region is

painted with grey color while the slip region is painted with white color.

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2.3.1 Cylindrical roller bearings

There are many types of bearings used in nowadays engineering applications, but rolling element bearings are sure one of the common forms. This type of bearing places rolling elements, such as balls or rollers, in between two rings, which are known as races. Since the rolling friction coefficient is well-known much smaller than the sliding friction coefficient, this type of bearing can carry a larger normal load. Compared with the ball bearings, the cylindrical roller bearings have a large contact area. Thus, the cylindrical roller bearings can carry a larger load, but instead, have a limit on the operating speed.

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3 Method

3.1 Problem analysis

First of all, an analysis of the whole cutter wheel was performed to understand working mechanisms within the cutter wheel. With this obtained knowledge, an equivalent and straightforward tribo-system was further established for the scuffing analysis. Various theories were used to create certain criteria that determine the minimum load that the tribo-system can withstand before the occurrence of scuffing. Furthermore, an analysis of the bearings was performed to determine all components and parts that are interacted with the bearings under operating. Based on this obtained knowledge, a simplified model was established. This simplified model was used for the FEM calculation later on. Lastly, the assumed load cases were also established cooperatively with the co-worker.

3.1.1 Cutter wheel analysis

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Figure 3.1 a). The whole cutter head. b). A close look at the disc cutters with different

configurations.

The drive unit consists of a support ring, a bearing housing, a hydraulic motor, and a

transmission unit. The cutter head is bolted with the support ring, while the support ring

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Figure 3.2 Drive unit. a). In LH, the hydraulic motor and the bearing housing could be

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Figure 3.3 a). The cutter wheel is able to turn right and left with ±25°. b). The cutter

wheel is able to lift and lower with 26.5° and -6.5°. [24]

For the convenience of description, a coordinate system was established for the cutter wheel. In this coordinate system, 3 o'clock is defined as the forward direction, and 6 o'clock is defined as the ground. The loads applied to the cutter wheel during excavation include a forward thrust force Fr, a side load Fa, a static weight Fmg, and a rotation torque. The rotation torque was neglected since this torque doesn’t apply to the bearings but the hydraulic motor. Figure 3.4 illustrates the mentioned loads together with the coordinate system of the cutter wheel.

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3.1.2 Scuffing analysis

The surfaces that are damaged by scuffing are located at Component A and Component B. As mentioned in Chapter 1.5, more details of Component A and Component B aren’t allowed to present. However, the equivalent and straightforward tribo-system was established and allowed to present. Since both the damaged surfaces and components are quite identical, the same model was used to analyze the failure mechanism. This model is named Two-rings case and shown in Figure 3.5.

Figure 3.5 The red circle indicates the surfaces that were damaged by scuffing. a) The

cross-section of the model. b). The side view of the model.

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ring, but the surface within the red circles is more severe. The other damaged surfaces are believed to be second-hand damaged by the metal chips created by the scuffing surfaces. Thus, the failure mechanism isn’t the same for all damaged surfaces. Thereby, the failure analysis was focused on the scuffing surfaces.

There is a lubrication system within the bearing housing used to reduce the friction and wear in the bearings under operating. Mobil SHC 630 is the lubricant oil used in this lubrication system. The bearings are temperature-controlled by the lubricant oil so that the temperature of the bearings is kept at 50° during excavation. Thus, the temperature effect was neglected in this analysis. The lubricant oil flows into the bearing housing from top to bottom without any pressure. The "top to bottom" doesn’t mean from 12 o'clock to 6 o'clock. In fact, the scuffing surfaces on the top aren’t directly lubricated. The yellow ring is rotated clockwise with 7.5 rpm while the green ring is fixed. Table 3.1 shows the information about the making materials of the rings.

Table 3.1 Some mechanical properties of the materials

Yellow ring Green ring

Material S275JR S355JR62

Elastic modulus [GPa] 210 210

Poisson's ratio 0.3 0.3

Hardness [HB] 121 - 163 150 - 190

Mass [kg] 156.25 N/A

Surface roughness [μm] 0.6 - 0.8 1.6 - 6.3

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● Assumption One: The surfaces were poorly lubricated or unlubricated and subjected only to the static weight of the yellow ring. In this assumption, the Archard wear equation was used to calculate the wear rate. Equation 8 was related to this calculation. Table 3.2 shows the parameters used in this calculation.

Table 3.2 The wear coefficient K, the normal load F, and the hardness H

The parameters Values

K 0.5 ・10-2 - 1.5・10-2

F [N] 1532.8

H [MPa] 1186.6 - 1598.5

The hardness used in the calculation is the hardness of the yellow ring since it has a lower hardness. The exact value is unknown, but according to the supplier, the hardness is varied between an interval shown in Table 3.2. Therefore, it is necessary to consider the maximum and minimum value of hardness. S275JR is mild steel, while S355JR62 is micro-alloyed steel, but the chemical composition is similar for both materials. Thereby, the wear coefficient of this tribo-system can be considered to be a similar case of “mild steel on mild steel”, which has a wear coefficient of 1.5・10 -2 _{[14]. Since} S355JR62 is

harder than mild steel, the wear coefficient in reality for this tribo-system may be lower. 0.5・10-2 is a guess value for this assumption.

● Assumption Two: The surfaces were well-lubricated and subjected only to the static weight of the yellow ring. In this assumption, the lubrication theory was used to calculate the critical load, since the lubricant film is presented. Two theories were found to calculate the critical load.

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Table 3.3 All input parameters for the calculation associated with the Reynold equation Parameters Values λ 3 Ra [μm] 1.7 - 6.35 hlmin [μm] 5.1264 - 19.0518 hlmax [m] 1.99487・10-3 - 1.98095・10-3 e [m] 9.94874・10-4 - 9.80948・10-4 c [m] 0.001 ε 0.994874 - 0.980948 w [mm] 27.6 d [mm] 1930 R [mm] 965 S 7.76612 - 0.559257 ρ [g/cm3_] _0.87 v [mm2_/s] _188.08 η [Pa・s] 0.16363 U [m/s] 0.353691 R 965 mm

λ < 3 is the necessary condition that scuffing will occur, but not sufficient condition.

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2. The lubrication number theory was used to calculate the critical mean contact pressure. Besides, the four methods that are presented by Strand were used to calculate the maximum contact pressures. After that, the contact type was assumed as line contact, and the maximum contact pressures were converted to the mean contact pressures by using Equation 11. Furthermore, the critical mean contact pressure was compared to the mean contact pressures. Equation 1, 2, 3, 4, 5, 10, and 11 were related to this calculation. Table 3.4 shows the parameters used in this calculation.

Table 3.4 All input parameters for the calculation associated with the lubrication

number Parameters Values η [Pa・s] 0.16363 U [m/s] 0.757909 Ra [μm] 1.7 - 6.35 μm L 0.0003 F [N] 1532.8 w [mm] 27.6 d [mm] 1930 E’_[GPa] _115.385 R’_[mm] _932.19 Δ [mm] 2 Quota 3781.28 α [°] 1.48701

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● Assumption Three: The surfaces were well-lubricated and subjected to a dynamic load. The dynamic load was assumed to be an impact load created by the yellow ring. More specifically, the yellow ring was assumed to move up and down due to some external forces during excavation, and collide the surfaces so that the damage occurs. Equation 6 and 7 were related to this calculation. Table 3.5 shows the parameters used in Equation 2.6 and 2.7.

Table 3.5 Static deflection and falling heights

Parameters Values δstatic [mm] 3.0・10-4

h1 [mm] 0.05

h2 [mm] 0.5

h3 [mm] 2

The static deflection was obtained by numerical analysis since it is difficult to obtain by an analytical analysis. The detailed description of the numerical analysis would be presented in Chapter 3.2. The falling heights were unknown since they are also difficult to obtain by an analytical analysis. However, the maximum falling heights were believed to be less than 2mm. Thereby, some guess values between 0mm and 2mm were assumed to use in this calculation. In order to investigate the real movement of the yellow ring, one sensor was mounted on the surface of the green ring at 6 o'clock to collect the change in distance between two rings over time.

3.1.3 Bearings analysis

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Figure 3.6 This model is the result of the simplification of the original CAD model. a).

The cross-section shows the details of the included components, which are distinguished by different colors. b). The side view shows the overall picture of the

simplified model.

The grey component represents the support ring and is bolted together with the drive plate, which is represented with red color in the figure. The blue component represents

outer ring, which is actually the raceway of both the radial bearing and the axial

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3.1.4 Load cases analysis

As mentioned in Chapter 3.1.1, four hydraulic cylinders steer the movement of the cutter wheel. The movement in different directions is controlled by different hydraulic cylinders and is independent of each other. In order to avoid confusion with the coordinate system in the cutter wheel, a coordinate system was established for the steering carriage. This coordinate system is assigned as the global coordinate system and is denoted with x-y-z. Figure 3.7 shows the steering carriage and the global coordinate system.

Figure 3.7 The hydraulic cylinders that control different directions are distinguished

with three different colors.

The blue hydraulic cylinder controls the movement in the x-direction, i.e., forward and backward. The red hydraulic cylinders control the movement in the y-direction, i.e., right and left. The yellow hydraulic cylinder controls the movement in the z-direction, i.e., up and down. The forward thrust force provided by the blue hydraulic cylinder is denoted as GFTF, Global forward thrust force, and it is always parallel with the x-direction in the global coordinate system. The earlier mentioned forward thrust force

Fr is the reaction force subjected to the cutter wheel when the cutter wheel is pushed

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Figure 3.8 The steering angle in the x-y plane was also named as cutting angle and is

denoted with β. [24]

Fr is the largest when the cutting angle is zero. Fa is, in contrast, the smallest when the

cutting angle is zero. In cutting mode, the cutter wheel is kept still sideways and vertical while being pressed with the full operational stroke of the blue hydraulic cylinder forward. In this way, the reaction forces could be reduced minimum. Otherwise, Fr and

Fa aren’t only the force components of GFTF but also the force components of the global

side load. Epiroc has obtained the maximum allowable Fr and Fa by the other calculation model and summed them up in a document. However, they weren’t verified. The assumed load cases used in the FEM calculation were developed based on this document. The detailed description of the development of the assumed load cases was presented by Fagrell [5]. Due to the confidentiality of this document, the values of these loads wouldn’t be allowed to present. Therefore, this paper was focused on the analysis of how the contact pressure on the bearings is affected by the cutting angles, the configurations, and the size of the cutter wheel.

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Figure 3.9 The angle between the cutter wheel and the disc cutter is named as disc

angle and is denoted with γ.

When Fr and Fa are applied to a disc cutter, these forces provide a bending moment around the axle of this disc cutter. This bending moment is denoted with M and has to be considered. M is dependent on Fa, Fr, and the cutting angle. In order to simplify the calculation of M, a new coordinate system was established for the disc cutter. This disc cutter coordinate system is denoted with Qx and Qz. Qx is defined as the force paralleled with the disc cutter, while Qz is defined as the force normal to the disc cutter. In this coordinate system, only Qz provides the bending moment. Table 3.6 shows the ten configurations and their disc angles.

Table 3.6 The names of the configurations and the corresponding disc angles

Configurations Disc angles [°] Configurations Disc angles [°]

LH-C1 90 RH-C1 90

LH-C2 105.82 RH-C2 74.18

LH-C3 121.78 RH-C3 58.22

LH-C4 137.38 RH-C4 42.62

LH-C5 153.02 RH-C5 26.98

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This situation was defined as the worst-case, and the assumed load cases were based on this situation. Due to the limitation of time, only the worst cases were calculated in this project. However, the new FE model can calculate the case that has more than one disc cutter in contact with the rock.

3.2 Numerical analysis of the two-rings case

The model of the two-rings case was exported directly from Creo to Ansys without any modification. The materials in the FEM calculation were assigned as pure elastic materials with an E-modulus of 210GPa and Poisson's ratio of 0.3. There was no Force applied to the rings, while the static load was generated by applying Standard Earth

Gravity in the model. Since the gravitation in the Standard Earth Gravity wasn’t allowed

to change, the mass of the yellow ring was instead altered to generate the impact load. Figure 3.10 shows how the boundary conditions and mesh sizes were assigned in this model.

Figure 3.10 a). The boundary conditions were assigned at the surfaces that

distinguished in different colors. b). The global mesh size was coarse, but the mesh size of the contact surface was fine.

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Table 3.7 The load cases and their gravitational accelerations and mesh sizes Gravitational accelerations Mesh sizes [mm] Case 1 1・g 2 Case 2 5・g 2 Case 3 10・g 3 Case 4 20・g 3 Case 5 60・g 3 Case 6 120・g 3

Case 1 represents the case of static load, while Case 6 represents the case of 2mm falling height. The reason that 2mm wasn’t the mesh size used for all calculations was that the calculation time became much longer when the gravitational acceleration is larger than 10g.

3.3 New FE model

The simplified model was exported directly from Creo to Ansys without any modification. The materials used in the cutter wheel are various steels with different mechanical properties. However, this project doesn’t consider the plastic deformation problem on the other components except for the bearings. The difference in the yield strength of various steels is, therefore, not considered in this project. Since the density, elastic modulus, and Poisson’s ratio of the most steels is similar, it is reasonable to assign the same material properties for the whole model. Figure 3.11 shows the material properties assigned in the new FE model.

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The materials in the new FE model were assigned as pure elastic materials with an E-modulus of 210GPa and Poisson's ratio of 0.3. Since many components and parts were deleted due to the simplification, Fmg was less than in reality. In order to compensate for this difference, the density was increased from 7850kg/m3_{to 9656kg/m}3_.

In reality, the cutter head and the drive plate are bolted with the support ring. However, all small machine elements such as bolts and screws were deleted due to the simplification. The bolts were assumed to have infinite strength so that the sliding and separation of the contact surfaces would never happen. Bonded is a contact type that restricts sliding and separation on the contact surfaces, i.e., glued together [25]. Therefore, Bonded was chosen as the contact type between them. Moreover, Bonded was also chosen as the contact type between the support ring and the outer ring for the same reason. Since they are interference fit with each other in reality. The Bonded surfaces are treated as an entire in Ansys so that the penetration between the Bonded surfaces is impossible to occur [26]. In other words, the contact pressure doesn’t exist on such surfaces. Thereby, Bonded wasn’t the contact type between the bearings and the outer ring. Frictionless is a nonlinear contact type that can model gaps and more accurately model the true area of contact [25]. Therefore, Frictionless was instead chosen as the contact type between the bearings and the outer ring. Additionally, Adjust

to Touch was chosen as the option of Interface Treatment for the Frictionless contact

type to reduce the initial penetrations/gaps problem [27]. Lastly, Program Controlled or

Default was used for the other options that weren’t mentioned. Table 3.8 summarizes

the contact type of surfaces in the new FE model.

Table 3.8 Contact surfaces and their contact types

Contact surfaces Contact types Segments - Support ring Bonded Support ring - Outer ring Bonded Support ring - Drive plate Bonded

Radial Bearing - Outer ring

Frictionless

Axial Bearings - Outer ring

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The use of Frictionless may have a convergence problem due to the contact nonlinearity. A simple solution is to use a finer mesh on the contact surfaces [28]. However, a finer mesh leads also to a higher time-cost of calculation. It isn’t a big issue if the load cases that need to be calculated are few. Since the number of load cases needs to be calculated is 60, the time-cost issue becomes very sensitive. The tactic used was that use a coarse mesh to calculate all load cases first, then use a fine mesh to calculate the worst-case or the interesting cases. This tactic provided a good balance between time-cost and accuracy. Table 3.9 shows the load cases need to be calculated.

Table 3.9 There are 10 configurations for each cutting angle

Cutting angles [°] Configurations

-25 LH-(C1, C2, C3, C4, C5), RH-(C1, C2, C3, C4, C5) -14 LH-(C1, C2, C3, C4, C5), RH-(C1, C2, C3, C4, C5) -0 LH-(C1, C2, C3, C4, C5), RH-(C1, C2, C3, C4, C5) 0 LH-(C1, C2, C3, C4, C5), RH-(C1, C2, C3, C4, C5) 14 LH-(C1, C2, C3, C4, C5), RH-(C1, C2, C3, C4, C5) 25 LH-(C1, C2, C3, C4, C5), RH-(C1, C2, C3, C4, C5)

According to the confidential load cases document, the operating angles are -25°, -14°, -0°, 0°, 14°, and 25° [24]. The loads applied to the disc cutters are Qx, Qz, and M, in which they are actually a function of the cutting angle. Therefore, there were 6・10 = 60 load cases that needed to be calculated. The difference between -0° and 0° is the direction of

Fa. Table 3.10 summarizes the mesh size used in the coarse calculation. Figure 3.12

shows the mesh distribution in the new FE model for the coarse calculation.

Table 3.10 Mesh surfaces and their mesh sizes

Mesh surfaces Mesh sizes [mm]

Global Coarse (Program Controlled)

Outer ring 11

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Figure 3.12 Mesh distribution in the coarse calculation. a). The global mesh

distribution. b). The mesh distribution in the bearings. c). The mesh distribution in the outer ring. d). The mesh distribution in the rollers.

All elements in the new FE model were assigned with tetrahedral since this element type can fit better complex geometry [29]. The accuracy of contact pressure depends only on the mesh size on the contact surfaces, i.e., the surface of rollers and the corresponding surfaces on the outer ring. The mesh size on the other surfaces is, therefore, not important. A mesh size of 11mm on the contact surfaces is the coarsest mesh size that can achieve convergence. Table 3.11 summarizes the mesh size used in the fine calculation. Figure 3.13 shows the mesh distribution in the new FE model for the fine calculation.

Table 3.11 Mesh surfaces and their mesh sizes

Mesh surfaces Mesh size [mm]

Global Coarse (Program Controlled)

Outer ring 11

Fine-mesh rollers 1

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Figure 3.13 Two rollers were fine-meshed in the fine calculation.

Since the coarse calculation has poor accuracy, a fine calculation is necessary to perform in order to obtain a more accurate result. Based on this mind, a roller subjected to the largest radial load and a roller subjected to the largest axial load was chosen to be fine-meshed, as shown in Figure 3.13. In the fine calculation, Each Iteration was chosen for the option of Update Stiffness to prevent the stiffness problem.

The disc cutters weren’t included in the new FE model since it can reduce the complexity of this model. The simulation of the disc cutters was achieved by applying a

Remote Force and a Moment on a Remote Point. This method is a way of abstracting a

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Figure 3.14 Comparison between reality and the new FE model. a) The disc cutter is

fitted on an axle that is fixed on the frame of the disc cutter. b). The Remote Point is located at the centrum of the disc cutter and coupled to the surfaces on the frames. Since the forces applied to the disc cutters will finally come into the bearings, the simulation between the bearings and the corresponding components is essential. Boundary conditions assigned in the bearings represent the interaction between them and are the key to obtain an accurate result. Figure 3.15 illustrates the boundary conditions on the bearings.

Figure 3.15 The boundary conditions assigned on the bearings are distinguished in

different colors.

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3.4 Cutter wheel with a larger dimension

The existing cutter wheel has a diameter of 3.5m. As the wish of Epiroc, the diameter has increased to 4.25m for the performance of the feasibility study of a larger cutter wheel. In the new cutter wheel model, the cutter head became larger. Meanwhile, the drive unit kept the same. This change implies that the existing cutter wheel could be easily updated by only changing the cutter head. This change is also giving a possibility to quickly-change on the job site in order to suit different tasks and situations. Figure 3.16 compares the new cutter wheel and the existing cutter wheel.

Figure 3.16 The comparison between the existing cutter wheel and the new cutter

wheel. a). The new cutter wheel with a diameter of 4.25m. b). The existing cutter wheel with a diameter of 3.5m.

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Table 3.12 These load cases are expected to expose how the size of the cutter wheel

affects the contact pressure.

Cutting angles [°] Configurations

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4 Results

4.1 Theoretical analysis of scuffing

For the calculation based on Assumption One, Equation 8, and the values in Table 3.2 were used to obtain wear rates of the tribo-system that is defined in Chapter 3.1.2. Since the wear coefficient given in Table 3.2 has a maximum value and a minimum value, the wear rate has two corresponding results. Table 4.1 shows the obtained results for the calculation based on Assumption One.

Table 4.1 The maximum and minimum wear rate of the tribo-system

Values [mm3_/m]

Max. wear rate 1.93765・10-2

Min. wear rate 4.79453・10-3

These wear rates obtained above are at a level of 10 -2mm3/m and 10-3mm3/m, which is considered as a completely unacceptable level in engineering applications.

There were two theories used for the calculations based on Assumption Two. For the calculation based on the equation that is derived from the Reynolds equation, Equation 9, 12, 13, and the values in Table 3.3 were used to obtain critical loads for the broken of (E)HL in the tribo-system. Since the surface roughness given in Table 3.3 has a maximum value and a minimum value, the critical load has two corresponding results. Table 4.2 shows the obtained results for the calculation based on the equation that is derived from the Reynolds equation.

Table 4.2 The maximum and minimum critical load for the broken of (E)HL in the

tribo-system

Values [N] Min. critical load 1782.6 Max. critical load 24754.1

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For the calculation based on the lubrication number theory, Equation 1, 2, 3, 4, 5, 10, 11, and the values in Table 3.4 were used to obtain the critical mean contact pressure for the broken of (E)HL in the tribo-system. Equation 1, 2, 3, 4, and 5 gave the maximum contact pressures on the contact surfaces in the tribo-system. Equation 10 gave critical mean contact pressure. By the use of Equation 11, all of the maximum contact pressures were converted to mean contact pressures in order to compare with the critical mean contact pressure. Since the combined surface roughness given in Table 3.4 has a maximum value and a minimum value, the critical mean contact pressure has two corresponding results. Table 4.3 shows the obtained results for the calculation based on the lubrication number theory.

Table 4.3 The mean contact pressures and the critical mean contact pressures for the

broken of (E)HL in the tribo-system

Mean contact pressures obtained by Values [MPa]

Equation 1 0.0226002

Equation 2 0.028775

Equation 3 1.16178

Equation 5 1.10884

Min. critical mean contact pressure 65 Max. critical mean contact pressure 240

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For the calculation based on Assumption Three, Equation 6, 7, and the values in Table 3.5 were used to obtain dynamic loads in the tribo-system. Since there were three falling heights given in Table 3.5, the dynamic load has three corresponding results. Table 4.4 shows the obtained results for the calculation based on Assumption Three.

Table 4.4 The falling heights and the corresponding impact load factors and impact

loads

Falling heights [mm] Impact load factors Impact loads [N]

0.05 19.2848 29560

0.5 58.7437 90043.1

2 116.474 178533

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Figure 4.1 The radial displacement of the yellow ring on LH.

The vertical axis represents the radial displacement in millimeter, while the horizontal axis represents the rotation angle in degree. These curves with different colors represent the running time in second. The radial displacements are approximately varied between 3.3mm and 2.2mm. Moreover, the peak of these curves has shifted a little bit toward sides. Additionally, many small peaks are found along the curves.

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variation represents the irregularity rather than the true radial displacement. This phase-shifting means that the rotation speed of the yellow ring has changed during excavation. These small peaks occur because of that there are many screw holes located on the inside surface of the green ring. As a summary of Figure 4.1, there is no evidence showing that the yellow ring has a large radial displacement over time.

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Similar to Figure 4.1, the vertical axis represents the radial displacement in millimeter, while the horizontal axis represents the rotation angle in degree. These curves with different colors represent the running time in second. The radial displacements, in this case, are varied between 2.9mm and 2.3mm but don’t have any phase-shifting. Additionally, many small peaks are also found along the curves.

For the same reason above, this variation also represents the irregularity of the yellow ring on RH. Moreover, this figure shows also that the irregularity on RH is smaller. Additionally, no phase-shifting implies that the yellow ring on RH rotated at the same speed all the time. Lastly, the small peaks also represent the screw holes since the yellow ring on RH and LH are identical.

4.2 Numerical analysis of the two-rings case

The yellow ring applies a load with different magnitude to the green ring depending on the assumed gravitational accelerations. Further, the applied load would create maximum contact pressure somewhere in the contact surfaces. Since there are six assumed falling heights, the maximum contact pressure has six corresponding results. By the use of Equation 11, all of the maximum contact pressures were converted to mean contact pressures in order to compare with the critical mean contact pressure. The final results are shown in Table 4.5.

Table 4.5 Gravitational accelerations and the corresponding mean contact pressures

Gravitational accelerations Mean contact pressures [MPa]

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The mean contact pressure obtained by the gravitational acceleration of 1・g represents the mean contact pressure resulting from the static weight of the yellow ring. The other obtained mean contact pressures are the results of dynamic loads. The static weight has caused a mean contact pressure of 3.7MPa, while dynamic loads have caused mean contact pressures of from 1.4MPa to 13.2MPa. These results show that a small dynamic load has caused a lower mean contact pressure compared with the mean contact pressure resulting from the static weight. These results show also that the largest dynamic load isn’t large enough to penetrate the fluid film according to the lubrication number theory. Figure 4.3 shows the contact pressure distribution in the case of 1・g.

Figure 4.3 a). A close look at 12 o’clock of the rings. b). The contact pressure

distribution at the contact surfaces.

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Dimensioning of a cutter wheel bearings