Load modelling of a mobile miner

IN

DEGREE PROJECT MECHANICAL ENGINEERING, SECOND CYCLE, 30 CREDITS

STOCKHOLM SWEDEN 2020 ,

Load modelling of a mobile miner

RAFAIL-NIKOLAOS DIMITRIOU

KTH ROYAL INSTITUTE OF TECHNOLOGY

SCHOOL OF ENGINEERING SCIENCES

1 Abstract

In the mining industry, finite element analysis (FEA) and experimental tests are essential for the verification of mechanical properties and for performing parametric studies. Such practices require lots of resources and, sometimes, long computational time. Therefore, there is an increased need for alternative time efficient methods with a low requirement of resources. On the other hand, simulations of multi body mechanical models is a way to achieve that. The current work investigates a structural design that is modelled as a system of rigid bodies, which is firstly derived in Python and then simulated in Matlab. As input, structural properties from previous experimental and FEM studies were used.

The aim of the project is to provide an understanding of the mechanical behaviour of the

rock and the mining machine, during the cutting procedure. Parametric studies by altering

the structural parameters of stiffness and damping and by varying cutter-wheel placements

provide us with several significant results. A correlation between the structural parameters

and the fatigue loads is observed. The influence of each individual structural parameter

on the fatigue loads are explicitly described. Overall, this methodology leads to faster

calculations since there is a massive decrease in the number of degrees of freedom compared

to a finite element simulation, and therefore to the ability to perform extensive parametric

studies.

2 Sammanfattning: Lastmodellering av en gruvmaskin

Inom gruvindustrin ¨ ar finita elementanalyser (FEA) och experimentella tester avg¨ orande f¨ or att verifiera av mekaniska egenskaper och f¨ or att utf¨ ora parameterstudier. S˚ adana metoder kr¨ aver mycket resurser och ibland l˚ ang ber¨ akningstid. D¨ arf¨ or finns det ett ¨ okat behov av alternativa, tidseffektivare metoder med l˚ agt behov av resurser. ˚ A andra sidan ¨ ar simu- leringar av mekaniska modeller med flera kroppar ett s¨ att att uppn˚ a detta. Detta arbete unders¨ oker en konstruktion modellerad som ett system stela kroppar, som f¨ orst h¨ arleds i Python och sedan simuleras i Matlab. Som indata anv¨ andes strukturella egenskaper fr˚ an tidigare experimentella och FEM-studier.

Syftet med projektet ¨ ar att ge en f¨ orst˚ aelse f¨ or stenens och gruvmaskinens mekaniska be-

teende under brytningssprocessen. Parameterstudier genom att ¨ andra de strukturella parame-

trarna f¨ or stelhet och d¨ ampning och genom variation i sk¨ arhjulens placering ger oss flera

betydelsefulla resultat. En korrelation mellan de strukturella parametrarna och utmat-

tningsbelastningarna observeras. Inverkan av varje enskild strukturparameter p˚ a utmat-

tningsbelastningarna beskrivs uttryckligen. Sammantaget leder denna metod till snabbare

ber¨ akningar eftersom det finns en massiv minskning av antalet frihetsgrader j¨ amf¨ ort med en

finit elementsimulering, och d¨ arf¨ or till f¨ orm˚ agan att utf¨ ora omfattande parameterstudier.

Contents

1 Abstract 1

2 Sammanfattning: Lastmodellering av en gruvmaskin 2

3 Nomenclature 4

4 Aknowledgements 7

5 Introduction 8

5.1 Background . . . . 8

5.2 Mobile miner 22H . . . . 9

5.3 Aim of the Project . . . . 10

5.4 Problem description . . . . 11

6 Methods 13 6.1 Reference model . . . . 13

6.2 Model evolution . . . . 14

6.3 Analysis procedure (1) . . . . 14

6.4 Analysis Procedure (2) . . . . 16

6.5 Python procedure . . . . 17

6.6 Modal Analysis . . . . 18

7 Results-Discussion 18 7.1 Initial conditions of the study . . . . 18

7.2 Parametric studies concept . . . . 19

7.3 Penetration - Forces graphs for reference case . . . . 19

7.4 Parametric Study (1) . . . . 20

7.5 Parametric study (2) . . . . 24

7.6 Parametric study (3) . . . . 26

7.7 Parametric study (4) . . . . 27

8 Conclusions 29

9 Future work recommendations 30

10 Appendix 32

10.1 Python Code . . . . 32

3 Nomenclature

List of Symbols

α Contact angle λ Eigenvalues

φ Driving degree of freedom of the cutter-wheel BT S Brazilian tensile strength

c ₁ Translational damping factor of the rear part (N s/m) c ₂ Rotational damping factor of the boom (N m/s)

c ₄ Translational damping factor of the front part (N s/m) c e Out of plane tilt damping factor of the boom (N m/s) c _t Twist damping factor of the boom (N m/s)

c _4p Rotational damping factor of the front part (N m/s)

C _eq Total structural damping matrix in the equilibrium position CW Cutter wheel

D _c Individual cutter diameter E Elastic modulus

e n,i normal vector of each cutter e _t,i tangential vector of each cutter F Total forcing matrix

F _x , F _y , F _z Applied forces in x, y, z directions F cutter,i Applied force in each cutter

I Penetration depth

k 1 Translational stiffness of the rear part (N/m) k ₂ Rotational stiffness of the boom (Nm)

k ₄ Translational stiffness of the front part (N/m) k _e Out of plane tilt stiffness of the boom (Nm) k _t Twist stiffness of the boom (Nm)

k _4p Rotational stiffness of the front part (Nm)

K _eq Total structural stiffness matrix in the equilibrium position M Total mass matrix

M ₁ Mass of the rear part (kg) M ₂ Mass of the boom (kg) M ₃ Mass of the cutter wheel(kg) M ₄ Mass of the front part (kg)

M _x , M _y , M _z Applied moments in x, y, z directions M _eq Total mass matrix in the equilibrium position M swing , M tilt Internal swing-tilt boom moments S Cutter spacing

U CS Uniaxial compressive strength V Eigenvectors

W _t Cutter width tip

z ₁ Coordinate of mass M ₁

z ₄ Coordinate of mass M ₄

List of Figures

1 Drilling Scheme [1] . . . . 8

2 3-Dimensional CAD model of the mobile miner 22H [4] . . . . 9

3 Main movements-Local coordinate system [3] . . . . 9

4 Schematic representation of the parts of the machine [4] . . . . 10

5 Conceptual design-optimization . . . . 10

6 Project flowchart . . . . 11

7 2-D 3 Degrees of freedom reference model, Top view [3] . . . . 13

8 3-D 7 degrees of freedom model, Top view . . . . 14

9 3-D 7 degrees of freedom model, Side view . . . . 14

10 Cutter-wheel with vectors explanation . . . . 15

11 Applied external and internal Forces [3] . . . . 16

12 Penetration and Applied forces in one cutter . . . . 19

13 Applied forces in one cutter . . . . 20

14 Boom swing-tilt moments . . . . 20

15 Structural representation of the system . . . . 20

16 Equivalent fatigue load in y direction . . . . 21

17 Force in y-direction versus tilt stiffness . . . . 21

18 Equivalent fatigue load in x direction . . . . 21

19 Equivalent fatigue load in z direction . . . . 22

20 Power spectral density of total forces in y-direction for combination 64 . . . 23

21 Power spectral density of total forces in y-direction for combination 54 . . . 23

22 Equivalent fatigue load in x direction . . . . 24

23 Equivalent fatigue load in y direction . . . . 24

24 Equivalent fatigue load in z direction . . . . 25

25 Power spectral density of total forces in y and x directions for combination 180 25 26 Power spectral density of total forces in y and x directions for combination 67 25 27 Equivalent fatigue load in x direction . . . . 26

28 Equivalent fatigue load in y direction . . . . 27

29 Equivalent fatigue load in z direction . . . . 27

30 Equivalent fatigue load in x direction . . . . 27

31 Equivalent fatigue load in y direction . . . . 28

32 Equivalent fatigue load in z direction . . . . 28

List of Tables 1 Eigenfrequencies in Hz for case 64 . . . . 23

2 Eigenfrequencies in Hz for case 54 . . . . 23

3 Eigenfrequencies in Hz for case 180 . . . . 26

4 Eigenfrequencies in Hz for case 67 . . . . 26

4 Aknowledgements

This Master Thesis project consists the final step of my postgraduate studies in Solid Me-

chanics at KTH in Stockholm. The project was carried out in Epiroc’s Applied Mechanics

department in Rocktec division in ¨ Orebro. The industrial supervisor was PhD specialist

Robert Pettersson and the examiner-supervisor at KTH was Dr. Arne Nordmark. I would

like to thank both supervisors for their constant support and guidance, the whole group that

I belonged to and the measurement technique group for their valuable aid as well.

5 Introduction

5.1 Background

Mining can be distinguished in two main categories, surface and underground mining. In the current project an underground mining procedure is analysed. Mobile miners have huge mobility capability. They use wheels or crawler tracks in order to move and have one operator. While cutting, with a conveyor system the pieces of rock are transferred to the back of the machine. They are also equipped with rock bolters that provide safety by minimizing the possibility of the rock of the top falling on the ground and causing human damage.

The mobile miner is considered a tunnel boring machine ”TBM” and it has a circular cross section with cutters placed circumferentially, that provides great cutting ability. They are used in ”hard rock” applications, where the uniaxial compressive strength exceeds 80 Mpa.

However, tensile strength and fracture behaviour are also important factors to be considered.

A drilling scheme is presented in figure (1)

Figure 1: Drilling Scheme [1]

Fracture behavior, regardless the excavation technique is created by applying stresses in the

rock in order to force the rock fail and create chips. Especially, during the mobile miner

excavation, pressure bubble is created under the rock surface, that results in formation of

a crater due to compressive stress and spalling by tensile stress among the kerfs from the

contact of multiple disc cutters [1].

5.2 Mobile miner 22H

Figure 2: 3-Dimensional CAD model of the mobile miner 22H [4]

Figure (2) shows a CAD representation of the machine that is going to be studied. This machine is used for low-profile mining, where the tunnel can be as low as 2.2 meters and the total advance that can be achieved is 10-12 meters per day [2]. The cutter-wheel has diameter of 3.5 m and the main movements apart from the rotational around its center, are the tilt and the swing. Tilt consists of rotation around the x-axis and the swing around y-axis as shown in figure (3). The cutter-wheel rotates mainly around 12 rpm and is driven by hydraulic motor. The swing is provided by swing cylinders. The global directions are also presented.

Figure 3: Main movements-Local coordinate system [3]

The cutter-wheel contains 32 cutters that are divided in four segments, that are identical in

pairs. In figure (4) the two main type of cutters, and the spacing between the four different

segments are shown. The face cutters, like the name implies face the rock and the gage

cutters have relative inclination angle.

Figure 4: Schematic representation of the parts of the machine [4]

For the design of such a machine the applied loads (static and fatigue), the machine properties such as the needed power of the motor and the cutter properties must be considered. As well as that, the tunnel dimensions, geometry and the rock properties are also very important.

Finally, the desirable and expected cutter life and advance rate of the machine must be also taken into account. In this way an optimization of the machine can be implemented. These concepts are shown schematically in figure (5).

Figure 5: Conceptual design-optimization

5.3 Aim of the Project

The aim of the current study of miner 22H is to gain understanding of the dynamics of the

mobile miner 22H by modelling the dynamics of the machine with a multibody system of rigid

bodies in order to find lower fatigue loads in the machine so as the life of the machine will be increased. Firstly, the necessary experimental and FEM data were collected from previous studies, in order to start building the model. The method for that was the exploitation of the symbolic mechanics of Python’s toolbox. This method enables the formulation of Lagrange equations and then equations of motion can be obtained. Matlab will then be used for the necessary results such as different parametric studies and possible optimization routine that is not part of this project. The procedure is presented in the figure (6).

Figure 6: Project flowchart

5.4 Problem description

Through many previous investigations it was inferred that the cutter-wheel design and more-

over the cutter placements and structural properties of the machine have important effect on

the load variations that occur during the cutting procedure. Therefore, it is very important

to find a specific individual cutter placement or machine structural properties that cause less

fatigue loads. In this project cutter head movement, cutters placement and the advance of

the cutter-wheel will be varied.

Structural properties include the stiffness of different machine parts and the damping factors

that both are obtained through simulations in FEM software and experimental measurements

such as experimental modal analyses. The necessary geometrical properties of the miner were

extracted from mechanical drawings.

6 Methods

6.1 Reference model

As mentioned in introduction, the first step is to define a mechanical system of rigid bodies that can represent the machine. The first approach was done by deriving a 3 degrees of freedom (d.o.f) system that is 2-dimensional and takes into account the translation movement of the front part of the miner and the swing movement of the boom as shown in the figure (7) with green arrows. The local reference frames and the structural stiffness-damping factors for the front part (M ₁ ), the boom (M ₂ ) and the cutter-wheel (M ₃ ) for each body need to be introduced for the symbolic presentation. Swing stiffness (k ₂ ) is obtained as the stiffness of the swing cylinders and the translational stiffness (k ₁ ) as the part’s stiffness clearly. The two corresponding dampings (c ₁ ) , (c ₂ ) were obtained from modal analysis performed in Ansys software. The third degree of freedom (φ) is driven by the motor that is attached in the center of the cutter-wheel.

It can be mentioned here that the equations of motion that describe this 3.d.o.f model, could be analytically derived “by hand” time efficiently, due to its size, as well.

Figure 7: 2-D 3 Degrees of freedom reference model, Top view [3]

6.2 Model evolution

The next step is the progressive extension of the reference model so as more precise results can be obtained. This step involves the thorough study of previous analyses both in FEM and experimental modal analysis in low modes. Those studies identified the need of taking into account also the translation of the rear part of the miner, the out of plane tilt, namely the up-down rotation of the boom, the twist of the boom and finally a “second” type swing that appears in mass (M ₄ ) that now represents the front part mass. Now the red arrows represent the 4 more d.o.f.s. For these movements also new stiffness and damping parameters must be applied. The final model is presented in figure (8) that presents the top view and in figure (9) that presents the side view of the model.

Figure 8: 3-D 7 degrees of freedom model, Top view

Figure 9: 3-D 7 degrees of freedom model, Side view

6.3 Analysis procedure (1)

First priority is to account for which procedure will be implemented in order to calculate the external forces and moments that are applied in the cutter-wheel during cutting. Here, method from Colorado School of Mines (CSM) was applied that will be explained below.

The contact angle (α) is given by the equation:

α (pen) = cos ⁻¹ ( ( ^D ₂

− pen)

D

2

) (1)

Here D c is the cutter diameter and pen is the penetration depth.

The nominal force in each cutter is given by the following formula:

F c , nominal = C c W t

D c

2 α ³ ∗ v u u t

U CS ² ∗ BT S ∗ S α

q W _t ^D ₂

(2)

Here U CS is the uniaxial confined strength, W _t is the cutter width tip, BT S is the Brazilian tensile strength, S is the cutter spacing and C _c is a cutter load constant. The values were taken from studies at the test site.

In order to fully investigate the forces in the whole cutter-wheel cutters the current load- penetration relationship was used.

F cutter,i = e n,i ∗ F c , nominal(pen i ∗ e n,i ) + e t,i ∗ F c , nominal tan (pen i ∗ e n,i ) (3)

In equation (3) the vectors e _n,i , e _t,i and pen _i are explained in figure (10)

Figure 10: Cutter-wheel with vectors explanation

The cutter-wheel is considered a rigid body and the displacement of its center of mass is given by ∆x, ∆y, ∆z. Also the twist, the tilt and the swing of the boom affect the calculation of the cutter forces, because they have large impact on each individual cutter’s position.

So according to these properties the vectors pen ~ _i and ~ e _n,i , ~ e _t,i can be calculated for each individual cutter, by taking into account the generalised coordinates.

Moreover, with the use of the CSM method the 3 external forces (F _x , F _y , F _z ) and moments

(M x , M y , M z ) on each cutter in the 3 direction can be calculated and then the total forces

can be obtained. As well as that, the calculation of the internal boom swing (M _swing ) and

tilt (M _tilt ) moments for the later parametric studies is vital. In figure (11) a schematic

representation of the loads is given.

Figure 11: Applied external and internal Forces [3]

6.4 Analysis Procedure (2)

In order to calculate the equations of motion there is a need of obtaining the total forces, that act on the cutter- wheel as shown in figure (11). This is done by Lagrange formulation.

The formulation of Lagrange equation is achieved by defining the reference frames, the positions-velocities of each mass, the potential and kinetic energy and finally the generalised forces such as damping forces and external loads.

The Lagrangian is given by the difference of potential energy V and the kinetic energy T as:

L = T − V (4)

and the equations of motion or the total force and mass matrix of the system of rigid bodies is given by:

d dt ( ∂L

∂ ˙ q _i ) − ∂L

∂q _i = Q _i (5)

In (5) ˙ q _i and q _i are the velocities and displacement vectors and Q _i are the generalised forces.

From (5) the equations of motion can be derived and then be rewritten in the form of (6) and (7), where the mass matrix (M) and the force vector (F) can be computed.

˙

q = u (6)

and :

M ˙u = F (q, u) (7)

The obtained M, F matrices are inserted to Matlab in order to derive numerically the ac-

celerations of each d.o.f. Then, by a time-stepping integration solver and more specifically

with ode45 the time displacements of each d.o.f can be obtained.

6.5 Python procedure

In order to apply this formulation, Python offers a very helpful symbolic environment for representing mechanics. The code is given in Appendix. Briefly review of the most important commands will be given below [7]:

1. Firstly, all the variables that describe each property need to be assigned to symbols either static or dynamic with the commands symbols and dynamicsymbols.

2. For the creation of the reference frames, that are shown in figure (8) the Ref erenceF rame command was used.

3. For the correct orientation of each triad command orient was used that needs to get supplied by the preferred orientation angle and the previous frame.

4. Then, there is the need of defining the four points that represent the centers of mass of each body and their respective positions and velocities. This was achieved by the use of P oint, set.pos, set.vel, and v1pt _theory or v2pt _theory , depending on if the relative velocity has to be defined with the aid of 1-point or 2-point theorem.

5. Afterwards, it is essential to define the Inertia according to x,y,z for each point with the use of inertia command, that actually represents a Rigid body. Then, in order to build the final rigid body, the concentrated mass is defined with the command mass and then with RigidBody commands the system is finally created.

6. Moreover, it is now feasible to formulate automatically the kinetic energy of each mass by implementing kinetic energy command. However, the potential energy has to be defined manually and this is done by programming the gravity part and the elastic spring energies part.

7. The Lagrangian equation can be formulated by running the command Lagrangian and then the Lagrange method can be calculated by running LagangeM ethod after having defined all the non conservative loads as well. Mass and force matrices can be obtained by executing LagangeM ethod.mass matrix and LagrangeM ethod.f orcing.

8. Finally for the modal analysis part, linearisation command (linearize) was used in order to derive the stiffness, mass and damping matrices.

9. Additional useful command is the pos.f rom in order to find the position vectors in

each axis and to formulate potential energies and the penetration vector.

6.6 Modal Analysis

Modal analysis was also conducted so as to verify the FEM results and to enhance the under- standing of the parametric studies. For this reason, the eigenvalue problem was solved, by formulating the mass, damping and stiffness matrices, evaluated in the equilibrium positions.

In order to estimate the equilibrium position gravity-spring forces need to be analysed. The state-space matrix form of the problem is obtained by linearisation of (6) and (7):

A ˙z = Bz (8)

where

z = (q, u) (9)

So in the current problem it is:





I 0

0 M _eq



 ∗ λ ∗ V =





0 I

−K _eq C _eq



 ∗ V (10)

In equation (10) I is the identity matrix and M _eq , K _eq ,C _eq are the mass, stiffness and damping matrices evaluated in the equilibrium position.

From the eigenvalues λ and eigenvectors V , the eigen-frequencies, the modal damping and the mode shapes can be extracted.

7 Results-Discussion

7.1 Initial conditions of the study

The starting point of the analysis was set for swing of the boom of 22 °, tilt angle of 10.5°, cut- ter penetration of 0.01m, and angular velocity of the cutter-wheel of 12 rpm. The simulation time was set to 20 sec, therefore each cutter is in contact for 4 times with the rock.

It must be mentioned that the equilibrium positions of each body are set close to the initial

conditions. Fatigue analysis in order to calculate the fatigue loads on the cutter-wheel and

boom joint was performed with a rainflow method. The transient part of about 2 sec is

omitted in the parametric analysis.

7.2 Parametric studies concept

In current project four main parametric studies were performed. The first analyses the fatigue loads for three different values of each of the six different stiffnesses, one 10% above the reference value and one 10% below the reference value. The second investigates damping factors studies of 35%, 25%, 15% of the reference obtained factors were achieved. The third investigates the removal of gage cutters for more realistic representation. The final study analyses a new cutter placement pattern.

There will be 729 combinations and the equivalent fatigue load in each direction will be presented. It was observed that for each case and for all the parametric studies the loads in the 3 directions x, y, z can provide all the information needed for the moments both internal and external as well, since they show similar relationship. Thus, only 3 graphs will be presented in each study. These graphs contain the equivalent fatigue loads in x, y, z directions in vertical axis, with respect to each case combination in horizontal axis. Finally, modal analysis and power spectral density spectrums of the sum of total forces in each direction will be presented in order to identify any possible “match” of the resonances.

7.3 Penetration - Forces graphs for reference case

In figures (12) and (14) the penetration and the forces and moments in x, y, z directions and also in tangential and normal directions are shown for one cutter. Finally, the two types of internal moments are presented. It is observed that the time functions are periodic.

(a) Penetration in one cutter (b) Cutter moments in x, y, z directions

Figure 12: Penetration and Applied forces in one cutter

(a) Applied forces in x, y, z directions (b) Applied forces in tangential and normal di- rection

Figure 13: Applied forces in one cutter

(a) Boom swing internal moment (b) Boom tilt internal moment

Figure 14: Boom swing-tilt moments

7.4 Parametric Study (1)

As described in previous section, in this parametric study there will be stiffness alteration of ± 10 %, comparing to the reference values.

The figure (15) shows the structural properties as a reminder.

Figure 15: Structural representation of the system

Figure 16: Equivalent fatigue load in y direction

Figure 17: Force in y-direction versus tilt stiffness

Figure 18: Equivalent fatigue load in x direction

Figure 19: Equivalent fatigue load in z direction

From (16) appears that the force in y direction (F _y ) and the moments around z, x axis and the boom tilt moment, M _z , M _x , M _tilt respectively, are mainly affected by the out of plane tilt stiffness (k _e ), and when it decreases, the loads increase. This is explained by the seperate graph(17). In this graph three different regions are indicated and the transition point between them are the three different tilt stiffness values. From graph (18) it is seen that the equivalent fatigue force in x direction (F _x ) is mostly sensitive to the swing movement of the front part (M ₄ ), and it reaches its maximum when the corresponding swing stiffness (k _4p ) has the lowest value.

The swing moment in the boom is highly affected by (k _4p ) and (k ₄ ) that represents the translation of M ₄ . The extreme case appears when (k _4p ) is minimum and simultaneously k ₄ is maximum. From graph (19), it is induced that moment around y direction (M y ) and force in z direction (F _z ) are dependent mainly in k ₄ and they both increase when k ₄ increases.

For the cases with minimum and maximum loads in y direction modal analysis and power spectral density spectrums for the total forces will be presented below:

For the combination 64, where the loads are the maximum, it is defined which state of

structural properties it is through Matlab script. It can be seen that a resonance of 12 Hz

appears.

Figure 20: Power spectral density of total forces in y-direction for combination 64

The eigenfrequencies from Modal analysis for this case are : Table 1: Eigenfrequencies in Hz for case 64 mode 1 mode 2 mode 3 mode 4 mode 5 mode 6

1.7 2.5 5.8 8.6 16.6 17.1

For the combination 54 the PSD is :

Figure 21: Power spectral density of total forces in y-direction for combination 54

In figure (21) no clear peak is appearing.

The eigenfrequencies from Modal analysis for this case are:

Table 2: Eigenfrequencies in Hz for case 54 mode 1 mode 2 mode 3 mode 4 mode 5 mode 6

1.7 2.8 5.7 7.8 16.2 16

It appears that there in not an identification, and this is due to the non-linear boundary condition in rock-cutterwheel interaction. Also the structure is highly damped as seen from PSDs. That is why the damping will get reduced significantly in next parametric study.

7.5 Parametric study (2)

In this section the equivalent fatigue loads will be presented when the damping factors are lowered to 35%, 25%, 15% compared to the reference case.

Again all the eight types of loads follow the same behaviour with the forces in x, y, z directions. The graphs are shown in next three figures.

Figure 22: Equivalent fatigue load in x direction

Figure 23: Equivalent fatigue load in y direction

Figure 24: Equivalent fatigue load in z direction

In this study, it is shown that fatigue loads in x, z direction even follow similar behaviour.

This explains the two different groups of results.

The loads F _z , F _x , M _y , M _swing are mainly sensitive to the damping of the swing of the boom (c ₂ ) and with the damping of translation of M ₄ only when c ₂ has its lowest value. The maximum case appears when both c ₂ and c ₄ are 15% of the reference case.

The remaining loads (M _z , M _x , F _y , M _tilt ) are sensitive primarily by c ₄ and they are increased when c ₄ has lower values.

Again, it is important to inspect the frequency spectrum and modal analysis.

Figure 25: Power spectral density of total forces in y and x directions for combination 180

Figure 26: Power spectral density of total forces in y and x directions for combination 67

The modal analysis for the two combinations give the following results:

Table 3: Eigenfrequencies in Hz for case 180 mode 1 mode 2 mode 3 mode 4 mode 5 mode 6

2.2 2.7 5.6 8.2 18 19.8

Table 4: Eigenfrequencies in Hz for case 67 mode 1 mode 2 mode 3 mode 4 mode 5 mode 6

2.1 2.7 5.6 8.2 17.9 19.5

Again there is no clear overlap of the eigenfrequencies but the lower damping provides more distinguishable resonance behaviour, as seen in figures (25) and (26).

7.6 Parametric study (3)

In this case a more realistic cutting behaviour was investigated. During the cutting procedure the gage cutters are not always in contact with rock, and this happens when the cutter-wheel changes position vertically. The current parametric study is identical with the first one in terms of stiffness alterations. Again, two main groups are distinguished.

The three main figure groups appear below:

Figure 27: Equivalent fatigue load in x direction

Figure 28: Equivalent fatigue load in y direction

Figure 29: Equivalent fatigue load in z direction

In this case, from figure (27) the twist boom stiffness (k _t ) affects significantly the forces in x direction (F _x ), the boom tilt moment M _tilt and the boom swing moment M _swing and while k _t increases the (M _t ) decreases and the F _x increases. The rest of the loads (F _y , F _z , M _x , M _y , M _z ) are mainly affected by out of plane tilt stiffness (k _e ), as presented in figures (29) and (28).

7.7 Parametric study (4)

In the final parametric study, a new cutter pattern was proposed and was tested under stiffness alteration once again.

Figure 30: Equivalent fatigue load in x direction

Figure 31: Equivalent fatigue load in y direction

Figure 32: Equivalent fatigue load in z direction

From graph (30) and (32), it is observed that F _x , F _z forces and boom swing moment M _swing are sensitive on stiffness k ₄ and on k ₁ . While k ₁ and k ₄ decrease all the 3 loads increase.

Finally from (31) it is seen that the rest of the loads (M _x , M _y , M _z , M _tilt ) are dependent

significantly on twist stiffness k _t , and they reach their maximums when k _t decreases.

8 Conclusions

The conclusions that are derived from this project are the following:

ˆ A methodology is presented on how to create a general multibody system that repre- sents a miner and then using different numerical simulations, the dependence on the parameters studies can be investigated faster than FEM.

ˆ For the reference design stiffness alterations and for the gage cutters removal, mainly an increase of out plane tilt stiffness is required so as to lower the fatigue loads. Also a decrease of the front part translational stiffness and an increase of the front part swing stiffness is required in order to lower the fatigue loads.

ˆ Concerning the damping factors parametric study, it is suggested to increase the damp- ing values of front part translation and of boom swing moment in order to lower the fatigue loads.

ˆ The new possible cutter placement requires an increase of the boom twist stiffness.

ˆ The new cutter placement results in lower forces in the x and z directions, moment around the y axis and boom swing moment, but meanwhile in higher forces in y and moments around the x, z axis, and boom tilt moment as well. That shows the need of further trade-off investigations.

ˆ The results also suggest that the modal analysis is not useful due to non-linear bound- ary conditions or rock-machine interaction.

ˆ In order to represent the results from real measurements, there is a need of increasing

the stiffness factors and decreasing the damping factors. This occurs from comparison

of real boom moments measurements with the results of this project (tilt, swing of the

boom).

9 Future work recommendations

ˆ Further validation with experimental results in order to verify the model and get more insight about the appropriate structural parameter adjustments, is needed.

ˆ Implementation of optimization routine in order to suggest machine structure proper- ties and cutter placement.

ˆ Consideration of more general load models than CSM that approach the surface irreg- ularities and fracture behaviour of the rock.

The current Master thesis project was very challenging and appealing for myself. Based on

the literature research and review of previous studies, I believe that a time-efficient method

is developed. However, further improvements and investigations are needed.

References

[1] Robert Pettersson: Presentation: Msc Thesis- Conseptual modelling, Epiroc, ¨ Ore- bro, 2019.

[2] Epiroc: Presentation: Mobile Miner technical specifications [Online, Available:

https://www.epiroc.com/en-uk/products/mechanical-rock-excavation/mobile-miner- 22h], Epiroc.

[3] Robert Pettersson: Technical Report Loads,dynamics and performance in mechan- ical rock excavation, Epiroc ,Rocktec Division, Applied Mechanics ¨ Orebro, 2019.

[4] Martin J. Persson: Technical Report Cutter and Structure Load Measurements on Reef Miner in Kvarntop Mine, Epiroc ,Measuring Technique Lab., ¨ Orebro, 2016.

[5] Annie Melbro, Egil Jersenius: Design optimization of a Mobile Miner A master thesis at Epiroc Rock Drills AB, Link¨ oping University, 2018.

[6] Johnny Lyly, Sverker Hartwig, Gunnar Nord: Technical Report Epiroc mobile miner- hard rock cutting is now a reality, Epiroc, ¨ Orebro.

[7] Welcome to SymPy’s documentaion URL: docs.sympy.org/latest/index.html

10 Appendix

10.1 Python Code

With t h i s c o d e t h e m u l t i b o d y system o f t h e f o u r r i g i d b o d i e s i s r e p s e n t e t e d s y m b o l i c a l l y i n t h e Python e n v i r o n m e n t .

R e f e r e n c e s t o s e c t i o n 5 . 5 a r e i n c l u d e d .

from sympy . s o l v e r s i m p o r t s o l v e , c h e c k s o l i m p o r t numpy , s c i p y . i o

i m p o r t sympy a s sp

from sympy . s o l v e r s i m p o r t s o l v e

from sympy . p h y s i c s . m e c h a n i c s i m p o r t LagrangesMethod , L a g r a n g i a n , m e c h a n i c s p r i n t i n g

from sympy . p h y s i c s . m e c h a n i c s i m p o r t ReferenceFrame , P a r t i c l e , Point , mprint , dot

from sympy . p h y s i c s . m e c h a n i c s i m p o r t dynamicsymbols , k i n e t i c e n e r g y , p o t e n t i a l e n e r g y , msubs

from sympy . p h y s i c s . m e c h a n i c s i m p o r t RigidBody , i n i t v p r i n t i n g # , m e c h a n i c s p r i n t i n g

from sympy i m p o r t symbols , p p r i n t , Matrix , MatrixSymbol from sympy i m p o r t i n i t p r i n t i n g , s i m p l i f y , f a c t o r , expand

# l a t e x

from IPython . d i s p l a y i m p o r t d i s p l a y

from sympy . p h y s i c s . m e c h a n i c s . f u n c t i o n s i m p o r t i n e r t i a from sympy . p h y s i c s . v e c t o r . p r i n t i n g i m p o r t v p p r i n t , v l a t e x i m p o r t math

from math i m p o r t c o s , s i n

from sympy . u t i l i t i e s . c o d e g e n i m p o r t c o d e g e n

z = dynamicsymbols ( ’ z ’ ) #Ref 1 zd = dynamicsymbols ( ’ z ’ , 1 )

t h e t a = dynamicsymbols ( ’ t h e t a ’ )

thd = dynamicsymbols ( ’ t h e t a ’ , 1 )

p h i=dynamicsymbols ( ’ phi ’ ) p h i d=dynamicsymbols ( ’ phi ’ , 1 )

e=dynamicsymbols ( ’ e ’ ) ed=dynamicsymbols ( ’ e ’ , 1 )

tw=dynamicsymbols ( ’ tw ’ ) twd=dynamicsymbols ( ’ tw ’ , 1 )

z p=dynamicsymbols ( ’ z p ’ ) z p d=dynamicsymbols ( ’ z p ’ , 1 )

t h e t a p=dynamicsymbols ( ’ t h e t a p ’ ) t h e t a p d=dynamicsymbols ( ’ t h e t a p ’ , 1 )

omega=dynamicsymbols ( ’ omega ’ )

e v a l u e , t h e t a v a l u e , Me, m1 , m2 , m3 , m4 , g , k1 , c1 , k2 , ke , kt , k4 , k4p , c2 , ctw , c4p , ce , c t h e t a p , L2 , L3 , Lg2 , t h e t a 0 ,

Cz , Ctheta , Ce , Ctw ,

Cz4 , Ctheta p , z0 , y , y0 , y1 , y2 , y3 , y4 , y5 , y6 , y7 , y8 , y9 , y10 , y11 , y12 ,

y13 , y14 , e0 , tw0 , zp0 , t h e t a p 0 , I8 , I9 , I10 , I11 , I12 , I30 , I31 , I40 , I44 ,

I41 , i , j , I3 , Mextx ,

Mexty , Mextz , Fx , Fy , Fz , t = symbols ( ’ e v a l u e t h e t a v a l u e Me m1 m2 m3 m4 g k1 c1 k2 ke k t k4

k4p c2 ctw c4p c e c t h e t a p L2 L3 Lg2 t h e t a 0 Cz Ctheta Ce Ctw C4p C t h e t a p z0 y

y0 y1 y2 y3 y4 y5 y6 y7 y8 y9 y10 y11 y12 y13 y14 e0 tw0 zp0 t h e t a p 0 I 8 I 9 I 1 0 I 1 1

I 1 2 I 3 0 I 3 1 I 4 0 I 4 4 I 4 1 i j I 3 Mextx Mexty Mextz Fx Fy Fz t ’ )

I 2 = symbols ( ’ I2 ’ )

i n i t v p r i n t i n g ( u s e u n i c o d e=True )

N = R e f e r e n c e F r a m e ( ’N’ ) #Ref 2

A = R e f e r e n c e F r a m e ( ’A’ ) B = R e f e r e n c e F r a m e ( ’ B ’ ) C = R e f e r e n c e F r a m e ( ’ C ’ ) D = R e f e r e n c e F r a m e ( ’D’ ) T = R e f e r e n c e F r a m e ( ’ T ’ ) P = R e f e r e n c e F r a m e ( ’ P ’ )

A. o r i e n t (N, ’ Axis ’ , [ 0 , N. x ] ) #Ref 3

P . o r i e n t (A, ’ Axis ’ , [ t h e t a p , A. y ] ) D. o r i e n t (P , ’ Axis ’ , [ e , P . x ] )

B . o r i e n t (D, ’ Axis ’ , [ t h e t a , D. y ] ) T . o r i e n t (B, ’ Axis ’ , [ tw , B . z ] ) C . o r i e n t (T, ’ Axis ’ , [ phi , T . y ] )

O=P o i n t ( ’O’ ) #o r i g i n p o i n t on t h e w a l l #Ref 4 O. s e t v e l (N, 0 )

O1 = P o i n t ( ’ O1 ’ ) # mass m1

O1 . s e t p o s (O, z * N. z ) O1 . s e t v e l (N, zd * N. z )

O4 = P o i n t ( ’ O4 ’ )

O4 . s e t p o s (O1 , z p * P. z ) O4 . s e t v e l (P , z p d *P. z ) O4 . v 1 p t t h e o r y (O1 , N, P)

O2 = P o i n t ( ’ O2 ’ ) # boom c e n t e r o f mass

O2 . s e t p o s (O4 , =L2 * B. z )

O2 . v 2 p t t h e o r y (O4 , N, B)

O3 = P o i n t ( ’ O3 ’ )

O3 . s e t p o s (O2 , =L3 * B. z )

O3 . v 2 p t t h e o r y (O2 , N, B)

I 1=i n e r t i a (A, I8 , I9 , I 1 0 ) # i n e r t i a o f mass m1 #Ref 5 I 4=i n e r t i a (P , I40 , I44 , I 4 1 )

#RigidBody ( name , m a s s c e n t e r , frame , mass , i n e r t i a ) mass1=RigidBody ( ’ mass1 ’ , O1 , A, m1 , ( I1 , O1 ) )

I 2=i n e r t i a (T, I11 , I2 , I 1 2 )

mass2=RigidBody ( ’ mass2 ’ , O2 , T, m2 , ( I2 , O2 ) ) #Ref 5

I 3=i n e r t i a (C, I30 , I3 , I 3 1 )

mass3=RigidBody ( ’ mass3 ’ , O3 , C, m3 , ( I3 , O3 ) )

mass4=RigidBody ( ’ mass4 ’ , O4 , P , m4 , ( I4 , O4 ) )

mass1 . k i n e t i c e n e r g y (N) #Ref 6 mass2 . k i n e t i c e n e r g y (N)

mass3 . k i n e t i c e n e r g y (N) mass4 . k i n e t i c e n e r g y (N)

# t o t a l k i n e n e r g y

Ttot=k i n e t i c e n e r g y (N, mass1 , mass2 , mass3 , mass4 )

# p o t e n t i a l e n e r g i e s f o r mass1 , mass2 and t o t a l

mass1 . p o t e n t i a l e n e r g y =1/2.0 *k1 *( z=z0 )**2 +

1 / 2 . 0 * k4p *( theta p =thetap0 )**2

mass2 . p o t e n t i a l e n e r g y =1/2.0 *k2 *( theta=theta0 )**2

+1/2.0 * ke *( e=e0 )**2 +1/2.0* kt *(tw=tw0 )**2 + m2*g* dot (O2 . pos from (O) ,N. y ) mass3 . p o t e n t i a l e n e r g y= m3 *g* dot (O3 . pos from (O) ,N. y )

mass4 . p o t e n t i a l e n e r g y =1/2.0 *k4 *( z p=zp0 )**2 + m4*g* dot (O4 . pos from (O) ,N. y )

Vtot=mass1 . p o t e n t i a l e n e r g y + mass2 . p o t e n t i a l e n e r g y

+ mass3 . p o t e n t i a l e n e r g y+ mass4 . p o t e n t i a l e n e r g y

# I c o u l d not do i t l i k e t o t k i n e t i c e n e r g y a u t o m a t i c a l l y

# L a g r a n g i a n

La=L a g r a n g i a n (N, mass1 , mass2 , mass3 , mass4 ) #Ref 7

# G e n e r a l i s e d f o r c e s Cz= =c1 *zd

Ce= =ce *ed Cz4= =c4p* z pd Ctheta= =c2 *thd

C t h e t a p= =ctheta p * theta pd Ctw= =ctw*twd

f l = [ ( C, +Mextx *N. x+Mexty*N. y+Mextz*N. z ) , (O3, Fx *N. x+

Fy *N. y+Fz*N. z ) , (C, Me *C. y ) ,

(T, ( =Me) * T. y ) , (T, Ctw * T. z ) , (B, =Ctw * B. z ) , (B, Ctheta * B. y ) , (D, ( =Ctheta )*D. y ) , (D, Ce * D. x ) , (P , =Ce*P. x ) , (P, Ctheta p * P. y ) , (A, =Ctheta p * A. y ) , (O4, +Cz4 * P. z ) ,

(O1 , =Cz4 * P. z ) , (O1, Cz * N. z ) , (O, =Cz * N. z ) ]

q = [ z , e , t h e t a , z p , t h e t a p , tw , p h i ]

LM = LagrangesMethod ( La , q , frame=N , f o r c e l i s t = f l )

L eq = LM. f o r m l a g r a n g e s e q u a t i o n s ( )

M=LM. m a s s m a t r i x F=LM. f o r c i n g

mprint ( msubs (F , { Mextx : 0 , Mexty : 0 , Mextz : 0 , Fx : 0 , Fy : 0 , Fz : 0 , g : 0 , t h e t a p : t h e t a p 0 ,

t h e t a p . d i f f ( ) : 0 , e : e0 , e . d i f f ( ) : 0 , t h e t a : t h e t a 0 , t h e t a . d i f f ( ) : 0 ,

tw : tw0 , tw . d i f f ( ) : 0 , p h i . d i f f ( ) : 0 ,

z : z0 , z . d i f f ( ) : 0 , z p : zp0 , z p . d i f f ( ) : 0 } ) )

F=s i m p l i f y (F) M=s i m p l i f y (M)

mprint (F) mprint (M)

O3Z=dot (O3 . p o s f r o m (O) ,N. z ) #Ref 9 O3Y=dot (O3 . p o s f r o m (O) ,N. y ) #p r o j e c t i o n O3X=dot (O3 . p o s f r o m (O) ,N. x )

## NOW MODAL ANALYSIS

Fmodal=(msubs (F , { Mextx : 0 , Mexty : 0 , Mextz : 0 , Fx : 0 , Fy : 0 , Fz : 0 , t h e t a p . d i f f ( ) : 0 , e . d i f f ( ) : 0 , t h e t a . d i f f ( ) : 0 , tw . d i f f ( ) : 0 , p h i . d i f f ( ) : 0 , z . d i f f ( ) : 0 , z p . d i f f ( ) : 0 } ) )

op ={Mextx : 0 , Mexty : 0 , Mextz : 0 ,

Fx : 0 , Fy : 0 , Fz : 0 , z : 0 , e : e v a l u e , t h e t a : t h e t a v a l u e , z p : 0 , t h e t a p : 0 ,

tw : 0 , p h i : 0 , zd : 0 , ed : 0 , thd : 0 , z p d : 0 , t h e t a p d : 0 ,

twd : 0 , p h i d : 0 , zd . d i f f ( ) : 0 , ed . d i f f ( ) : 0 , thd . d i f f ( ) : 0 , z p d . d i f f ( ) : 0 , t h e t a p d . d i f f ( ) : 0 , twd . d i f f ( ) : 0 ,

p h i d . d i f f ( ) : 0 }

#Ref 9

M7, K7 , B7 , i n p v e c = LM. l i n e a r i z e ( q i n d =[ z , e , t h e t a , z p , t h e t a p , tw , p h i ] , q d i n d=

[ zd , ed , thd , z pd , t h e t a p d , twd , p h i d ] , o p p o i n t=op ) M7=s i m p l i f y (M7)

K7= =s i m p l i f y (K7)

### t o t a l d i s p l a c e m e n t s f o r m a s s e e s 1 2 4 O1Z=dot (O1 . p o s f r o m (O) ,N. z )

O1Y=dot (O1 . p o s f r o m (O) ,N. y ) O1X=dot (O1 . p o s f r o m (O) ,N. x )

O4Z=dot (O4 . p o s f r o m (O) ,N. z ) O4Y=dot (O4 . p o s f r o m (O) ,N. y ) O4X=dot (O4 . p o s f r o m (O) ,N. x )

O2Z=dot (O2 . p o s f r o m (O) ,N. z ) O2Y=dot (O2 . p o s f r o m (O) ,N. y ) O2X=dot (O2 . p o s f r o m (O) ,N. x )

#[(M name , M code ) ] = c o d e g e n ( ( ”M” , M) , ’ Octave ’ )

#[( F name , F c o d e ) ] = c o d e g e n ( ( ” F” , F ) , ’ Octave ’ )

TRITA SCI-GRU 2020:272

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