A study of wear and load behaviour on bucket teeth for heavy-duty cable shovels
Jamal Choudhry
Mechanical Engineering, master's level 2020
Luleå University of Technology
Department of Engineering Sciences and Mathematics
ACKNOWLEDGEMENT
This thesis is the last and final project that concludes my time as a mechanical engineering student at Lule˚a University of Technology in Sweden. During the time of this thesis, i have gained incredible experience that allowed me to develop myself further as an engineer. With this being said, i would like to express my gratitude towards some key people in my life that helped me immensely throughout this thesis.
First and foremost i want to thank my examiner and supervisor prof. P¨ar Jons´en for his excellent guid- ance and patience. A special thanks to my supervisor Andreas Svanberg at Boliden for his brilliant support and time throughout the whole project. I also want to thank the team at Boliden for providing me with excellent feedback and support.
Lastly i would like to thank my parents and family for their massive encouragement and support through- out my studies, without them i would have never gotten this far. A thank you to my dear wife and best friend, Komal, for all her love and understanding.
Jamal Choudhry Lule˚a, Sweden May, 2020
ABSTRACT
Many of today’s engineering advancements rely on minerals such as copper, gold and iron. For this reason, the mining industry plays an important role for the development of society and technological wonders. Mining excavators are commonly used tools for extracting the minerals from the mine. Min- ing excavators are large machines used to breakdown, penetrate and load the rock ores onto trucks that transport the minerals. During the dynamic loading, the excavator bucket experiences significant amount of wear and tear that negatively affects the production by increasing the downtime. The bucket teeth are arguably the most worn parts of the bucket and are responsible for significant amounts of down- time. This thesis aims to provide a better understanding of the load and wear on the bucket teeth of large scale mining excavators used in Bolidens Aitik copper mine in Sweden. Because of how much wear and tear the bucket teeth are exposed to, there is a need to better understand the wear behaviour of the teeth and for the whole bucket in general. This understanding can then be used to improve the ser- vice life of the teeth and other parts of the bucket and thus increase work efficiency and reduce downtime.
This project was divided into two parts. The first part consisted of regular field measurements to follow the wear on the bucket for about two weeks of digging and loading. The gathered data was then analysed to provide a better understand about the wear behaviour. The second part was to develop a numerical model that could predict the wear on the bucket and could be verified by the field measurements.
The field measurements consisted of seven 3D laser scans of the bucket starting with brand new teeth. At the time of the last scan, the buckets total loaded tonnage was approximately 542 kton and the excavator had operated in total of approximately 195 hours. After the raw data from the scans was gathered and analysed, various information about the wear behaviour on the teeth was achieved. The 3D scanned data was also used to provide a complete wear development cycle which allowed to track the wear of any point in the bucket. The method could also be used to create animations of the teeth as they were being worn.
From the results, it was concluded that the wear rate for the teeth slowed down and even converged as the geometry changed due to wear. When comparing all nine teeth on the bucket, it was also found that the middle teeth on the bucket were most exposed to wear. The most worn tooth was found to lose around 50 kg of weight after approximately 117 operating hours, which accounts for 40 % of the original weight.
The animations from the complete wear development results also showed how the individual teeth and the whole leading edge with all nine teeth were being worn as the buckets loaded tonnage increased from 0 to 542 kton.
The numerical model consisted of simulations of loading with the rocks being modelled with the Discrete Element Method (DEM). These were divided into four cases, the first being with the bucket with all new teeth. The second bucket with a mixture of new and worn teeth. The third bucket with all worn teeth and then finally the fourth bucket in which a new tooth geometry was tested. The numerical model showed promising results and potential for being a reliable way to predict the wear on the bucket. The results showed that both the penetration force and wear for the middle teeth was higher than the other neighbouring teeth. It also showed that the completely worn teeth had a lower wear rate than the new teeth which is in agreement with the results from field measurements. Other factors such as tooth shape and length were also observed to have a significant impact on the wear and penetration force. Lastly, the new teeth geometry also showed potential for design improvements in terms of wear resistance but can be further optimised. From the new teeth geometry, a suggestion was given for using an existing tooth system that might be more wear resistant.
SAMMANFATTNING
M˚anga av dagens tekniska framsteg f¨orlitar sig p˚a mineraler som koppar, guld och j¨arn. Av denna anled- ning spelar gruvindustrin en viktig roll f¨or samh¨allsutvecklingen och det ¨ar h¨ar lingr¨avarens roll kommer in. Lingr¨avare ¨ar stora maskiner som anv¨ands f¨or att lasta de mineralrika bergmalmerna p˚a truckar som sedan transporterar malmen. Under den dynamiska lastningen upplever skopan betydande m¨angd slitage som negativt p˚averkar lastningsprocessen. Skopans t¨ander ¨ar utan tvekan de mest slitna delarna p˚a skopan och ansvarar f¨or ofta f¨orekommande och l˚anga driftsstopp. Denna avhandling syftar till att ge b¨attre f¨orst˚aelse f¨or belastning och slitage p˚a skopt¨anderna hos storskaliga lingr¨avare som anv¨ands i Bolidens Aitik koppargruva. P˚a grund av hur mycket slitage som skopt¨anderna uts¨atts f¨or, finns det ett behov av att b¨attre f¨orst˚a t¨andernas och ¨aven hela skopas n¨otningsutveckling. Denna f¨orst˚aelse kan sedan anv¨andas f¨or att f¨orb¨attra livsl¨angden p˚a t¨anderna och andra delar av skopan och d¨armed ¨oka drifttiden.
Detta projekt delades upp i tv˚a delar. Den f¨orsta delen bestod av regelbundna f¨altm¨atningar f¨or att f¨olja slitage p˚a skopan i cirka tv˚a veckor. De insamlade data analyserades sedan f¨or att b¨attre f¨orst˚a n¨otningen. Den andra delen var att utveckla en numerisk modell som kunde f¨oruts¨aga n¨otningen p˚a skopan och kunde verifieras med f¨altm¨atningarna.
F¨altm¨atningarna bestod av sju stycken 3D-laserskanningar som utf¨ordes under tv˚a veckors period. Efter att datat samlades in och analyserades, uppn˚addes information om t¨andernas n¨otningstrender. 3D- skannade data anv¨andes ocks˚a f¨or att uppn˚a en fullst¨andig n¨otningsutvecklings cykel som kunde anv¨andas f¨or att skapa animationer av t¨anderna n¨ar de slits. Fr˚an resultaten drogs slutsatsen att n¨otningsthastigheten f¨or t¨anderna bromsades och till och med konvergerades n¨ar geometrien f¨or¨andrades p˚a grund av slitage.
En j¨amf¨orelse av alla nio t¨ander p˚a skopan visade att de mittersta t¨ander p˚a skopan var de som var mest utsatta f¨or slitage. Den mest slitna tanden visade sig f¨orlora ungef¨ar 50 kg vikt efter ungef¨ar 117 drifttim- mar, vilket motsvarar ungef¨ar 40% av den ursprungliga tandvikten. Animationerna fr˚an de fullst¨andiga n¨otningsutvecklingsresultaten visade ocks˚a hur de enskilda t¨anderna och hela framkanten av skopan med alla nio t¨ander n¨ots n¨ar den lastade tonnaget ¨okade fr˚an 0 till 542 kton.
Den numeriska modellen bestod av simuleringar av lastning d¨ar malmen modellerades med Discrete Element Method (DEM). Dessa delades in i fyra fall, varav den f¨orsta var med skopan med alla nya t¨ander. Den andra skopan med en blandning av nya och slitna t¨ander. Den tredje skopan med alla slitna t¨ander och sedan slutligen den fj¨arde skopan d¨ar en ny tandgeometri testades. Simuleringsresul- taten visade att b˚ade kraften och slitage f¨or de mittersta t¨ander var h¨ogre ¨an de andra angr¨ansande t¨anderna. Det visade ocks˚a att de helt slitna t¨anderna hade en l¨agre n¨otningshastighet ¨an de nya t¨anderna, vilket ¨overensst¨ammer med resultaten fr˚an f¨altm¨atningar. ¨Aven l¨angden och spetsighet p˚a t¨anderna visade p˚averka n¨otningen och penetrations-kraften. Den nya tandgeometri visade ocks˚a po- tential f¨or designf¨orb¨attringar n¨ar det g¨aller slitstyrka men kan optimeras ytterligare. Fr˚an den nya t¨andergeometri, gavs ett f¨orslag om att anv¨anda ett befintligt tandsystem som kan vara mer slitstarka.
Contents
1 Introduction . . . . 1
1.1 Tooth types and functions . . . . 2
1.2 Current maintenance methods . . . . 5
1.3 Aim and objective . . . . 5
1.4 Problem description . . . . 5
1.5 Limitations and simplifications . . . . 5
2 Theory . . . . 7
2.1 Finite Element Method . . . . 7
2.1.1 Explicit time integration . . . . 7
2.1.2 Finite element method steps . . . . 8
2.1.3 Contact algorithms in FEM . . . . 8
2.2 Discrete Element Method . . . . 9
2.2.1 The Discrete Element Method in LS-DYNA . . . . 9
2.3 Archard’s wear law . . . . 11
2.3.1 Numerical modelling of wear . . . . 11
2.4 Hausdorff distance . . . . 11
2.4.1 Application to point clouds . . . . 12
3 Method . . . . 14
3.1 Studying the wear development . . . . 14
3.1.1 Field measurements . . . . 14
3.1.2 Importing the point clouds . . . . 15
3.1.3 Finding the wear length . . . . 16
3.1.4 Finding the wear volume . . . . 17
3.1.5 Complete wear development . . . . 19
3.2 Numerical model . . . . 23
3.2.1 Preparation 1 - Bucket model . . . . 23
3.2.2 Preparation 2 - Rock model . . . . 25
3.2.3 Material Properties . . . . 26
3.2.4 Contact and implementation of Archard’s wear law . . . . 26
3.2.5 Discrete element properties . . . . 27
3.2.6 Final simulation set-up . . . . 27
3.2.7 Post simulation analysis . . . . 29
4 Results and discussions . . . . 30
4.1 Field measurements . . . . 30
4.1.1 Wear length and wear volume . . . . 30
4.1.2 Complete wear development . . . . 32
4.2 Numerical model . . . . 36
4.2.1 Case 1 - All new teeth . . . . 36
4.2.2 Case 2 - All worn expect for teeth 4 and 5 . . . . 38
4.2.3 Case 3 - All worn teeth . . . . 39
4.2.4 Case 4 - New teeth geometry . . . . 41
4.2.5 Comparison between all cases . . . . 42
4.2.6 Wear intensity map . . . . 45
4.2.7 Tooth system suggestion . . . . 46
4.3 General Discussion . . . . 47
4.3.1 Field measurements . . . . 47
4.3.2 Numerical model . . . . 48
5 Conclusions . . . . 50
5.1 Future work . . . . 51
References . . . . 52
Appendix . . . .
BOLIDEN 5 Contents
1. INTRODUCTION
Many of today’s engineering advancements rely on minerals such as copper, gold and iron. For this reason, the mining industry plays an important role for the development of society. One of the most common way to extract the minerals from the mine is by use of mining excavators. Mining excavators are commonly used machines used for loading the rock ores onto trucks for transportation.
In Boliden’s copper mine Aitik, large scale mining excavators are used for loading of the copper ore from the already blasted rock ore onto the trucks. The trucks then transport the rocks to the crusher, where it is further broken down from about 0-1.5 m to smaller than about 0.4 m. The mine has three active rope shovel excavators, two of which are P&H (which is now Komatsu) 4100 and the Bucyrus (CAT 7) 495. The excavators have an operating time of around 7000 hours per year and load approximately 12000 ktons of rocks during the same time.
Figure 1 shows the Bucyrus 495 being currently used in the mine.
Figure 1: The Bucyrus 495 rope shovel loading the rocks into a truck.
The Bucyrus 495 is an electric rope shovel. The crowd and hoist line are used to move and adjust the position of the bucket and the excavator is also free to move and rotate (known as swinging). Figure 2 shows the basic movements and maneuvers of this excavator.
BOLIDEN 1
Figure 2: The crowd can move in and out and the hoist line can be lifted up and down, which can be used to adjust the bucket position. The crowding and hoisting movements are indicated by the red arrows. The bucket can be moved forward and backwards and relative rotation (known as swinging) is of course also possible (Khorzoughi, 2017).
During the operational life of these excavators, the bucket teeth are subjected to large scale impact and resistive forces. This is the main reason for failure of machine parts such as of the bucket teeth. The failure of tooth can typically occur due to impact, wear and abrasion subjected due to the large impact force during the dynamic loading. Other factors such as rock size, penetration velocity, penetration angle, and the material of the bucket teeth also play key roles in the wear and tear of the bucket teeth. The key role that the bucket teeth has is to penetrate into the muck pile and assist in the loading of the ore inside the bucket. Due to this, it is one of the most worn and torn part of the bucket. Inappropriate choice of the teeth followed by a bad maintenance schedule can lead to poor penetration process and cause more damage to the whole bucket, which will lead to decreased productivity and higher maintenance costs (Singla, Shibe, & Grewal, 2011).
The bucket is of the type ESCO 55 YD Cable Shovel Dipper, it is approximately 5 m wide and 3 m high and has a capacity of 42 m3 which means that it can carry around 70-100 tons of rock ore. The bucket teeth used are also manufactured by ESCO and are called the Ultralok system which is a series consisting of different tooth types and other leading edge parts used for various applications.
1.1 Tooth types and functions
Figure 3 shows a list of the commonly used Ultralok tooth systems manufactured by ESCO. The teeth are categorised by different tooth shapes.
BOLIDEN 2 1.1 Tooth types and functions
(a) The S is a standard tooth for excava- tors in general purpose applications.
(b) The AP is a heavy-duty penetration tooth for wheel loaders for highly abrasive applications and is used on excavators in the concrete recycling and demolition in- dustry.
(c) The A tooth is designed for working in extreme abrasion applications.
(d) The C is a chisel tooth primarily for use on excavators. It is used in applica- tions where good penetration is required.
(e) The F is a flared tooth for general pur- pose digging and continuous edge applica- tions.
(f ) The H is a heavy tooth for extremely abrasive applications and is primarily de- signed for excavators
(g) The P is a pick tooth for extremely hard to penetrate materials, and is pri- marily designed for excavators but can be used on wheel loaders.
(h) The T is a twin pick tooth for max- imum performance in hard to penetrate materials. The configuration reduces the chance of rocks wedging between the tines;
and is designed for use in the corner posi- tions in conjunction with P style teeth to cut clearance for the buckets sides Figure 3: Figures (a)-(h) show the different tooth types and their functions (ESCO, 2009).
BOLIDEN 3 1.1 Tooth types and functions
The ESCO Ultralok U90R-A Rock Point tooth system are used in the ESCO 55 YD Cable Shovel Dipper operated by the Bucyrus 495 in Bolidens copper mine Aitik. This is a variant of the heavy tooth described in Figure 3f. The bucket teeth are made through casting and the most commonly used material are alloy steels (Fern´andez, Vijande, Tucho, Rodriguez, & Mart´ın, 2001) & (Suryo, Bayuseno, Jamari, &
Ramadhan, 2018).
Figure 4 shows the excavator bucket with its worn teeth. The particular images were taken approximately 72 operating hours after installation of brand new teeth.
(a) (b)
Figure 4: Figure (a) shows the bucket called ESCO 55 YD Cable Shovel Dipper. The coloured boxes show the name of the parts that will be studied in this project. Figure (b) shows one of the worn bucket tooth roughly 72 operating hours after installation.
The rock fragmentation can vary in size at different mining sites. Generally speaking, the size can range from extremely fine to somewhere around 1.5 m. Due to this, the wear and tear on the bucket teeth can vary on a large scale depending on where the digging is taking place. Figure 5 shows an example of how the rock fragmentation can look at different excavation sights.
(a) (b)
Figure 5: Figure (a) shows a sight where coarser/larger rocks are residing, while figure (b) shows a sight where smaller rocks are residing.
BOLIDEN 4 1.1 Tooth types and functions
1.2 Current maintenance methods
Today’s way of keeping track of tooth wear is basically completely manual. Inspections of wear com- ponents are done several times a week and worn parts are repaired or replaced. Most of the time it is however the operators that request the replacement of the teeth. From estimated calculations based on experience from the service and maintenance staff, it is concluded that the teeth are worn about 0.01 m per day. This also implies that new teeth are either completely worn out or failure occurs after about two weeks time of continuously loading of the bucket. The same type of bucket teeth are used on all rope shovel excavators, and the choice between teeth (a)-(h) shown in Figure 3 is also based on experience from the service and maintenance staff.
1.3 Aim and objective
The bucket teeth are one of the most worn and torn parts of mining excavators. This is due to the direct contact the teeth has with the rocks, which causes impact and abrasion on the surfaces of the teeth. The main aim of this thesis is to better understand the wear development of the bucket teeth and other wear parts of the bucket used in the Bucyrus 495 rope shovel. The information could then be used to help reduce downtime and improve the service life of the teeth. The objective is to develop and evaluate a methodology for field measurements of wear with high accuracy. The second objective is to develop a numerical method for prediction of force and wear on the bucket teeth during the dynamic loading. The second objective also includes the condition to predict the consequence of wear on the penetration force and a suggestion for a tooth design improvement in terms of wear resistance should also be given. This information could then be used in the future to improve the service life of the teeth by planning better maintenance schedules and help reduce downtime.
1.4 Problem description
A brief problem description for the different parts of the thesis is given below:
• Understanding the current maintenance routines on the bucket teeth.
• Perform field measurements on the bucket.
• Analyse the data from measurements and understand the wear behaviour.
• Numerical modelling to predict wear.
– Numerical rock model with the Discrete Element Method (DEM).
– Calibration of the rock model.
– FEM model of the bucket.
– Simulation of loading with the multi-physics, FEM software LS-DYNA.
• Analyse the results from numerical model and compare it with the field measurements.
• Simulated different tooth geometry.
1.5 Limitations and simplifications
This section will cover some of the simplifications that were made during this thesis. The simplifications and limitations are presented below:
• One series of field measurements were performed during a period of two weeks.
• The exact material of the bucket and teeth were not known, so data for common steel alloy was used in the numerical model.
BOLIDEN 5 1.2 Current maintenance methods
• The rock model did not include complex rock geometries as this was considered beyond the scope of the thesis.
• Four loading scenarios will be used to study the wear behaviour in the numerical model to save time.
• Archard’s wear law was used to numerically predict the wear. This is because it was the simplest wear model that was also assumed to be the most appropriate for this project.
BOLIDEN 6 1.5 Limitations and simplifications
2. THEORY
This section will present the basic theory used in this project.
2.1 Finite Element Method
Originally developed to study load, strains and stresses for the aerospace industry, the finite element method (FEM) has been developed and evolved to be used in a broad field of continuum mechanics.
This is a numerical analysis tool that can be used to approximate the solutions to partial differential equations and thus solve many engineering problems. The main idea behind this method is to transfer an infinite continuous domain with infinite number of unknown solutions into finite discrete elements and thus solving partial different equations that consist of finite number of unknowns. The elements are connected by the so called nodal points that lie on their circumference and this connection is described by the interpolating shape functions. In the finite element method, the equation of motion is solved for the unknown nodal displacements (Borst et al., 2012). The equation of motion is written as
M ¨d + C ˙d + Fint= Fext, (1)
where M is the mass matrix, C is the damping matrix, Fint is the internal forces, Fext is the external forces acting on the system and d are the nodal displacements, ˙d are the nodal velocities and ¨d are the nodal accelerations. Explicit and implicit time integration techniques can be used to numerically solve for the nodal displacements. The next section will give a brief overview of how the explicit time integration method works.
2.1.1 Explicit time integration
In the explicit time integration, the central difference method is used to solve the equations of motion.
Let n denote the time step, ∆t the time step size, dnthe nodal displacement at time step n, ˙dnthe nodal velocity at time step n and ¨dn the acceleration at time step n. The acceleration and velocity vector can then be expressed as
d¨n = 1
∆t2(dn−1− 2dn+ dn+1), d˙n = 1
∆2t(dn+1− dn−1),
(2)
which can then be re-written as
d¨n= 1
∆t( ˙dn+1
2 − ˙dn−1 2), d˙n+1
2 = 1
∆t(dn+1− dn).
(3)
If damping is neglected, equation (1) can then be combined with equations (3) to give the three equations of motion in discrete form used in finite element code with explicit time integration. Let the brackets represent the matrices and the braces represent the vectors. The equations of motion in discrete form are:
{ ¨d}n= [M ]−1({Fext}n− {Fint}n), (4) {d}n+1= {d}n+ ∆t{ ˙d}n−1
2 + ∆t2{ ¨d}n, (5)
{ ˙d}n+1
2 = { ˙d}n−1
2 + ∆t{ ¨d}n, (6)
where [M ] is the mass matrix, {Fext} are the external forces and {Fint} is defined as
{Fint}n= [K]{d}n. (7)
[K] in equation 7 is the stiffness matrix. This solution is conditionally stable and the stability criteria depends on the crticial time step size.
The critical timestep is choosen from Courant’s stability condition for explicit time integration
BOLIDEN 7
∆t ≤ Le
c . (8)
Where Le is the smallest element length and c = q
E
ρ is the speed of sound wave propagating in the material. E is the elastic modulus and ρ is the material density (Logan, 2007).
2.1.2 Finite element method steps
This section will cover the different steps that are used in a finite element method routine. The steps witch are covered give only a brief explanation and the author refers to D. E. Smith T. G. Byrom K.
H. Huebner (2001) for a more comprehensive explanation.
1. Discretize the Continuum. The first step is to divide the body into region of elements.
2. Select Interpolation Functions. The second step is to assign every element a node and then to choose interpolating functions that describe stresses and strains between each node.
3. Find the Element Properties. This is where the matrix equations are established to determine the element properties. This is done by three ways: the direct approach, the variational approach, or the weighted residuals approach. The material parameters are used to establish the mass and stiffness matrix during this step.
4. Assemble the Element Properties to Obtain the System Equations. This is where the system of elements describing the body are assembled. The matrix equations for each elements are combined to form the matrix equations for the whole system. This is done to find the overall properties of the whole system instead of just individual elements.
5. Impose the Boundary Conditions. The system of equations from the last step must be modified to account for boundary conditions.
6. Solve the System Equations. The next step is to solve the system of equations for the nodal displacements and other nodal variables. The nodal values can be solved by either explicit or implicit time integration. In cases where the material is changing within the cross section or a non-linear material model is used, the stiffness matrix [K] needs to be updated with every time step (Logan, 2007).
7. Make Additional Computations If Desired. Nodal displacements can be used to calculate other element properties such as strains and stresses.
2.1.3 Contact algorithms in FEM
A common way to define contacts in commercial FEM programs is to identify the locations that should be checked where penetration of slave nodes into master segments can occur. The search for penetrations is made at every time step and by using any of a number of different contact algorithms. The most commonly algorithms are penalty based and these use normal interface springs to resist the penetration by applying a force that is proportional to the penetration depth. The common formulations used in the non-linear FEM program LS-DYNA are:
• Standard penalty formulation. This is the most general and most used interface algorithm.
• Soft constraint penalty formulation. This formulation is different than the standard and was devel- oped to handle contact between very soft and hard materials.
• Segment-based Penalty Formulation. This is segment based contact algorithm that is used as a general purpose penalty formulation for shell and solid elements.
The penalty contact stiffness is usually chosen as the minimum of the master segment and slave node stiffness. Penetration can still arise in problems where high contact pressure occurs and a way to solve this is to manually scale the contact stiffness. In cases where the soft material has very low density, the time step size should also be reduced to avoid instability (LSTC, 2020).
BOLIDEN 8 2.1 Finite Element Method
2.2 Discrete Element Method
Originally introduced by Cundall in 1971, the Discrete Element Method (DEM) is a numerical method that is used for computing the motion and behaviour of a large number of particles. The method was developed for the analysis of rock mechanics problems and other granular material flow. In this method, each element is treated as a separate rigid particle and can have its own independent motion. The contact between each particle is described by a soft-contact approach and overlapping between the contact points is allowed. The contact forces between the particles are calculated from the contact overlaps and this combined with other external forces can generate particle motion calculated by Newton’s second law of motion. As the particles move around, they interact with each other and generate new interface forces so that new motion of particles are continuously created (Chen, Zeng, & Yin, 2013). In Chen, Zeng, and Yin (2013), the following assumptions are presented for the calculation process involving DEM:
1. Each particle is treated as a rigid body and the deformation of the whole system is calculated as the sum of the deformations in contact points between all particles.
2. The contact between each particles occurs only at contact points.
3. The contact uses a soft contact formulation which allows some overlap. The overlap should be much smaller than the particle size and its motion.
4. The particles should only be allowed to interact at the contact points. The time step size should be small enough to ensure this.
5. Both the velocity and acceleration should be constant for each time step. These should be calculated using Newton’s second law of motion.
6. The time step size should be small enough to stop any disturbance to propagate further than the neighbouring particles.
2.2.1 The Discrete Element Method in LS-DYNA
In LS-DYNA, particles are treated as rigid bodies and given a mass and inertia which are calculated from the radius and density. The particles are interacting with each other with penalty based contact formulations using a set of springs and dampers. Figure 6 shows the springs and dampers used for modelling the contact between the particles. The area in red shows the elastic contributions, the area in blue shows the damping contributions and the area in green shows the frictional contributions.
Figure 6: The mechanical particle-particle penalty contact in which the elastic (red), damping (blue) and frictional contributions (green) are shown (Karajan, Lisner, Han, Teng, & Wang, 2012).
The particle to particle contact has three different collision states as shown in Figure 7.
BOLIDEN 9 2.2 Discrete Element Method
Figure 7: The figure shows possible collision states. dint is the interaction distance while dcritis the rupture distance where the adjacent particles are no longer in contact (H. Ten J. Wang N. Karajan, 2013).
The collision states depend on the interaction distance which is calculated as
dint= r1+ r2− |x1− x2|. (9)
The rupture distance dcrit defines the distance where the adjacent particles are no longer in contact.
The most important quantity to influence the penalty based contact are the elastic contributions from the springs. The normal force contribution is calculated as
Fn= Kn· dint, (10)
where Kn is the normal spring constant. The normal damping force is calculated as
Fn= Dn· ˙dint, (11)
where Dnis the normal damping coefficient. Using Coulomb’s friction law, the frictional force contribution is then calculated as
Ff r= µf r· Fn, (12)
where µf ris the friction coefficient and Fn in this case is the sum of all normal forces.
Particle-surface interaction is instead achieved by applying the frictional force to the particle perime- ter and thus accounting for the moments induced on the particle (Karajan, Lisner, Han, Teng, & Wang, 2012). The particle-surface contact is shown in Figure 8.
Figure 8: Particle-surface interaction. A friction force Ff ric is applied at the perimeter of the particle to account for the induced moment (H. Ten J. Wang N. Karajan, 2013).
To account for surface roughness and non-spherical particles, rolling friction can be added (H. Ten J. Wang N. Karajan, 2013). In order to avoid contact instabilities, three different time step sizes are computed and compared to each other.
Raleigh time step:
dtRaleigh = πrsphere 0.1631(ν + 0.8766)
r ρ
E/(2 + 2ν). (13)
Hertz time step:
BOLIDEN 10 2.2 Discrete Element Method
dtHertz= 2.87
"
((4/3)ρπr3sphere)2 rsphereE2Vmax
#0.2
. (14)
Cundall time step:
dtCundall= 0.2π v u u t
((4/3)ρπr2sphere
E
3+6νN ORM K. (15)
Where rsphereis the smallest particle radius, ρ is the density, E is the elastic modulus, ν is the Poissons ratio, Vmax is the maximum expected relative particle velocity and N ORM K is the stiffness penalty parameter. Numerical stability can then be ensured by setting the time step to 20% of the minimum size from the methods mentioned above (Jensen, Fraser, & Laird, 2014).
dt = 0.2min(dtRaleigh, dtHertz, dtCundall). (16)
2.3 Archard’s wear law
Originally developed by Holm and Archard in the 1950s, the Archard’s law of wear is one of the most famous wear description of a sliding system (Archard., 1953). The law is perhaps one of the most fundamental yet empirical model to describe wear. Archard’s wear law states that the wear depth rate ˙h at a contact point is proportional to the sliding velocity ˙d and pressure as
˙h = kp ˙d
H, (17)
where k is a dimensionless wear factor, p is the contact pressure and H is the hardness parameter (Almqvist & R`afols., 2019). Another way to re-write this would be
h = W EARC · fn· d
A , (18)
where W EARC = k/H is the dimensional wear coefficient, fnis the normal interface force, h is the wear depth and A is the area of contact segment (LSTC, 2018). Because the dimensional wear coefficient is of interest, the next step is to solve for it. Multiplying by the contact area on both sides gives the wear volume
w = W EARC · fn· d, (19)
Where w is the wear volume and d is the relative sliding distance. Solving for the paremeter W EARC gives
W EARC = w
fn· d. (20)
2.3.1 Numerical modelling of wear
Developing numerical models for predicting wear in large scale applications is a challenge. The strategy used in this project is to combine different numerical methods to couple the model for solid structure, granular material flow and the wear behavior/model. FEM can be used to model the solid structures and particle methods such as DEM can be used to simulate the granular material flow. The methodology used in this project will be based on Forsstr¨om and Jons´en (2016) and Forsstr¨om, Lindb¨ack, and Jons´en (2014), which are previous projects that developed numerical models to accurately predict wear.
2.4 Hausdorff distance
Suppose that two set of points are to be compared. The Hausdorff distance is a method used to compare the two sets and calculate the distance between them. The Hausdorff distance is defined as the maximum distance of a set to the nearest point in the other set. Mathematically it is written as
BOLIDEN 11 2.3 Archard’s wear law
h(A, B) = max
a∈A{min
b∈B{d(a, b)}}, (21)
where h is the Hausdorff distance between the set A and set B and d(a, b) is the euclidean distance between points a and b (Gr´egoire & Bouillot, 1998).
2.4.1 Application to point clouds
This algorithm is useful when calculating distances between point clouds obtained from 3D scans of the geometry of a single object which is changing with time. Let the reference point cloud be the set B containing points and their three dimensional coordinates, and similarly let the compared point cloud be the set A containing its own points. Now assume that C is a set with points that represent a state between A and B and that its points depend on a variable t. The set A is valid for tA and B is valid for tB. This variable can be seen as an independent variable such as time or tonnage. If the Hausdorff distance between A and B is hAB, it is possible to use linear interpolation between A and B to find the Hausdorff distance hAC that corresponds to any t = tAC such that tA< tAC < tB by scaling hAB. It is then possible to use hAC to find sets C that correspond to tAC. Mathematically it is written as
C(tAC) = A − hAC(tAC), (22)
where hAC is calculated as
hAC(tAC) =tAC− tA
tB− tA
hAB, (23)
where A is the compared point cloud with n points
A =
ax1 ay1 az1
... ... ... axn ayn azn
, (24)
and hAB is the matrix containing the Hausdorff distance between A and B
hAB=
hx1 hy1 hz1 ... ... ... hxn hyn hzn.
. (25)
The variable tAC can be seen as the tonnage/load. Figure 9 shows how this method can be applied to study the wear development of a tooth. The variable C is a point cloud that represents a state between B and A. In this study, this variable will represent the point cloud of the shape of the worn teeth at an arbitrary tonnage/load. The variable B represents the shape of the tooth as when it was worn (reference) while the variable A represents the new tooth (compared). As the load increases, the point cloud C changes linearly until finally resembling A, which was the exact shape of the worn teeth as when the 3D scan was performed on it.
Figure 9: The interpolation of a tooth. The worn tooth (in grey) is the reference point cloud and the new tooth (in black) is the compared point cloud. The point cloud in blue is interpolated between the reference and compared point clouds.
BOLIDEN 12 2.4 Hausdorff distance
If the same teeth has been 3D scanned at different times, the interpolation can be done between each and every scan to obtain more realistic results of the wear development.
BOLIDEN 13 2.4 Hausdorff distance
3. METHOD
This section will cover the methodology used in both the analysis of the field measurements and devel- opment of the numerical model for predicting wear.
3.1 Studying the wear development
In this section, the methodology for understanding the wear behaviour on the bucket teeth are presented.
The section is divided into firstly the field measurements and secondly the analysis of data. The analysis of the data will cover the method for finding the wear length and wear volume of each tooth. Later, the method for developing a global wear development in which the wear of any point on the teeth can be tracked will also be described.
3.1.1 Field measurements
In order to correctly understand and capture the wear behaviour on the bucket teeth, field measurements were performed on the whole bucket. This was done using the Leica RTC360 3D-laser scanner. The scans were performed on intervals ranging from two to three days for about two weeks and the primary focus was to capture the wear development of each tooth. The major advantage of working with 3D scans is the possibility to follow the wear of the bucket on a global scale with higher accuracy compared to conventional measurement methods such as with a yardstick (3SPACE, 2020). After each measurement, various data such as loaded tonnage, operating hours and type/size of loaded material was recorded. The output of the scans were point cloud files that represented the geometry of the bucket including all teeth.
Figure 10 shows the laser scanner being used to scan the bucket.
(a) (b)
Figure 10: The Leica RTC360 laser scanner can be seen in Figure (a). The scanner can also be seen scanning the bucket in Figure (b).
In order to capture the whole shape of the bucket, the scanner was placed both inside and outside of the bucket. Because the teeth were the primary focus for this study, the scanner was also placed directly beneath the teeth as seen in Figure 10. This was done to ensure that the whole tooth geometry was correctly captured.
Before each scan took place, three reference points were installed and mounted on each side of the bucket.
The reference points were later used for alignment purposes during the analysis. Table 1 describes the corresponding loaded tonnage just before each scan was performed.
BOLIDEN 14
Scan 1 2 3 4 5 6 7
Tonnage [kton] 0 60 150 197 275 403 542
Operating hours [hours] 0 36 72 90 117 156 195 Table 1: The corresponding loaded tonnage and operating hours for each scan.
The first scan on the bucket was performed when all teeth were replaced and brand new. Depending on the amount of wear, some of the teeth were replaced during the time of these scans. Two teeth were replaced between scan 4 and 5 and two other were replaced between scan 6 and 7. Unfortunately, the exact loaded tonnage and operating hours just before the replacements were not noted. Due to this, it was necessary to go back to the daily loaded tonnage report and estimate the total loaded tonnage during the replacements. The first replacement is estimated to be at 295 kton with an error tolerance of
±30 kton and the second replacement at 445 kton with an error of ±50 kton. When studying the wear development, the loaded tonnage was considered more interesting than the operating hours. The reason behind this was that the excavator spent some time moving between different dig sites, meaning that the excavator was not continuously digging and loading during the whole operating time.
3.1.2 Importing the point clouds
The software used for handling and analysing the geometry was CloudCompare. This is an open source 3D point cloud processing software that can also handle triangular meshes. Because the range of the scan was up to 50 m, the scans revealed quite a lot of data around the bucket which was of no interest and the first step was to perform an image clean-up. After the clean-up, only the point cloud representing the geometry of the bucket was present. Initially, the total number of points was around 30-80 million points but this was reduced to around 20-60 million points depending on the precision and quality of the scans. A total of seven scans were taken on the same bucket over a period of two weeks. Figure 11 shows one of the scans opened in CloudCompare.
Figure 11: Point cloud representing the bucket obtained from the first scan.
After the image clean-up, the individual scans were imported in CloudCompare and the point picking alignment tool was used to align the point clouds of the bucket with each other. The alignment procedure is important to correctly measure the wear length and wear volume of the teeth on every scan. The points
BOLIDEN 15 3.1 Studying the wear development
used for the alignment were the three reference points mounted on each side on top of the bucket. See Figures 12a and 12b. Figure 13 shows an example of two teeth aligned with each other.
(a) Reference points on the top left side of the bucket. (b) Reference points on the top right side of the bucket.
Figure 12: Reference points (black and white circles) on both sides of the bucket. The reference points are used to correctly align the point clouds with each other.
3.1.3 Finding the wear length
When the point clouds of the same tooth (but from different scans) were correctly aligned with each other, a comparison between each scan was possible to perform. Each tooth could then be segmented out from the reference scan and a comparison could be made with the same tooth segmented from the next scan. Because they were aligned, the segmentation of all teeth was done simultaneously to ensure that the cut was made at the same place and a correct comparison could be made, see Figure 13a. After the teeth were segmented out from the rest of the bucket, their orientation was changed to be aligned with the global coordinate system. Figure 13 shows the point clouds of the teeth and their orientation before and after the alignment with the global coordinate system.
(a) Before alignment with the global coordinate system. The same teeth (but from different scans) are segmented out from the rest of the bucket.
(b) After alignment with the global coordinate system.
Figure 13: The teeth were first aligned with each other and then segmented from the rest of the bucket. The segmentation is done simultaneously to ensure that the cut was made at the same place for all teeth, see Figure (a). After this step, the orientation for the teeth was changed to match with the global coordinate system as shown in Figure (b).
After the alignment procedure was done, the point clouds were ready to be compared. As it can be seen
BOLIDEN 16 3.1 Studying the wear development
from Figure 13b, the teeth were aligned so that the long side was parallel with the x-axis, which is the red colored axis seen on the bottom-right side of the figure. The wear length could then be measured as the difference in the x direction by using the point picking function in CloudCompare. This procedure was done for each of the nine teeth between all seven scans.
3.1.4 Finding the wear volume
Because the point clouds of the same tooth (but from different scans) were aligned with each other and segmented from the rest of the bucket simultaneously, this ensured that the cut was made at the same place for all teeth and a correct volume comparison could be made between them. The wear volume could then be simply calculated as the difference between the volumes of the different teeth. The volume of the point clouds representing the teeth was found by first creating a triangular mesh of the segmented teeth and then calculating the volume of the mesh. The first step was to segment out the teeth from the rest of the bucket using the segmentation tool in CloudCompare. The segmentation was done after the alignment procedure of both point clouds to ensure that the cut was made at the same place for all the different teeth. Figure 13 shows an example of two aligned teeth being cut and segmented at a common cross section of both teeth. The second step was to calculate the point normals using the triangulation local surface model. After the normals were calculated, the Poisson surface reconstruction feature was used to generate a triangular mesh with a octree depth level of 12. Figure 14 shows the teeth being meshed.
Figure 14: The triangular mesh of the segmented teeth.
The meshes can also be visualized separately. See Figure 15.
BOLIDEN 17 3.1 Studying the wear development
(a) Mesh of the reference (new) tooth (b) Mesh of the compared (worn) tooth
Figure 15: The meshes of both the reference and compared teeth. Even though the point clouds were aligned and segmented at a common place, the meshes of the rear side are not flat and match poorly together. The teeth have different colors because the green paint on the new tooth in Figure (a) wears off.
As it can be be seen from Figure 15, the rear side of the two meshes are not flat and match poorly together. This is because the point clouds were previously segmented from the rest of the bucket. This means that there are no points in the rear side filling the gap and creating a bounding volume, which leads to an improper mesh generation. This creates a problem if accurate volume calculations are to be performed. A solution to this was to segment the teeth again at a cross-sectional plane where the wear is almost non-existent. The resulting shell meshes can then be converted to solid meshes for the purpose of measuring volumes. Figure 16 shows the segmented mesh in CloudCompare.
Figure 16: The figure shows the segmented meshes. Again, the segmentation is done simultaneously for all meshes to ensure that the cut is made at the same place for all teeth. The result is that a hole is created at the rear which creates a problem if volumes are to be measured.
The resulting meshes can now be exported as STL mesh files to another program called Autodesk Mesh- mixer to convert them to solid bodies in the editing section. Meshmixer is an free to use program that specializes in working with 3D scanned geometries and triangular meshes. See Figures 17a and 17b.
BOLIDEN 18 3.1 Studying the wear development
(a) Segmented tooth imported in Meshmixer. (b) Tooth being converted to a solid body.
Figure 17: Solid body conversion of the tooth mesh. This procedure is done for all segmented meshes seen in Figure 16. The result is that the rear side of the tooth mesh is completely flat, meaning that the rear sides match well compared to the original meshes seen in Figure 15. As seen in Figure (b), the tooth volume can now be measured because it is converted to a solid and gets a completely flat rear side/cross section.
The resulting solid bodies now have matching rear sides and are now ready for comparison. The wear volume is taken as the difference in the volumes of the meshes.
3.1.5 Complete wear development
Finding the wear length and wear volume of each individual tooth is a good way to obtain a general understanding of how the wear behaviour looks like. But if a more detailed understanding of the wear development is wanted, a more comprehensive analysis should be done to find the complete wear behaviour of each tooth and even the whole bucket. A complete wear development should give information of how each point in the geometry is approximately moving due to wear. In other words, it should be possible to track the wear of any point as the loaded tonnage increases. To be able to do this, the C2C (Cloud- to-cloud) distance was first calculated between each point clouds and the information was then imported into a MATLAB code. This code was written to apply the method of linear interpolation between each point clouds to estimate the wear anywhere in-between as explained in 2.4.1. MATLAB is a highly advanced programming software that uses a matrix-based language. For this reason, it was also the most appropriate to deal with large data and point clouds. This method was applied for every individual tooth and for the whole leading edge of the bucket as well. The method allowed for the wear of any point to be tracked between each scan/measurement and gave an even better understanding of the wear development.
In CloudCompare, the Cloud-to-cloud distance feature computes the closest distance between the points in the reference and compared point clouds using an algorithm based on the Hausdorff distance. Figure 18 shows how the result looks like after this feature is applied.
BOLIDEN 19 3.1 Studying the wear development
Figure 18: Wear intensity on a tooth calculated in CloudCompare. The colorbar shows the wear range in unit meters.
This result is then saved and exported as a text file to MATLAB. A code was written to read this data and use the method of linear interpolation as described in 2.4.1 to estimate the wear of any point at any given tonnage between the scans. It was also possible to visualize and animate the tooth being worn as the tonnage increases as seen in Figure 19.
BOLIDEN 20 3.1 Studying the wear development
0 50 100 150 200 250 Tonnage [kton]
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
Wear length in x-dir [m]
0 50 100 150 200 250
Tonnage [kton]
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
Wear length in x-dir [m]
0 50 100 150 200 250
Tonnage [kton]
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
Wear length in x-dir [m]
0 50 100 150 200 250
Tonnage [kton]
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
Wear length in x-dir [m]
Figure 19: Results from MATLAB describing and animating the wear on a tooth. The colorbar next to the teeth shows the wear range in unit meters while the figure on the right shows the wear length in the x-direction according to Figure 13b. The axis on the left figures have also unit meters. The tonnage refers to the total loaded tonnage of the bucket.
BOLIDEN 21 3.1 Studying the wear development
