the UWA Profiles and Research Repository...In the middle of the nineteenth century scientists started to add the lubrication to the friction contact. The first study of fluid lubrication

vi

TABLE OF CONTENTS

ABSTRACT ............................................................................................................................. iii

General introduction .................................................................................................................. 1

1.1 Frictional sliding ............................................................................................................... 2

1.2 Stick-slip phenomenon ..................................................................................................... 4

1.3 Friction models ................................................................................................................. 7

1.4 Friction law ..................................................................................................................... 11

1.5 Thesis overview .............................................................................................................. 14

Frictional sliding in rock cutting .............................................................................................. 19

2.1 Introduction .................................................................................................................... 20

2.2 System parameter identification of laboratory equipment “Thor” ................................. 22

2.3 Vibrational scratch test performance comparison .......................................................... 33

2.4 Conclusions .................................................................................................................... 42

Self-oscillations as a mechanism of stick-slip in frictional sliding .......................................... 46

3.1 Introduction .................................................................................................................... 47

3.2 Constant friction law ...................................................................................................... 48

3.3 Propagation of sliding zone over a fault with rate-independent friction ........................ 52

3.4 Rate-dependent friction law ............................................................................................ 58

3.5. Comparative analysis between constant and rate-dependent friction coefficient .......... 63

3.6. Propagation velocity of sliding zone for rate-dependent friction .................................. 64

3.7 Conclusions .................................................................................................................... 64

Effect of external vibrations on frictional sliding .................................................................... 70

4.1 Introduction .................................................................................................................... 71

4.2 Effect of normal vibrations ............................................................................................. 72

4.3 Effect of longitudinal vibrations ..................................................................................... 90

4.4 Conclusions .................................................................................................................... 98

vii

Negative stiffness produced by particle rotations and its effect on frictional sliding ............ 101

5.1 Introduction .................................................................................................................. 102

5.2 Analytical modelling .................................................................................................... 103

5.3 Physical modelling ....................................................................................................... 107

5.4. Friction force fluctuation associated with rotation of particles ................................... 111

5.5 Conclusions .................................................................................................................. 115

General conclusions and discussion....................................................................................... 118

6.1 Conclusions .................................................................................................................. 119

6.2 Recommendation for the future work ........................................................................... 122

APPENDIX ............................................................................................................................ 124

viii

ACKNOWLEDGEMENTS

Foremost I wish to express my sincere appreciation to all my supervisors – Professor

Elena Pasternak, Winthrop Professor Arcady Dyskin for their continuous support of my PhD

research. Their constant guidance, encouragement, scientific advice and immense knowledge

helped me in all the time of research and writing of this thesis. I am also very grateful to Soren

Soe for his priceless advices and sharing industrial experience with me.

I am greatly indebted to my main teacher in the past: Professor V.A. Zhovdak who was

the first person who believed in me and inspired me to start doing PhD research.

I would like to thank my fellow PhD students Maxim Esin, Habibullah Chowdhury,

Yuan Xu, Junxian He for their friendship, support and understanding during this amazing

journey.

I acknowledge the financial support of the University of Western Australia for

providing me the Scholarship for International Research Fees, Australian Postgraduate Award,

Overseas Travel Award and Safety-Net, Top-Up Scholarship, Ad Hoc Scholarship, which

made my studies possible at UWA. This research was supported by an Australian Government

Research Training Program (RTP) Scholarship.

I appreciate and acknowledge the Top-up Scholarship from the Deep Exploration

Technologies Cooperative Research Centre (DET CRC) and I am very thankful for the unique

experience and possibility to be a part of this program.

I would especially like to thank the administrative staff of the School of Mechanical

and Chemical Engineering and Graduate Research and Scholarships Office for their support at

different stages of the PhD program.

I am very much indebted to my parents for their unconditional support, endless love

and encouragement to pursue my interests. I deeply miss my beloved brother Arsen who is not

with me to share this joy.

Finally, I would like to acknowledge the most important person in my life – my husband

Roman who supported me in every possible way to see the completion of this work. Thank you

for all your love, sense of humour, understanding, great support and for believing in me.

1

CHAPTER 1

General introduction

Chapter 1. General introduction

2

It has long been recognised that frictional effects play an important role in stick-slip

motion which is undesirable phenomena in engineering applications. Thus, stick-slip motion

produces cycling load and hence can cause fatigue in the materials involved, reducing the life

of certain working parts. The stick-slip motion is believed to be a mechanism of earthquakes.

For this reason, controlling the stick–slip friction phenomenon is very important for many

engineering applications. One of the methods of controlling the stick-slip motion (friction

reduction) is application of additional harmonic loads.

The main aims of the thesis are (1) to study the elastic dynamic effects arising at the

contact surface during friction sliding and cutting and (2) to determine the effect of

vibrations on friction reduction and the efficiency of cutting with the influence of rock

fragments in front of and under the cutter.

1.1 Frictional sliding

Friction plays a key role in many physical systems from the ancient time, and is still a

hot topic in research today [1], [2]. Generally, the history of frictional sliding can be divided

into four stages [3]:

Classical formulation of friction law (from Leonardo da Vinci to Coulomb);

The theories of lubrication (Tomlinson’s theory of friction);

The theories of wear (the Bowden and Tabor’s synthesis, Tribology);

Nanotribology (new fields of research).

Leonardo da Vinci was one of the first scientists who described friction phenomenon

[4]. He noticed that friction force is proportional to the total weight and does not depend on

mass distribution. In 1699 Guillaume Amontons formulated first two classic friction laws,

according to which the friction force is independent on the contact area and proportional to the

normal load. About a 100 year after, Coulomb noticed the difference between static and

dynamic friction [5]. Later on, other scientists, such as Isaac Newton, Mikhail Lomonosov,

Leonhard Euler attempted to formulate the friction laws [6], [7]. In 1750 Euler introduced the

Greek letter µ for the friction coefficient [2]. Amontons-Coulomb law is widely used in dry

friction case. Friction coefficient is introduced as the ratio of the friction force to the normal

load (Figure 1.1)


3

Figure 1.1– Amontons-Coulomb friction law

where fF is the Coulomb force, is the friction coefficient, c is cohesion, NF is the

compressive force, T is the shear force, U is the displacement.

f NF F c (1.1)

However, Amontons-Coulomb law does not give the full picture of friction. It was

identified that the coefficient of friction is not a universal characteristic. The German scientist

Stribeck proved that static friction force is different from kinematic friction. He found that the

friction force decreases from the static friction to the Coulomb friction as the velocity increases

(Figure 1.2) [8].

Figure 1.2––Stribeck curve (Pervozvanskii, 1998 [8])

where fF is the Coulomb force, NF is the compressive force, SF is the static friction force, V

is velocity.


4

In the middle of the nineteenth century scientists started to add the lubrication to the

friction contact. The first study of fluid lubrication was presented by the French scientist

Gustave-Adolphe Hirn. Others significant publications in lubrication were written by N.P.

Petrov, B. Tower, O. Reynolds, W.B. Hardy, G.A. Tomlison [3]. Film thickness is an important

parameter in lubricated friction and was studied by H. Olsson, K. Astrom, M. Gafvert, and P.

Lischinsky. Tomlinson (1929) developed a molecular theory of friction in which the friction

explanation is related to the ideas of energy conservation and dissipation.

The first important works on the mechanisms of wear have been conducted by the

following researches: Holm, Burwell, Strang, Archad, Hirst. It is important to note the Bowden

and Tabor’s synthesis. Bowden and Tabor introduced the concept of real contact area which is

made from asperities of the contacting surfaces. They undertook a great amount of experiments

on metals and wood, and demonstrated that friction between two contact bodies is dependent

on the real contact area [3].

Study of friction was very popular in the beginning of the twenties century. As a result,

tribology science was created (from the Greek tribos, meaning rubbing). Friction control in

engineering systems is the main aim of tribology [9]. In the late 1980s nanotribology was

created as a result of the development of new technologies. These technologies allowed friction

measurements at micro and nano-metric scales and provide more accurate simulations [3].

1.2 Stick-slip phenomenon

Friction is often described as a resistance to relative motion of two surfaces that slide

upon each other. From an engineering point of view there are two forms of friction: static and

kinetic friction. The static friction (also referred to as a “stiction”) keeps a stationary object at

rest and increases with the time of contact. The kinetic friction (also referred to as a

sliding/dynamic friction) acts against the relative motion of two contacting surfaces and slows

down the moving object. Stick-slip is the spontaneous jerking motion that can occur because

of alternating changes between static and kinetic friction. Stick-slip vibration is present in

tribological processes, in technological, geological or biological areas, and appears in everyday

life. In most cases the stick-slip motion can lead to many engineering problems.

In geotechnical engineering (mining petroleum and geothermal) self-excited stick-slip

oscillations often occur in drilling, which may lead to fatigue problems, wear of drilling


5

equipment, premature tool failure and poor rate penetration [10], [11], [12], [13], [14]. Stick-

slip vibration is considered as one of the main limiting factors in the quality of drilling

performance, limit tool life and productivity.

Stick-slip vibration is considered to be a mechanism of earthquake triggering [15].

Brace and Byerlee in 1966 suggested that earthquakes might be caused by the observed stick-

slip instabilities in the relative sliding of geological materials [16]. A number of studies were

devoted to investigation of the fault slip (including stick-slip) behaviour of the rock around the

fault [17], [18], [19]. It was observed that topographic features and geometric properties of

fault play an important role in earthquake dynamics [20]. Fournier and Morgan noted the

influence of fault structure on the slip behaviour [21]. They used the discrete element method

(DEM) for simulation the slip behaviour of faults. For this purpose, a shear strain was imposed

upon a 2D bonded particle assemblage (as a predefined fault). Fournier and Morgan’s study

showed that increasing normal stress leads to stick-slip behaviour of a fault. Nasuno et al

investigated the influence of frictional force in granular layers [22], [23]. They observed the

stick-slip motion of fault during the experiments with the transparent cover plate pushed by a

leaf spring. The stick-slip motion was also observed under the set of 2D numerical experiments

of shear in granular layers (simple spring-block model for sliding on a frictional base) [24].

Vibrations induced by stick-slip movement can be a reason of serious problems in many

industrial applications (robot joints, turbine blade joints, electric motor drives, and others) and

daily life [25]. Creaking doors, squeaking chalks on a blackboard, the motion of a windscreen

wiper on a dry glass, squealing noise of tramways in narrow curves are common examples of

undesired stick-slip processes in daily life [26], [27], [28], [29]. Vibrations induced by stick-

slip motion can cause a number of problems such as fatigue and failure, excessive noise,

instability, energy loss, and loss of accuracy of measurements [12], [13]. Overall, the economic

loss caused by friction and as a result the wear of equipment has been estimated at 5% of gross

national product [30], [31].

Friction in some cases is desirable. Friction plays an important role in mechanical

machinery such as anti-skidding systems, brakes and clutches [13]. Friction and the associated

stick-slip motion is important bowed instruments [32]. The emitted sound from violin is a result

of friction-induced self-sustained oscillations of violin string. Popp and Steller noticed that the

self-excited oscillation system consists of an energy source (bow motion), an oscillator (string)


6

and a switching mechanism triggered by the oscillator [33]. The stick-slip frequency depends

on the speed of the bow and the length of the string [34].

Figure 1.3–– Violin and stick-slip motion

(Popp and Steller, 1990 [33])

Animals, insects, microorganisms experience the friction and lubrication forces which

are similar to machineries movement [35]. In nature, some insects (crickets, locusts, and

cicadas) can produce sound using stick-slip friction.

Figure 1.4––Cicada and grasshopper

(from songsofinsects.com; www.insectsofscotland.com)

The spiny lobsters can effectively produce warning sounds against predators by using

the stick-slip mechanism. They rub their antennae over smooth surfaces on its head [36], [37],

[38].


7

Figure 1.5–– Spiny lobster

(from www.msc.org)

1.3 Friction models

A wide number of friction models (static and dynamic type) were proposed in the

literature to describe the friction phenomenon. Static models are entirely velocity dependent.

However, the static models do not properly describe the system behaviour near the zero

velocity. For this situation, dynamic systems are used.

Static models

Coulomb model

The Coulomb model is one of the most basic and simplest friction models. It describes

friction as a function of the difference in the velocities of the sliding surfaces. However,

Coulomb model does not describe friction accurately enough for velocities equal to zero or

crossing zero. Thus, a specification of the friction force at zero velocity needed (stiction

models). Stiction force describes friction force at the rest.

Stiction model

The system of equations below describes the friction force for zero velocity which is a

function of the external force.

if 0 and | | <sgn( ) if 0 and | |

e e s

s e e s

F V F FF

F F V F F

(1.2)

where Fe is the external force, FS is the stiction force, sgn(Fe) is the signum function.


8

Karnopp model

The force balance equation of a dynamic model

( ) ( ) ( ) ( )ext spring frictionmx t F t F t F t (1.3)

where m is the block mass, Fext is the external force, Fspring is the spring force, Ffriction is the

friction force.

Since in numerical computation it is impossible to get exactly “zero” velocity, Karnopp

developed a force-balanced model with a small-velocity region (DV) in order to determine the

stick phase [39]. In other words, the mass is assumed to have a small velocity during the stick

phase.

Figure 1.6 – Karnopp friction model: Ff is the friction force, V is the velocity, DV is

the limited velocity

(1.4)

where Fc is the Coulomb friction coefficient, Fv is the viscous friction coefficient, Fs is the static

friction coefficient, DV is the limited velocity, k is stiffness, sgn( )x is the signum function.

Here function sgn( )x is defined as follows

sgn( ) , | | | |

( ), | |

| |< sgn( ),

| |

c v

fr extext s

s extext s

F x F x x DVif x DV and

F F kxF kx F

if x DV andF F kx

F kx F


9

1 for x 0sgn( ) 0 for x 0

1 for x 0v

(1.5)

Coulomb kinetic friction force is used outside the small-velocity region (V>DV, slip

stage), inside the small-velocity region (V<DV, stick stage) the friction force is based on a force

balance.

Karnopp’s force-balance model has a problem with determination of the small velocity

region. The procedure for estimating the small velocity region parameter is fully described in

Romano and Garcia [40]. The value of the limited velocity is very small. It varies from 0.025%

of the average velocity [39] to an arbitrary value of 1% of the maximum velocity [40].

Dynamic models

According to the literature dynamic models can predict friction more accurately than

static models [41]. Dynamic models are more complex and require more parameters.

Dahl model

The Dahl model was developed to simulate the control systems with friction. The

starting point is the stress-strain curve in classical solid mechanics. The stress-strain curve is

obtained by solving the following differential equation:

(1 sgn )c

dF F vdx F

(1.6)

where F is the friction force, Fc is the Coulomb friction force, σ is the stiffness coefficient and

α is a parameter that determines the shape of the stress-strain curve, sgn( ) is the signum

function.

The value α=1 is the most common case. In this model, the friction force is a function

of the displacement and the sgn of the velocity.

LuGre model

The LuGre model is described by equations


10

0| |( )

dz vv zdt g v

(1.7)

0 1 ( )F z z f v (1.8)

where ν is the velocity between two surfaces in contact, z is the average deflection of the

asperities, F is the predicted friction force, g(v) is the velocity-dependent function, 휎 is the

stiffness, 휎 is the micro damping, 푓(푣) is the general form for the memoryless velocity-

dependent term.

For constant velocity, the steady-state friction force Fss is given by

( ) ( )sgn( ) ( )ssF v g v v f v (1.9)

| / |( ) ( ) sv vc s cg v F F F e

(1.10)

where g(v) models the Stribeck effect, 푓(푣) is the viscous friction. Fs corresponds to the stiction

force, Fc is the Coulomb friction force, sgn( )v is the signum function. The parameter 푣

determines how quickly g(v) approaches Fc.

Bliman -Sorine Model

Bliman and Sorine have developed different dynamic models based on experimental

data of Rabinowicz.

In their models the magnitude of the friction depends only on sgn(v) and the space

variable s defined by:

0

( )t

s d (1.10)

Friction does not depend on system’s speed and is a function of the path only. The

models are expressed as linear systems in the space variable s [41].


11

1.4 Friction law

The exact formula for the friction force has not been yet proposed. For this reason most

of the available friction models are empirical [41] and based on fault simulation in laboratory.

Even though an experimental fault models are simplified, the general laboratory faults

behaviour is similar to the real one [42].

An important factor in friction phenomenon is the non-uniformity of the contact

surfaces (especially the contact surfaces of a fault) whereby the contact area can be divided

into a number of short sliding zones interacting with each other. A simple spring and slider

model was proposed to account for the interaction and to explain the experimental results

observed in the laboratory by Burridge and Knopoff (BK model) in 1967 [43]. The BK model

consists of several blocks connected by a linear springs kc to each other and connected to a

driving plate by a leaf springs kL (Figure 1.7).

The Burridge-Knopoff model equation has a following form:

1 1( ) ( 2 )i L i c i i i im x k Vt x k x x x f (1.11)

where m is the mass of each block, fi is the local friction force acting on the block i, V is the

velocity of the plate.

Figure 1.7– Burridge and Knopoff (BK) model

Since the introduction of the model, numerous modifications of the model have been

applied in order to study dynamics of the system by numerical simulations [44], [45], [46],

[47], [48], [49], [50].

The behaviour of the numerical model is controlled by the chosen friction law. The

investigation and formulation of friction law play an important role in modelling of earthquake


12

processes. The complexity of stick-slip behaviour typical for the earthquakes is successfully

described by the phenomenological state- and rate-dependent friction law formulated by Rice

and Ruina [51], [52], [53] based on Dieterich’s experiments on rocks. The experiments with

rocks are performed in order to obtain the empiric parameters (a, b, Dc) needed to study

frictional behaviour. Since then a variety of laboratory experiments with different materials

(e.g. paper, polymer glass, etc.) have been conducted using the state- and rate-dependent

friction law model [54], [55], [56].

The one state variable friction law is a simple form among the state variable laws and

considered in detail in Chapter 4. However, a two-state variable law provides more accurate

description of friction.

The Dieterich rate- and state-dependent one state variable friction law has the following

form [57]:

*

**

( ln( ) ln( ))c

VVA BV D

(1.12)

where V is the instantaneous sliding velocity, 푉∗is an arbitrary positive reference velocity for

which steady state friction is µ*, θ is the state variable, τ is the shear stress, σ is the normal

stress, Dc is a characteristic slip distance. The magnitude of Dc depends on surface roughness.

The values A, B and Dc are positive empirical constants.

A two-state variable law has the next form

1 2 * 1 2*

( , , ) ln( )VF V AV

(1.13)

11 1 2 1 1

1 *

( , , ) ( )[ ln( )]d V VG V Bdt L V

(1.14)

22 1 2 2 2

2 *

( , , ) ( )[ ln( )]d V VG V Bdt L V

(1.15)

where 푉∗ is an arbitrary positive constant, 휏∗ is positive constant and dependent on choice of 푉∗,

A, 퐵 , 퐵 are positive empirical constants, 퐿 , 퐿 are the slip distances, 휃 , 휃 are the state

variables, τ is the shear (frictional) stress, σ is the normal stress.


13

Model parameters determined from the experimental data

According to the numerous laboratory experiments the presence of quartz formation in

rock is conductive to stick-slip behaviour [58]. The Westfield granite is the most common type

of the rock which is used for study of stick-slip phenomenon and contains 30% of quartz [59].

A lot of publications [60], [51], [42], [61], [62], [63], [59], [64], [57], [65], [66] have been

observed and all empirical parameters used in friction tests are listed below in Table 1.1.

Table 1.1 Model parameters based on experimental data

k a b σ (MPa) V0 V* µ0 Dc notes 0.5

MPa/µm 0.013 0.0145 5 1 µm/s 1-2 µm

0.006 -

0.008 (r)

0.003 -0.005

(s)

0.007 (r)

0.009-0.014

(s)

10 0.25-25 µm/s

1 µm/s 40-50 µm (r) 4-25

µm (s)

r-roughest surface

s-smoothest surface

0.01-0.02

1.0-2.0 6.07 Stepwise manner

10-6 cm/s to 10-3

cm/s

0.6-0.8

0.0015 0.0065 25-100 1-1000 µm/s

60 µm Amplitude 0.1-10 MPa,

T=0.1-200s

V0=10 µm/s

V*=30 0.001 µm/s

81 mm T=5 s L/V0=2.7 yr

0.0015

MPa/µm 0.0125-0.0150

0.0161-0.112

50 V=0.001 µm/s to

3162µm/s

4-7 µm

0.092 15 0.01mm/s 0.75 0.6-2

(c.r) 5-20 (n-c

r.)

0.1-2.6 mm/sec

0.7

c.r. cohesive rock

n-c r. non-cohesive rock

0.005 to 0.007

0.006 to 0.008

100 10-9 m/s

(a-b) ~0.3-3 µm/sec

<0.3µm/s <=100 µm

San Andreas fault velocity

0.5-4.5 mm/year 0.005 0.008 100 0.001

µm/s 0.6 1 µm


14

1.5 Thesis overview

Despite the extensive research many problems of modelling of friction forces is not

solved and remain unclear. Several questions arise in this regard. Some of them are what is

the effect of elasticity? What is the role of the debris, gouge (negative stiffness)? What is the

role of applied vibrations and is it possible to reduce friction?

The thesis is organized as a series of 6 chapters and has the following structure:

Chapter 1 – “General introduction”. This chapter contains a literature review of friction

sliding.

Chapter 2 – “Frictional sliding in rock cutting”. This chapter presents the analysis of

the laboratory data in order to increase performance output of a drilling rig.

Chapter 3 (“Self-oscillations as a mechanism of stick-slip in frictional sliding”) is

devoted to the development of mathematical modelling of the oscillator with constant and rate-

dependent friction law. The importance of dynamic self-oscillations of the material (or rock)

surrounding the sliding surface is discussed.

Chapter 4 (“Effect of external vibrations on frictional sliding”). Friction is undesirable

effect in drilling mechanisms and should be avoided as it can be a reason of noise generation,

wear and damage of constructions. Oscillations of various frequencies and amplitudes are

widely used in engineering in order to influence friction [67]. Applying the longitudinal and

transverse vibrations is the effective method to reduce friction between the machine elements

or cutter and rock. The analysis of influence of harmonic load on reduction of friction force is

provided.

Chapter 5 (“Negative stiffness produced by particle rotations and its effect on frictional

sliding”). According to the previous research [68], [69] the rolling of non-spherical particles

may lead to the effect of apparent negative stiffness. The appearance of negative stiffness

elements at the cutter/rock interface can significantly affect the longitudinal vibrations and

even transform the movement to aperiodic. The aim of this chapter is to consider the theoretical

model of loss of stability in the example of an inverted pendulum and compare it with the

physical model results. Analysis of the effect of a set of rotating non-spherical grains/debris

sitting between sliding surfaces and their influence on friction is provided.


15

Chapter 6 (“General conclusions and discussion”). This chapter presents the

conclusions and recommendations for the future work.

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16

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CHAPTER 2

Frictional sliding in rock cutting

Chapter 2. Frictional sliding in rock cutting

20

2.1 Introduction

The problem of drill string vibrations is one of the main reasons of reduction of drilling

performance. Drilling systems produce different types of oscillations during the operation:

lateral, axial and torsional. Many prior investigations have concluded that the bottom hole

assembly (BHA) is the part that is most significantly affected by vibrations. The torsional

vibrations can lead to the stick-slip vibration in operation of mechanism (in drilling systems

due to the contact between the drill bit and the rock).

Stick-slip is undesirable effect in drilling. It can cause fatigue problems, reduce life of

the bit, cause unexpected changes in drilling direction, and even result in failure of the drill

string. Other undesirable effect includes heating of certain working parts of the bit, reducing

cutting efficiency, and associated energy dissipation resulting in inefficient drilling [1]. For

this reason, the avoidance of the stick-slip vibrations (which includes friction reduction) is one

of the main challenges in the drilling industry. Thus, the friction reduction is used to reduce the

process forces in drilling [2].

Belokobyl’skii and Prokopov firstly introduced the concept of self-excited torsional

vibration in drillstrings [3]. Since then a significant amount of research dedicated to the

analytical and dynamic models of stick-slip oscillation in drilling were introduced

Bit-rock interaction law without considering the frictional and cutting forces at

bit-rock interface [4];

Dynamic models of a rotating drillstring [5];

Stick-slip oscillations with two degrees of freedom [6];

Systems with a Hertzian contact [7];

Bit-rock interaction taking into account the frictional and cutting forces at bit-

rock interface [8];

Phenomenological model of cutter/rock interaction in the ductile regime [9].

The latter model was expanded and is characterized by three main parameters: the

intrinsic energy, associated with the cutting forces, the inclination of forces, acting on the

cutting face and the friction coefficient [10]. This model is relevant to the present chapter.

Temporal variations of the friction and cutting forces as well as their ratio are important

characteristics of the rock cutting process. The friction force is a parasitic force that dissipates


21

energy and leads to reduced efficiency of cutting. We investigated experimentally the

variations in these forces.

There are two commonly identified ways to reduce the friction: lubrication [11], [12],

[13], [14], [15] and application of ultrasonic vibration.

The second method is very effective and used widely nowadays. It was observed that

friction forces can be reduced due to normal contact motions [16], [17], [18]. Sakamoto [17],

[18] assumed that friction force is proportional to the area of contact. It was found that a

maximum average friction reduction around 10% occurs without any loss of contact in the

presence the dynamic loading [19].

It was also found that dry friction in the contact area between two surfaces under the

influence of ultrasonic vibrations is converted into the viscous friction [20]. The average

viscosity value decreases with the growth of amplitude [21].

Vibration cutting (VC) is used in many technical fields. It involves the use of high or

low frequency vibrations applied to the cutting tool in order to achieve better cutting

performance [22]. Moreover, the decrease of cutting forces and improvement of the drilling

quality are observed in case of applying of ultrasonic vibrations. Astashev and Babitsky pointed

out that the reduction of cutting force depends on two factors as: contact between the tool and

sample; the intense of vibrations [23]. Jin and Murakawa noticed that applying ultrasonic VC

can prevent the chipping of the cutting tool and extend the tool lifetime [24]. It was also found

that the mean thrust force can be reduced under the influence of ultrasonic VC [25].

Different experiments were conducted in CSIRO laboratory to determine the optimal

range in vibration of a drill bit to increase performance output of a drilling rig [26], [27], [28].

The system parameters identification of laboratory equipment “Thor” is presented in Section

(2.2); the vibrational scratch test performance comparison of “Wombat” is given in Section

(2.3); discussion and conclusions are presented in Section (2.4).


22

2.2 System parameter identification of laboratory equipment “Thor”

Subject of research

The main aim of “Thor” experiments was the explore the effect of imposed axial

vibrations on the cutting response and on cleaning and cooling system. The present section has

the purpose to analyse the experimental results of several types of tests (amplitude, stiffness

and cutting) and identify the system parameters.

The laboratory equipment “Thor” was designed to perform cutting test with single

segment. The cubic size of single segment is (10 x10x10 mm). This segment is fixed on a

cutting assembly (Figure 2.1), which is attached to the XY table of the lathe through a

dynamometer.

The cutting action takes place when the face of the segment is in the contact with the

cross-sectional segment of rotating rock pattern and moving into a longitudinal direction.

A list of all specifications for Thor is summarized in the Table 2.1.

Table 2.1 - Thor specifications (T. Richard, L. F. Franca, 2011 [29])

Parameters Range Precision Noise level

Rotary speed (control) (Ω) 100 to 3000 RPM 4 RPM NA

Feed rate (control) (V) 0.01 to 4 mm/s 2 µm/s NA

Tool position (measurement)

(U)

0-500 mm 0.5 mm 0.1 %

Force (measurement) (Fn, Fs) 5 kN 1 N 0.03%

Rock-Laser distance (Uw) 1 mm 1µm 0.1%


23

Figure 2.1 - Laboratory drilling rig – “Thor” (T. Richard, L. F. Franca, 2011 [29])

Power Piezo Stack Actuators P-235-80 PICA

P-235 is preloaded, high-load piezo actuator (PA) intended for static and dynamic

applications. It consists of PICA™ Power piezoelectric ceramic stacks and a frictionless

internal spring preload. P-235 is ideal for machining applications and active vibration

cancellation.

Table 2.2 - Technical data of Power Piezo Stack Actuators P-235-80 PICA ([30])

P-235.80 Unit Tolerance

Operation voltage 0 to 1000 V

Motion and positioning

Closed-loop travel 120 µm

Open-loop resolution ** 2,4 nm typ.

Closed-loop resolution *** 1,2 nm typ.

Linearity 0,2 % typ.


24

Table 2.2 - Technical data of Power Piezo Stack Actuators P-235-80 PICA ([30])

P-235.80 Unit Tolerance

Mechanical properties

Static-large signal stiffness*** 210 N/µm 20

Unloaded resonant frequency 3,9 kHz 20%

Push/pull force capacity 30000/

3500

N Max.

Shear force limit 147 N

Torque limit (on tip) 2 Nm

Drive properties

El. capacitance 5100 nF 20%

Dynamic operating current

coefficient

65 µm/(Hz x µm) 20%

Miscellaneous

Mass (with cable) 1400 g 5%

*Requires SGS sensor. SGS versions are shipped with performance reports. ** Measured with an interferometer. The resolution of piezo actuators is not limited by stiction or friction. *** Dynamic small-signal stiffness is 50% higher Piezo ceramic type: PICATM Power Operating temperature range: -40 to +80 0C Recommended controller/driver: B, I, J. For maximum lifetime, voltages in excess of 750 V should be applied only for short durations.


25

Figure 2.2 - Power Piezo Stack Actuators P-235-80 PICA ([30])

Segment selection

Three types of segment were used during the tests.

Matrix segment (without diamond);

Leached segment (no comet tail);

Normal segment (comet tail).

Experimental procedures on “Thor”

Three different types of test have been conducted in an experimental drilling rig “Thor”:

amplitude test, stiffness test and cutting test.

Amplitude tests

During the amplitude test we varied only two input parameters – amplitude (range from

5-25µm) and frequency (50-200 Hz). For this purpose, we used Piezo Actuator, which located

in Black Box (Figure 2.3). The results of this test are given in Tables 2.4, 2.5 and 2.6.


26

Figure 2.3 – The diagram of amplitude test

Table 2.3 illustrates insignificant difference between input amplitude parameters and

real amplitude parameters, which we obtained by using signal data. This can be a reason of

some technical issues of “Thor” installation.

Table 2.3 –Amplitude parameters

Amplitude (input), µm Amplitude (real), µm Frequency, Hz

5 3 50

25 19

5 3 100

25 18

5 3 150

25 16.5

5 3

200 15 8

25 15

As can be seen from the data there is no time shift in system, except of insignificant

shift under the condition F=200 Hz (Table 2.4, 2.5 and 2.6).


27

Table 2.4 - Frequency shift

INPUT parameters OUTPUT parameters

Amplitude

(µm)

Input

frequency

(Hz)

Frequency

(Hz)

Amplitude

(µm)

Input

frequency

(Hz)

Frequency

(Hz)

8,5*105 5 50 9,8*105 5 50

3,7*105 25 50 3.7*105 25 50

6.5*104 5 100 6.3*104 5 100

3.8*105 25 100 3*105 25 100

5.5*104 5 150 5.8*104 5 150

3.2*105 25 150 3.2*105 25 150

5.8*104 5 198 7.8*104 5 198

2,3*105 15 198 2,3*105 15 198

3.3*105 25 199 2.8*105 25 199

Table 2.5 - Time shift

Frequency (Hz) Amplitude (µm) Points Sec Degree

200

5 23 0.001 2,7

15 47 0.0195 5,4

25 35 0.015 4


28

Table 2.6 – Signal shift

Frequency (Hz)

Amplitude (µm)

Shift

200

5

15

25


29

Stiffness tests

Several stiffness tests have been conducted with different segments (matrix segment,

leached segment and normal segment), see Table 2.7. The results of the tests are shown in the

Table 2.7 and in Appendices, Stiffness Results.

Table 2.7 – Stiffness of different systems

Name of the test Diagram K stiffness

[N/mm]

Test 5 (6A- comet tail)

8873

6A_t1_normal -groove

rock-system-high dis

7954

6A_t1- normal- -

groove rock-system-

low dis

8794

leach_t1-groove rock-

system low dis

13659

leached-flat rock-sys

3514

I matrix-metal bar-

system

9557

matrix_i1_t1-groove

rock-sys-high dis

12206

matrix_i1_t1-groove

rock-sys-low dis

17164

H matrix- flat rock-sys

14049


30

Table 2.7 – Stiffness of different systems

Name of the test Diagram K stiffness

[N/mm]

H matrix-flat rock-sys

15926

Metal bar-no segment

8076

Metal bar-AB core-

segment holder

10349

Metal bar-Limestone

(I) core-segment holder

4971

Metal bar leached

segment

7204

The results displayed in the Table 2.7 show that the value of stiffness of machine “Thor”

(Metal bar-no segment test) is small comparable with the stiffness’s of the other systems. This

may suggest that stiffness results are not representative enough for further analysis. We need

more data to analyse the stiffness of the machine. Probably, the best option is to conduct more

tests, decreasing the loading interval.

Cutting tests

A series of cutting tests (Table 2.8) were performed under kinematic control, meaning

that the penetration (or feed rate) V and angular velocity Ω are imposed. One type of tests is

conducted: step-test. In the step test d is decreased by stages (steps), see Figure 2.5. Cutting

tests have been performed with normal segment.

Figure 2.4 shows the schematic cutting process, where

V [mm/s] the feed rate;

Ω [rad/s] the rotary speed;

α – the radial distance between the rock sample centre line and the tool centre

line (or centre point of the contact surface);


31

d [mm/rev] the depth of cut per revolution;

pф- average diamond exposure.

The depth of cut per revolution

Vd 2 (2.1)

The total force F consists of two components of normal force FN and tangential FS to

the rock surface.

N SF F F (2.2)

Figure 2.4 – Rock-cutting process with leached segment: frontal view

(T. Richard, L. F. Franca, 2011 [29])

Figure 2.5 – Depth of cut per revolution


32

Table 2.8 – Series of cutting tests (PA-on)

Amplitude (µm) RPM (70) RPM (200)

5

10

12 20

22 25

The cutting response is presented in a force (F) versus cross-section area of the cut (Ac)

diagram; the results are shown in Figure 2.6 and 2.7.

Analysis of the results shows the decrease in normal force, PA-on (A=10, A=20 µm).

However, this difference is reducing from Ac=0.35 mm2. At the higher amplitude (A=25 µm)

there is no force difference in PA-off and PA-on condition see Figure 2.6.

Figure 2.6 – Evolution of the cutting response for RPM=70

-0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40

500

1000

1500

2000

2500

3000PA-on/off

Ac

Fn

PA-offPA-offPA-on, A=10PA-on, A=20PA-on, A=25

-0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4-100

0

100

200

300

400

500

600

700

800PA-on

Ac

Fs

PA-offPA-offPA-on,A=10PA-on,A=20PA-on,A=25


33

Figure 2.7 – Evolution of the cutting response for RPM=200

At higher RPM=200, the increase of amplitude (A=25 µm) leads to reduction of the

normal and tangential force, PA-on.

Discussion and conclusions

Three types of tests have been conducted in an experimental drilling rig “Thor”:

stiffness test; amplitude test and cutting test. It was observed the obtained data is not

representative for the stiffness analysis of machine. By the same time, the amplitude and cutting

tests data are also not sufficient for the system parameters analysis. For this reason, a new

experiments and data are needed for the further research.

2.3 Vibrational scratch test performance comparison

Materials and methods

The present section has the purpose to show and analyse the experimental results of the

scratching tests with different types of cutter. The aims of these experiments are

analysis the influence of geometrical and rock characteristics on cutting process;

determination of the optimal range of vibrations of a drill bit in order to increase

performance output of a drilling rig.

The apparatus used throughout experimentation was the Wombat rig available at the

CSIRO Kensington drilling mechanics laboratory.

-0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40

500

1000

1500

2000

2500PA-on/off

Ac

Fn

PA-offPA-offPA-on, A=12PA-on, A=22

-0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4-50

0

50

100

150

200

250

300

350

400

450PA-on

Ac

Fs

PA-offPA-offPA-on,A=12PA-on,A=22


34

Figure 2.8 - Wombat test rig (R. Aguiar, 2009 [31])

Wombat is a scratch test apparatus developed to measure vertical and tangential

components of the force acting on cutter while making a groove at a constant depth on the rock

surface. A cutter is mounted onto a cutter holder (Figure 2.9) that is able to change the back-

rake angle in 5-degree increments. This cutter holder is directly mounted onto a load cell. The

cutter holder held the cutter at a fixed back rake angle, 020r (approach angle 00 ). A

stepper motor was used to move the cutter across the face of the rock sample at a velocity of

4mm/s.

Figure 2.9 - Cutter mount (S. Banks, 2010 [32])

A scratch test is widely used to study the rock cutting process. The cutting action takes

place when the face of the cutter is in the contact with the rock sample (Tuffeau limestone) and

moving into a longitudinal direction. Experiments were performed at depth d of cut ranging

from 0.2 mm to 0.6 mm (what corresponds to ductile regime).


35

Each test is typically conducted under a fixed cutting velocity v=4 mm/s and a back-

rake angle 020r . The tangential SF and normal NF components of total force F are the

outcome recorded parameters.

The energy is dissipated into two processes that are taking place during the rock cutting

Pure cutting (rock fragmentation) ahead the cutting face;

Frictional cutting (contact) along the wear flat rock interface.

The total cutting force F (Formula 2.2) consists of two components normal NF

(Formula 2.3) force and shear SF (Formula 2.4) forces. These forces are normal and parallel to

the rock surface respectively (Figure 2.10).

Figure 2.10 – Normal and shear forces acting on rock (B. Besselink, 2008 [33])

N c fF F F (2.3)

S c fF F F (2.4)

where cF is the cutting force and fF is the wear flat force.

The normal and tangential cutting forces are decomposed in:

N fF d F (2.5)

S fF d F (2.6)

Here, is the specific energy; is the width of cutter; is the friction coefficient;

fF is the wear flat force; is the ratio between vertical and horizontal components of the force

acting on the cutting face.


36

Experimental procedure

Two types of scratch tests were conducted. The first being straight cut tests which is

the base result for comparison. The second test was to conduct multiple tests each with

incremental changes to the approach angle of the cutter. The second set of experiments was

conducted to simulate a vibrational input on the drill bit. Rather than use a linear actuator

multiple experiments were conducted with varying approach angles. As the approach angle

was incremented the output data was non-continuous and were discrete values. As the standard

cutter mount was inadequate for the change in approach angle required for the experiments a

different cutter mount was used. The cutter mount allows for a 5-degree incremental change in

the back-rake angle, which can simulate a change in approach angle by keeping the back-rake

angle constant. The optimal range is found through analysis and comparison of two different

types of scratch tests.

Experiments were conducted with three different cutters to analyse the difference

between a straight cut and a sinusoidal cut, both also maintaining a constant depth of cut

through each experiment. Each cutter, both blunt and sharp, would cut a rock sample for a set

range of depth, from 0.2mm through to 0.6mm at a 0.1mm increment.

Also, the experiments were conducted over seven different approach angles. Ranging

from 15 degrees to -15 degrees at a 5-degree increment. At each 5-degree increment

experiments were conducted for the five different depths of cut. Each of the seven approach

angles will give one discrete value of force for a certain depth of cut.

Table 2.9 - Back rake and approach angle relation

r [deg] 35 30 25 20 15 10 5

[deg] -15 -10 -5 0 5 10 15


37

Figure 2.11 – Back-rake and approach angle (S. Banks, 2010 [32])

where r is the back-rake angle, is the wear flat angle, V is the velocity of the cutter,

/dU dt is the cutter vertical velocity, is the approach angle.

Instead of incrementing the approach angle the back-rake angle was incremented. This

was to keep the rock sample straight while allowing for a change in the approach angle and

theoretically obtaining the same force values.

20r (2.7)

As the depth of cut was easily changed from one experiment to another, the various

depths of cut would be scratched for a set approach angle and after the various depths are cut

the approach angle would be incremented and the next set of experiments were conducted.

In the incremental approach angle data set the range of approach angles included zero.

These force values obtained, when conducting the experiments for an approach angle of zero,

were used as the straight cut data to compare with the data from the overall incremental

approach angle data set.

Three types of cutters were used during the scratching tests to allow for analysis on how

the wear flat angle and wear flat length may affect the performance of a drill bit. Two blunt

cutters (4B and 5B) and a sharp cutter were used throughout the experiments (Figures 2.12 and

2.13). The perfectly sharp cutter has a single cutting face; the blunt cutter has the wear flat (the

bottom of the cutting face was removed to create an additional face). The cutting face was

made of a polycrystalline diamond compact (PDC) layer. According to the literature

observations, the PDC cutters play an important role in drilling performance [34]. In particular,

the use of PDC cutters showed better efficiency (reduction in friction coefficient and decrease

in SF and NF for cutting the rock) comparable with the standard cutters [35]. The experiments


38

will be coupled with the analysis of frictional forces, acting on the wear flat of cutter and the

rock (Figure 2.12).

a) b)

Figure 2.12 - The cutter geometry and forces acting on it

a) Sharp cutter, b) Blunt cutter (B. Besselink, 2008 [33])

Several cutters were used throughout experimentation. Two blunt cutters and a sharp

cutter. These various cutters were used to compare and analyse the difference in both shear

and normal forces acting on the cutter. The two cutters differed in wear flat angle and length

of wear flat but had the same width.

Figure 2.13 – Cutters used throughout experimentation

Table 2.10 - Properties of the cutters used in the experiments

Cutter

Type

Thickness of

Diamond Carbide

Layer [mm]

Wear Flat

Angle

[degrees]

Wear Flat

Length

[mm]

Cutter

Width

[mm]

4B 0.7 18.75 1.82-1.99 10 5B 0.7 15.67 0.98 10

Sharp 0.67 ~0 ~0 10


39

Experimental results

Straight cut test

The straight cut data were taken from the vibrational data set for each depth of cut.

From the table below, we can see that both the normal and shear forces increased linearly as

the depth of cut increased for the straight cuts.

Table 2.11 - Table of the normal and shear forces acting on the cutters for the straight cut

Depth of Cut Force 4B [N] 5B [N] Sharp [N]

0.2mm Normal 36.35 48.24 25.36 Shear 25.49 30.04 32.35

0.3mm Normal 47.63 56.87 33.74 Shear 34.53 37.38 44.6

0.4mm Normal 58.26 70.73 40.49 Shear 44.28 48.58 56.14

0.5mm Normal 67.68 80.92 45.9 Shear 50.37 57.28 65.22

0.6mm Normal 76.44 92.28 51.02 Shear 59.34 66.9 76.29

a) b)

Figure 2.14 – Force variations during the straight cut (4B cutter)

a) 4B cutter; b) sharp cutter

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.0815

20

25

30

35

40

45

50

55

60

65Input signal-4B cutter

L (m)

Fn

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.0810

15

20

25

30

35

40

45Input signal Fs-4B cutter

L (m)

Fs


40

a) b)

Figure 2.15 – Force variations during the straight cut (sharp cutter)


The results displayed on Figures 2.14-2.15 show the fluctuation in normal NF and

shear SF forces.

Table 2.12 – Friction coefficient during the straight cut

d (mm) Sharp cutter 4B cutter 5B cutter

0.2 0.6356 0.7012 0.6228 0.3 0.8537 0.7248 0.6574 0.4 1.0288 0.7601 0.6869 0.5 1.1704 0.7442 0.7079 0.6 1.2865 0.7763 0.7250

a)

0 10 20 30 40 50 60 70 8020

25

30

35

40

45

50Input signal-sharp cutter

L (m)

Fn [N

]

0 10 20 30 40 50 60 70 8015

20

25

30

35

40Input signal-sharp cutter

L (m)

Fs [N

]

10 12 14 16 18 20 22 24 26 28 306

8

10

12

14

16

18

20

22

Fn

Fs


41

b)

c)

Figure 2.16 – Linear regression relation between normal and shear force

during the straight cut

a) 4B cutter; b) sharp cutter; c) 5B cutter (Iu. Karachevtseva, 2012 [27])

The growing of shear force SF is typical for cutting effect, however normal force NF

should stay constant (Figure 2.16). In our case the growth of NF may be caused by grains

rotation in front of the cutter. Difference in friction coefficient at different depth of cut (Table

2.12) may be caused by possible influence of powder or grains stuck (ductile regime).

Sinusoidal cut

From the forces measured through the many experiments conducted average forces can

be calculated and these discrete values for the range of approach angles and depths of cut can

be graphed and further analysed. Curves are plotted to fit the discrete values through the range

of approach angles for each depth of cut (see Appendix).

15 20 25 30 35 4020

25

30

35

40

45

50

Fn

Fs

0 10 20 30 40 50 60 700

5

10

15

20

25

30

35

40

45

Fn

Fs


42

The difference in the normal force acting on the cutters was not significant when the

approach angle was negative. However, there was a significant difference in normal force

acting on the cutters when the approach angle was positive. This was due to the significant

increase in normal force acting on the blunt cutter as the approach angle becomes positive. The

shear forces for both the sharp and blunt cutter were relatively similar with no substantial

difference in force.

As the approach angle increase both cutters have an increasing trend in their ζ values.

However, as the approach angle increases the blunt cutter’s ζ values increase significantly

greater than that of the sharp cutter. The values of ε and µ for both cutters had no significant

differences and both had a downward trend. All experimental graphs can be found in Appendix.

The friction force in the blunt cutters was close to zero prior to a positive approach

angle, but as soon as the approach angle is positive the friction forces increased dramatically

as the grains rub against the wear flat of the blunt cutter. As the sharp cutter has nearly no wear

flat the friction force stays roughly the same throughout the range of approach angles.

There is no obvious difference in between the 4B and 5B cutters. Both the shear and

normal friction forces acting on the 4B cutter is larger than the friction forces on the 5B cutter

when the approach angle is positive. The friction forces are larger for the 4B blunt cutter

because it had a significantly greater surface area that is in contact with the grains. The larger

the contact surface area the larger the friction forces will be.

As the region of interest in approach angle for the vibration is around -5 to 5 degree

further experiments using a linear actuator will allow for better definition and data in that

region. As the cutter mount only allows for 5-degree increments a linear actuator will allow for

much smaller increments in the order of a degree or less.

These experiments will allow further analysis into the exact range within the -5 to 5

degree range that would be optimal for the vibration. The increments should be kept as low as

possible, to allow for clarity, while still maintaining a high quality of work and

experimentation.

2.4 Conclusions

The most important conclusion from the data obtained in the laboratory experiments

(“Wombat”) that the shear force shows considerable fluctuations that cannot be attributed the


43

equipment but to the effect of cuttings/fragments rotating between the tool and the rock (see

Chapter 5 for the analysis).

The optimal region of the approach angle has been determined during the rock

scratching tests (“Wombat” laboratory experiments). The force variation observed during the

tests may affect the frictional forces. The reason of the observed fluctuations in normal and

shear forces is gouge rotation. Effect of external vibrations is investigated in Chapter 4. The

grains (gouge) rotation under compression can produce the effect of apparent negative stiffness.

The negative stiffness effect is investigated in the Chapter 5.

References

1. Y. A. Khulief, F. A. Al-Sulaiman, and S. Bashmal, Vibration analysis of drillstrings with self-excited stick–slip oscillations. Journal of Sound and Vibration, 2007. 299(3): p. 540-558.

2. H. Storck, W. Littmann, J. Wallaschek, and M. Mracek, The effect of friction reduction in presence of ultrasonic vibrations and its relevance to travelling wave ultrasonic motors. Ultrasonics, 2002. 40(1): p. 379-383.

3. S. V. Belokoby’skii and V.K. Prokopov, Friction-induced self-excited vibrations of drill rig with exponential drag law. Soviet Applied Mechanics, 1982. 18(12): p. 1134-1138.

4. W. R. Tucker and C. Wang, An Integrated Model for Drill-String Dynamics. Journal of Sound and Vibration, 1999. 224(1): p. 123-165.

5. A. S. Yigit and A. P. Christoforou, Coupled axial and transverse vibrations of oilwell drillstrings. Journal of Sound and Vibration, 1996. 195(4): p. 617-627.

6. S. J. Cull and R. W. Tucker, On the modelling of coulomb friction. Journal of Physics A: Mathematical and General, 1999. 32(11): p. 2103-2113.

7. D. P. Hess and A. Soom, Normal Vibrations and Friction Under Harmonic Loads: Part I—Hertzian Contacts. Journal of tribology, 1991. 113(1): p. 80-86.

8. T. Richard, C. Germay, and E. Detournay, Self-excited stick-slip oscillations of drill bits. Mechanique, 2004. 332(8): p. 619-626.

9. E. Detournay and P. Defourny, A phenomenological model for the drilling action of drag bits. International Journal of Rock Mechanics and Mining Sciences and Geomechanics Abstracts, 1992. 29(1): p. 13-23.

10. T. Richard, F. Dagrain, E. Poyol, and E. Detournay, Rock strength determination from scratch tests. Engineering Geology, 2012. 147-148: p. 91-100.

11. M. Wakuda, Y. Yamauchi, S. Kanzaki, and Y. Yasuda, Effect of surface texturing on friction reduction between ceramic and steel materials under lubricated sliding contact. Wear, 2003. 254(3): p. 356-363.

12. A. Ramesh, W. Akram, S. P. Mishra, A. H. Cannon, A. A. Polycarpou, and W. P. King, Friction characteristics of microtextured surfaces under mixed and hydrodynamic lubrication. Tribology International, 2013. 57: p. 170-176.


44

13. J. H. Choo, H. A. Spikes, M. Ratoi, R. Glovnea, and A. Forrest, Friction reduction in low-load hydrodynamic lubrication with a hydrophobic surface. Tribology International, 2007. 40(2): p. 154-159.

14. A. Martini, D. Zhu, and Q. Wang, Friction Reduction in Mixed Lubrication. Tribology Letters, 2007. 28(2): p. 139-147.

15. B. Bhushan, J. N. Israelachvili, and U. Landman, Nanotribology: friction, wear and lubrication at the atomic scale. Nature, 1995. 374(6523): p. 607.

16. D. M. Tolstoi, Significance of the normal degree of freedom and natural normal vibrations in contact friction. Wear, 1967. 10(3): p. 199-213.

17. T. Sakamoto, Normal displacement and dynamic friction characteristics in a stick-slip process. Tribology International, 1987. 20(1): p. 25-31.

18. T. Sakamoto, Normal displacement of the sliding body in a stick-slip friction process, in Proc. of JSLE Intl. Trib. Conf. 1985: Tokyo. p. 141-146.

19. D. P. Hess, A. Soom, and C.H. Kim, Normal vibrations and friction at a Hertzian contact under random excitation: Theory and experiments. Journal of Sound and Vibration, 1992. 153(3): p. 491-508.

20. V. K. Astashev, Nonlinear dynamics of ultrasonic processes and systems. Scientific and Tecnhological Journal VNTR, 2007. 2: p. 129-134.

21. A. A. Pervozvanskii, Friction: a known and mysterious force. Sorosovskii educational journal 1998. 2: p. 129-134.

22. G. L. Chern and J. M. Liang, Study on boring and drilling with vibration cutting. International Journal of Machine Tools & Manufacture 2007. 47: p. 133-140.

23. V. K. Astashev and V. I. Babitsky, Ultrasonic cutting as a nonlinear (vibro-impact) process. Ultrasonics, 1998. 36(1): p. 89-96.

24. M. Jin and M. Murakawa, Development of a practical ultrasonic vibration cutting tool system. Journal of Materials Processing Tech., 2001. 113(1): p. 342-347.

25. H. Takeyama and S. Kato, Burrless Drilling by Means of Ultrasonic Vibration. CIRP Annals - Manufacturing Technology, 1991. 40(1): p. 83-86.

26. Iu. Karachevtseva and L. Franca, System parameter identification of laboratory equiopment "Thor". 2012, CSIRO Earth Science and Resource Engineering, Drilling Mechanics Group. p. 26.

27. Iu. Karachevtseva and S. Oishi-Govindasamy, Vibrational scratch test perfomance comparison, in CSIRO Earth Science and Resource Engineering, Drilling Mechanics Group. 2013. p. 24.

28. E. Pasternak, A. V. Dyskin, and Iu. Karachevtseva, Determination of the effect of vibrations on the single cutter response. Experimental verification. Model development., in Interim Technical Report/Final Project Report, Project 1. 2: Fundamentals of rock fragmentation. 2014. p. 56.

29. T. Richard and L. F. Franca, Fundamentals of rock drilling processes, in CSIRO Earth Science and Resource Engineering. 2011.

30. P-235 PICA Power Piezo Actuators. Available from: https://www.piceramic.com/en/products/piezoceramic-actuators/stack-actuators/p-235-pica-power-piezo-actuators-101755/.

31. R. Aguilar, Wombat modal analysis, in SCIRO. 2009. 32. S. Banks, Geometry can be used to accurately model the normal force on the cutter, in

CSIRO. 2010. 33. B. Besselink, Analysis and validation of self-excited drill string oscillations. 2008,

Technische Universiteit Eindhoven.


45

34. F. Dagrain and T. Richard, On the influence of PDC wear and rock type on friction coefficient and cutting efficiency, in Eurock 2006: Multiphysics Coupling and Long Term Behaviour in Rock Mechanics, A. V. Cotthem, et al., Editors. 2006, Taylor & Francis: Liege, Belgium. p. 577-584.

35. R. H. Smith, J. B. Lund, M. Anderson, and R. Baxter, Drilling Plastic Formations Using Highly Polished PDC Cutters. SPE-30476-MS, 1995.

CHAPTER 3

Self-oscillations as a mechanism of stick-slip in frictional sliding

Chapter3. Self-oscillations as a mechanism of frictional sliding

47

3.1 Introduction

Friction is a well-known phenomenon arising at the contact surface plays a central role

in engineering applications. Friction modelling and simulation is important in a variety of

engineering disciplines such as geomechanics, contact mechanics, structural mechanics,

system dynamics and others [1].

The investigation of the friction law on geological faults is the key element in the

modelling of earthquakes. Rate- and state-dependent friction laws proposed by Dieterich,

Ruina and Rice [2], [3], [4] have successfully modelled frictional sliding and earthquake

phenomena. There are two types of frictional sliding between surfaces that include the tectonic

plates. The first type occurs when two surfaces slip steadily (V=V0 condition, where V - is

relative velocity, V0 - is the load point velocity) and is analogous to the fault creep [5]. In the

stable state, the sliding over discontinuities (faults and fractures) is prevented by friction.

Modelling of the frictional sliding is an important tool for understanding the initiation and the

development of rupture, and also, the healing of the faults. Many models and numerical

methods are developed to describe seismic activities and the supershear fracture/rupture

propagation [6], [7], [8], [9], [10], [11], [12]. The faults are continuously subjected to variations

in both shear and normal stresses and can produce sliding over initially stable fractures or

interfaces [13]. In the Earth’s crust, the increase in shear stress is an obvious consequence of

tectonic movement, while oscillations in the normal stress can be associated with the tidal

stresses or seismic waves generated by other seismic events. These can generate the second

dynamic state when the sliding occurs jerkily (slip, stick and then slip again). This type of

sliding is called “stick-slip” sliding which exhibit cyclic behaviour. It is assumed that the

mechanism of stick-slip lies in intermittent change between static and kinetic friction and the

rate dependence of the frictional coefficient [14].

Brace and Byerlee assumed that the stick-slip instabilities in the tectonic plates are

associated with the appearance of earthquakes [15]. Both types of sliding are usually

investigated using a spring-block model introduced by Burridge and Knopoff in 1967 [16]. The

BK model consists of an assembly of blocks, where each block is connected via the elastic

springs to the next block and to the moving plate.

The stick-slip motion in engineering systems can be a source of technological

(excessive energy consumption, fatigue, premature wear and failure of drilling components)


48

and economic problems. The economic losses caused by friction and as a result the wear of

equipment have been estimated at 5% of gross national product [17], [18].

Here we discuss the mechanisms of stick-slip and investigate the importance of

dynamic self-oscillations of the material (or rock) surrounding the sliding surface. The present

chapter is organized as follows: a simple spring-block model with constant friction is

considered in Section (3.2); the propagation of sliding waves over a fault with rate-independent

friction is modelled in Section (3.3); the velocity-dependent friction law for a single block

model is formulated in Section (3.4); discussion and conclusions are presented in Section (3.5).

3.2 Constant friction law

Study of mechanics of friction was very popular in the beginning of the twenties

century. As a result, the new science of friction – tribology was created. The friction control in

engineering systems is one of the main aims of tribology [19].

The friction modelling is a difficult and important (critical) issue for dynamic systems.

Developing of an appropriate and useful system model remains a very difficult task as the

friction has strongly nonlinear behaviour near zero velocity. A wide number of friction models

were proposed in the literature to describe the friction phenomenon.

All variables and constants used in equations in this section are listed below in the Table

3.1.

Table 3.1 - The list of variables and constants

Symbol Meaning

V0 load point velocity V relative velocity of block k single spring stiffness m block mass N normal force T shear force µ friction coefficient ω0 eigen frequency t time


49

The Coulomb model is one of the most basic and simple friction models. It describes

friction as a function of the difference in the velocities of the sliding surfaces (Figure 3.1).

sgn( ) sgn( )frF F v N v (3.1)

Here function sgn( )v is defined as follows

1 for 0sgn( ) 0 for 0

1 for 0

Vv V

V

(3.2)

The appearance of the sign function in the system of equations represents the fact that

friction always acts against velocity.

Figure 3.1 – Coulomb friction model

It is commonly assumed that friction has nonlinear dependence of the velocity.

However, there are many models in literature that consider constant friction parameters. Very

often a constant friction coefficient is taken for modelling of landslides [20], [21].

Adams [22], [23] considered self-excited oscillations with constant friction coefficient

for a simple beam model and for infinite planes. He examined the simple model (infinite planes

case) to analyse the source of the oscillation and then investigate the beam model. The loss of

stability of the beam is caused by buckling. Thus, the self-excited oscillations can lead either

to partial loss-of-contact or to stick-slip.

Oden and Martins also derived a constant friction law for model of contacting rough

surfaces [24]. It was concluded that stick-slip motion could be observed when the coefficient


50

of friction is constant and equal to its so-called static value. Results are limited by the first

period of oscillations.

Single degree of freedom frictional oscillator

Consider a simple model, whereby a single block is sliding on a rigid horizontal surface

(Figure 3.2). A block is driven by a spring whose other end is attached to a driver moving with

a constant velocity (load point velocity V0).

Figure 3.2 – The single degree of freedom block-spring model

A necessary condition for the block to slip is V>0 or T>µN. The system of equations

representing the motion of the block reads

0

( , )( )

mV f T NT k V V

(3.3)

Here function f (T, µN) is defined as follows

, and 0( , )

0, or 0T N T N V

f T NT N V

(3.4)

In order to represent the system of equations (3.3) in dimensionless form, it is

convenient to introduce a dimensionless time푡∗

* 20 0, kt t

m (3.5)


51

The governing system of equations for constant friction in the dimensionless form reads

* *

*

( , )1

V f T NT V

(3.6)

where the dot stands for the derivative with respect to dimensionless time t*; V*, T* and N*

are the dimensionless velocity, shear force and normal force respectively.

*

0

VVV

, *

0 0

TT

mV *

0 0

NN

mV (3.7)

In order to demonstrate the behaviour of the system at stick-slip-type regime, we

consider the block sliding under the following set of initial conditions

(0) 0, (0) 0V T (3.8)

Figure 3.3 represents the corresponding behaviour of the system (dimensionless

velocity vs. dimensionless time).

Figure 3.3 – Block sliding with constant friction coefficient

It can be observed that the system exhibits self-excited oscillations even with constant

friction coefficient, which somewhat resemble the stick-slip-type sliding. Furthermore, the

energy in the system does not change with time, obviously due to the constant energy influx

by velocity V0, where the excess of energy associated with the V0 is dissipated by friction. It

should also be noted that similar oscillation-type movements were observed in laboratory

experiments on sliding of two granite blocks under biaxial compression [25].

0 50 100 150-0.5

0

0.5

1

1.5

2

2.5

t*=to, (time)

V* =V/V

o, (ve

loci

ty)

velocity

0 50 100 1500

20

40

60

80

t*=to

T* =T/(V

o om

ega o m

)

force


52

3.3 Propagation of sliding zone over a fault with rate-independent friction

Stress wave propagation in frictional sliding (generalisation to 1D solid)

Earthquakes can lead to catastrophic structural failures and may trigger tsunamis,

landslides and volcanic activities [26], [27]. The earthquakes are generated at faults and are

either produced by rapid (sometimes ‘supersonic’) propagation of shear cracks/ruptures along

the faults or originated in the stick-slip sliding over the fault. The velocity of rupture

propagation is crucial for estimating the earthquake damage. The rupture velocities can be

classified by comparison its speed with the speeds of stress waves in the rupturing solid [28].

There are several types of rupture propagation: supersonic (V>VP), intersonic (VS<V<VP),

subsonic (V<VS), supershear (V>VS), sub-shear (VR<V<VS) and sub-Rayleigh (V<VR).

According to the data obtained from the seismic observation of crustal earthquakes, most

ruptures propagate with an average velocity that is about 80% of the shear wave velocity [29].

However, in some cases, supershear propagation of earthquake-generating shear ruptures or

sliding is observed [30], [31], [32], [33], [34], [35]. The above observations introduced the

concept of supershear crack propagation [36], [8], [37], [38]. However, due to the lack of strong

motion recording, there are still some debates regarding the data interpretation [39], [37]. For

instance, it was suggested that during the 2002 Denali Earthquake both ruptures were

propagated over a distance of up to 40 km at supershear speed [34]. However, the data was

based on a single ground motion record. The joint inversion of the combined data-set provides

a more robust description of the rupture. The recent studies, which are aimed at deriving the

kinematic models for large earthquakes, have shown the importance of the type of data used.

It has been shown that slip maps for a given earthquakes may vary significantly [40], [41].

The analytical [42] and numerical [43] research in fracture dynamics indicate that only

Mode II rupture (shear-induced slip occurring in the direction perpendicular to the crack front)

can propagate with intersonic velocity (Vs<V<Vp) for short durations, as long as the prestress

of the fault is high compared to both failure and residual stresses [38]. Intersonic Mode II crack

propagation was first confirmed in laboratory by Rosakis et al. [44].

In the present section, we analyse a simple mechanism of unusually high shear fracture

or sliding zone propagation, also referred as the p-sonic propagation of sliding area over a

frictional fault. The analysis is based on the fact, that accumulation of elastic energy in the

sliding plates on both sides of the fault can produce oscillations in the velocity of sliding even


53

if the frictional coefficient is constant. We note that Walker and Shearer [45] found evidence

of the intersonic rupture speeds close to the local P-wave velocity by analysing the Kokoxili

and Denali earthquakes seismic data. This section considers a highly simplified 1-D rod model

where many properties of real fault system have been neglected. (Considerable fault geometry

simplification is in use in analysing intersonic ruptures, e.g., Bouchon et al., [33])

All variables and constants used in equations in this section are listed below in Table

3.2.


Symbol Meaning Symbol Meaning V0 load point velocity τ shear stress V relative velocity of block c velocity of longitudinal wave (p-

wave) µ friction coefficient ω eigen frequency c1 propagation speed of rupture k the spring stiffness relating stress

and displacement discontinuity (the difference between the rod displacement and the zero displacement of the base)

t time J0 Bessel function of order 0 h thickness of an infinite rod

0J derivative of Bessel function

ρ volumetric rod density i imaginary unit σN uniform compressive load ξ independent variable σ longitudinal stress z integration variable τf friction stress f, g arbitrary functions E Young’s modulus a, b, c scalar parameters

We assume the constant friction law, which will permit us to obtain an analytical

solution. For this purpose, following Nikitin [46], we consider the simplest possible 1D model

of fault sliding, which takes into account the rock elastic response and the associated dynamic

behaviour. The model is shown in Figure 3.4. It consists of an infinite elastic rod of height

(thickness) h, and of unit length in the direction normal to the plane of drawing in Figure 3.4.

The linear density is and the rod is assumed to be able slide over a stiff surface. The sliding

is resisted by friction. The stiff surface can be described as a symmetry line such that instead

of the (horizontal) fault, only the upper half of the line is considered. The rod is connected to a

stiff layer moving with a constant velocity V0. The connection is achieved through a series of


54

elastic shear springs. Both the elastic rod and the elastic springs describe the model of the

elasticity of the rock around the fault, as shown in Figure 3.4. We assume that the system is

subjected to a uniform compressive load N such that the friction stress is kept constant, which

is assumed equal to f N const .

Figure 3.4 – Infinite elastic rod driven via elastic shear springs with velocity V0

Equation of movement of the rod reads

1 ( )fV

x h t

(3.9)

where is the longitudinal (normal) stress in the rod, is the contact shear stress, f is the

frictional stress, V0 is the load point velocity and V(x,t) is the velocity of point x of the rod at

time t, as shown in Figure 3.4.


55

According to the Hooke’s law

uEx

(3.10)

where u(x,t) is the displacement and E is the Young’s modulus of the rod. After differentiating,

we have

VEt x

(3.11)

The elastic reaction of the shear springs is expressed as

0 ( )k V Vt

(3.12)

where k is the spring stiffness relating stress and displacement discontinuity (the difference

between the rod displacement and the zero displacement of the base).

Defining 0V V V and solving the system of equations (3.9) - (3.12), we get the

following wave equation

2 22 2

2 2 V Vc Vt x

(3.13)

where c Eh is the velocity of the longitudinal wave (p-wave), ( )k h is regarded

as eigen frequency of the system consisting as a unit length of the rod considered as a lamp

mass on the shear springs.

It is observed that despite the presence of shear springs and friction between the rod

and the stiff surface, the waves propagate with the p-wave velocity determined by the Young’s

modulus and density of the rod. Therefore, according to the terminology described in the

introduction, the wave should be named p-sonic wave. It should be highlighted that while such

waves look like the shear waves, they are in fact compressive waves propagation along the rod,

hence denoted as the p-wave velocity.

In order to analyse the way the pulse propagates, equation (3.13) is complemented by

the initial conditions as


56

0 0( , ) ( ); ( )d VV x t f x F xdt

(3.14)

Solution of wave equation (3.13) can be found by using the Riemann method [47]

1 1( , ) [ ( ) ( )] ( , , )2 2

x ct

x ct

V x t f x ct g x ct x t z dz

(3.15)

where

2 2 2

1( , , ) ( , , )( )

x t z x t zc t z x

(3.16)

The integral from (3.15) can be found by using the Chebyshev-Gauss method

1

2 1( , ) ( , , ) ( , , ), cos2

x ct n

j jjx ct

jI x t x t z dz x t x ctn n

(3.17)

where

2 2 2 2 2 2 2 2 20 0

1 1( , , ) ( ) ( ) ( ) ( ) ( )x t z F z J i c t z x c t z x tf z J i c t z xc c i c

(3.18)

Propagation of initial sliding

Figures 3.5-3.6 represent the propagation of initial sliding under the different initial

conditions. Particularly, a triangular velocity impulse, equation (3.19) and zero acceleration

were used as initial conditions for Figure 3.5. As shown in Figure 3.6, linear and harmonic

functions are used for velocity and acceleration as initial conditions.

( ; , , ) max min , ,0x a c xV f x a b cb a c b

(3.19)

where x, a, b, c are scalar parameters.


57

Figure 3.5 –Propagation of initial sliding in the form of a triangular function V of

zero area

a) b)

Figure 3.6 –Propagation of initial sliding with different initial conditions

a) 2 ; cos( )d VV x xdt

b) sin( ); 1d VV x xdt

It is seen that the initial sliding (impulse) propagating with p-wave velocity keeps its

width but the amplitude reduces with time. It is also observed that as the impulse propagates,

it loses energy of friction.

05

1015

20

05101520-7

-6

-5

-4

-3

-2

-1

0

x 104

t, [sec]x, [m]

V

, [m

/sec

]

05

1015

20

05

1015

20-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

x 106

t, [sec]x, [m]

V, [

m/s

ec]

05

1015

20

0

5

10

15

20-5

0

5

10

15

x 105

t, [sec]x, [m]

V

, [m

/sec

]


58

3.4 Rate-dependent friction law

Friction is a complex phenomenon and difficult for modelling. For this reason, most of

available friction models are empirical, based on fault simulation and interpretation of the

friction laws studied in the laboratory [49]. Despite the fact that experimental faults are simple

models of natural faults, they have many analogies and similarity with the real faults [50]. The

main analogue is the similarity between stick-slip motion observed in the laboratory and

earthquake fault slip.

It has been agreed that the type of friction law is one of the most important factors

which affect in the accuracy of the friction model. The static-dynamic friction law is the

simplest law that produces stick-slip motion [51], [52]. In this case, the sliding initiates when

the static friction exceeds the dynamic friction, s dF F [16]. Initially, Burridge and Knopoff

used a simple friction law in their model, where the friction force depends on the relative sliding

velocity of the block.

Later on, it was indicated that the friction cannot be a single valued function of velocity

[53]. Some researches (Archard [54], Tabor [55], Bhushan [56], Chowdhury and Helali [57])

identified that friction coefficient also depends on a large variety of parameters, such as: sliding

speed, acceleration, temperature, humidity, adhesion, roughness of surfaces, normal force,

critical slip distance, contact geometry, presence of lubrication, vibration, interface condition

and physical properties of materials. Thus, the friction law has been improved and formulated

by numerous researches:

Static/dynamic friction law [58];

Velocity-weakening friction law [59];

Time dependent friction law [60];

Rate- and state-dependent friction law [4], [61], [62].

Rate- and state-dependent friction laws successfully modelled frictional sliding and

earthquake phenomena. These laws were originally developed for constant normal stress

conditions [13] and proposed by Dieterich [2], [50], Ruina [4] and Rice [3]. The laws were

based on the experiments with rocks in order to obtain the empiric parameters (a, b, Dc) which

can help to study frictional instability. These rate- and state-dependent friction laws are similar

to the one proposed by Rabinowicz [63] based on the experiments with metals.


59


3.3.


The one-state variable constitutive law is a simple form among the state variable laws

and is used for the present system.

The shear stress τ is a function of the slip rate V and the state of the surface θ

( ) ( , )t V (3.20)

This law was proposed by Dieterich for the range of positive velocity values and

expressed as

**

*

( ln( ) ln( ))c

VVA BV D

(3.21)

The magnitude of Dc does not depend on the change of the velocity but stand on surface

roughness. θ is describe microscopic features of the slipping surface [65], which can in most

cases be viewed as representing the real contact area [66].

The state variable can be written as

1c

d Vdt D (3.22)

Symbol Meaning Symbol Meaning

µ0, µ friction parameter k spring stiffness σ normal stress V0 load point velocity N normal force V* reference velocity τ shear stress V relative velocity of block T shear force m mass of block

A0, B0, C empirical constant θ state variable a, b, c empirical constant

(dimensionless) Θ0 initial state variable

S unit area Dc characteristic slip distance ω0 eigen frequency DV limited velocity δ slip displacement


60

Equation (3.22) indicates that state variable cannot change instantaneously and evolves

with time and slip distance.

It was experimentally established that the velocity-weakening frictional sliding is a

necessary condition for modelling of stick-slip failure and seismic cycle [67], [64]. The velocity

dependence of the steady-state friction is characterised by the parameter (a-b). If a<b the

friction exhibits velocity-weakening effect (the sliding friction decreases with velocity). For

a>b a velocity strengthening effect can be observed.

Figure 3.7 illustrates the friction response described by equations (3.21) and (3.22). As

can be seen from the Figure 3.7, starting after steady state slip stage (V0 and τ0), the frictional

stress τ increases to (τ0+A) or decreases to (τ0+ (A-B)) simultaneously with the suddenly

imposed increase/decrease of V.

Figure 3.7 – Schematic illustration of one state variable friction law

It is impossible to get an exact “zero” velocity in numerical computation for nonlinear

friction law. Therefore, we introduce a limited velocity (DV) in order to determine stick phase

(Karnopp model principle, [68]).

Necessary conditions of mass block movement

0 or , slip and , stick

V T NV DV T N

(3.23)

where DV is a limited velocity and is equal to 0.1% of average velocity [69].


61

Stick-slip sliding under the normal constant stress is considered in the following

section.

Stick-slip sliding

In case of stick-slip block movement (V(0)=0; T(0)=0) the governing system of

equations can be expressed as

** 0 0

*

* 0

[( ln( ) ln( ))]

( )c

VVmV S A B SV D

S k S V V

(3.24)

System of equations (3.24) can be written in next form (for 100% contact and unit area

S=1 m2):

**

*

0

[( ln( ) ln( ))]

( )c

VVmV T a b NV D

T k V V

(3.25)

The stick-slip motion is challenging for modelling as a procedure of determination

numerical value of state variable θ has not yet been identified [52]. The major problem is how

to describe the evolution of θ with slip history [66]. Nagata et al [66] calculate values of the

state variable θ by substituting the slip velocity and measured shear stress into the constitutive

law. Okubo and Dieterich [70] selected an initial state variable θ0=300 s as a reasonable time

between stick-slip events. According to the experimental data the state variable varies from 105

to 109 s [71].

For the purpose of parametric analysis, the general simplified friction law [71] for range

of positive velocity values is used

**

ln VcV

(3.26)

We introduce the empirical parameter c (eq. 3.27), c<0 for the velocity-weakening case

*

ln for stick-slipVc a constV

(3.27)


62

where *lnc

Vb constD

.

The governing system of equations has the following form

*

0

[ ]

( )

mV T c N

T k V V

(3.28)

The system of equations (3.28) can be written in the next form

0

( , )( )

mV f T NT k V V

(3.29)

where the function f (T, µN) is defined as follows

, and ( , )

0, or T N T N V DV

f T NT N V DV

(3.30)

The system of equations (3.30) in dimensionless form reads

* *

*

( , )1

V f T NT V

(3.31)

where the dot stands for the derivative with respect to dimensionless time *0t t ; V*, T* and

N* are the dimensionless velocity, shear force and normal force respectively.

*

0

VVV

, *

0 0

TT

mV , *

0 0

NN

mV (3.32)


63

Figure 3.8 – Block sliding with rate-dependent friction coefficient

As can be seen from Figure 3.8 a single degree of freedom block-spring model can

produce oscillations in the velocity of sliding that has the stick-slip behaviour.

3.5. Comparative analysis between constant and rate-dependent friction coefficient

Stick-slip sliding producing by the system with constant and rate-dependent friction

coefficient is coincide under the certain set of parameters (Figure 3.9). Which means that stick-

slip behaviour of the system does not depend on friction coefficient and is caused only by the

influence of the spring.

Figure 3.9 – Block sliding with constant and rate-dependent friction coefficient

0 50 100 150 2000

1

2

3

4

5

t*=to, (time)

V* =V/V

o, (ve

loci

ty)

velocity

0 20 40 60-0.5

0

0.5

1

1.5

2

2.5

t*=to, (time)

V* =V/V

o, (ve

loci

ty)

=0.8*=0.75

0 20 40 600

2

4

6

8

10

t*=to, (time)

T* =T/(V

o om

ega o m

), (fo

rce)

=0.75*=0.8

0 50 100 150 2000

20

40

60

80

t*=to, (time)

T* =T/(V

o om

ega o m

), (fo

rce)

force


64

3.6. Propagation velocity of sliding zone for rate-dependent friction

This section introduced the notion that the frictional movement resembling the stick-

slip sliding, which are often observed and usually attributed to the rate dependence of friction,

can be obtained with constant friction by taking into account the elasticity of the surrounding

and its self-oscillations. This understanding is applied to propagation of slip over infinitely

long fault leads to a simple model that predicts that the slip will propagate with p-wave velocity.

This conclusion is made under the assumption of constant (rate-independent) friction. Relaxing

this assumption, that is taking into account that f fVt

leads to the following equation

replacing equation (3.13)

2 22 2 '

2 2

11 ,f

t

d V V Vc V Vh d V t x t

(3.33)

It is seen that when the sliding rate changes slowly, the propagation speed of rupture c1

can be approximated as

12 2

111 f

t

dc c

h d V

(3.34)

Furthermore, it is observed that when the friction increases with the sliding rate, c1

becomes smaller than p-wave velocity. If the rate dependence of friction is lowered further, the

slip propagation can become intersonic.

3.7 Conclusions

A single degree of freedom block-spring model with both constant and rate-dependent

friction coefficient was modelled and studied to investigate the frictional sliding.

Stick-slip phenomenon of the system can occur in the absence of imposed oscillations

with constant friction coefficient. It was observed that the rate dependence of friction

coefficient also leads to the stick-slip behaviour of the system. Thus, the mechanism of stick-

slip motion is caused by self-excited oscillations (structural dynamics) and does not depend on

friction behaviour.


65

The accumulation of elastic energy in the sliding plates on both sides of the fault can

produce oscillations in the velocity of sliding even when the friction is constant. These

oscillations resemble stick-slip movement.

The sliding exhibits wave-like propagation over long faults. Furthermore, the 1D model

shows that the zones of sliding propagate along the fault with the velocity of p-wave (the

propagation speed can however be lower if the rate dependence of friction is taken into

account). The mechanism of such fast wave propagation is the normal (tensile/compressive)

stresses in the neighbouring elements (normal stresses on the planes normal to the fault surface)

causing a p-wave propagating along the fault rather than the shear stress controlling the sliding.

This manifest itself as a p-sonic propagation of an apparent shear rupture.

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42. R. Burridge, Admissible Speeds for Plane‐Strain Self‐Similar Shear Cracks with Friction but Lacking Cohesion. Geophysical Journal of the Royal Astronomical Society, 1973. 35(4): p. 439-455.

43. S. Das and K. Aki, A numerical study of two‐dimensional spontaneous rupture propagation. Geophysical Journal of the Royal Astronomical Society, 1977. 50(3): p. 643-668.

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45. K. T. Walker and P. M. Shearer, Illuminating the near‐sonic rupture velocities of the intracontinental Kokoxili M w 7.8 and Denali fault M w 7.9 strike‐slip earthquakes with global P wave back projection imaging. Journal of Geophysical Research: Solid Earth, 2009. 114(B2): p. B02304.

46. L. V. Nikitin, Statics and dynamics of solids with an external dry friction. 1998: Moscow Lyceum. 272.

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48. Iu. Karachevtseva, A. V. Dyskin, and E. Pasternak, Generation and propagation od stick-slip waves over a fault with rate-independent friction. Nonlin. Process Geophys., 2017. 24: p. 343-349.

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69

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CHAPTER 4

Effect of external vibrations on frictional sliding

Chapter 4. Effect of external vibrations on frictional sliding

71

4.1 Introduction

The frictional behaviour in the presence of vibrations is completely different from the

relatively smooth sliding [1]. Normal and tangential stress/force variation may play an

important role in frictional force. Mitskevich [2] was the first who considered the possibility

of average friction reduction under the influence of transverse vibrations in his theoretical

model. Later the influence of transverse vibrations on friction was experimentally studied by

[3], [4].

It was experimentally established that friction reduction depends on the direction of

vibrations. There are a lot of publications related to friction reduction by applying vibration in

the normal [5], [6], [7], [8] and tangential (to the plane of contact) directions [9], [10], [11],

[12].

In the Earth’s crust, friction is important in preventing faults from sliding, which is a

form of instability that generates earthquakes. However, the faults are continuously subjected

to stress oscillations which can eventually cause friction reduction and sliding over fractures

and ruptures [13]. Thus, these stress oscillations can affect the strength and stability of faults.

In the Earth’s crust the normal vibrations can be produced by tidal stresses or by the seismic

waves generated by other seismic events. When tidal normal stress is negative (directs inward

to a fault plane), the total normal pressure on the fault would increase and keep the system

stationary (stick phase). In case of reduction of normal force, the friction will decrease and

allow fault slipping (slip phase). The amplitudes of tidal stress on a seismic fault depend on

such factors as fault type, geographical location, orientation and tidal forces produced by sun

and moon [14].

The oscillations in the Earth’s crust are associated with the earthquake triggering and

leading to the stick-slip. Rate- and state-dependent friction laws successfully developed for

simulation of earthquakes and were proposed by Dieterich [15], [16], Ruina [17] and Rice [18].

Despite the fact that the rate- and state-depended friction laws were originally developed for

constant stress, there are a lot of publications related to the investigation of normal stress

variation in the rock friction models. Thus, Lockner and Beeler [19], [20] specified the

influence of amplitude and frequency on the triggering of stick-slip motion. Tworzydlo and

Hamzeh [21] also concluded that variable normal force can lead to unstable sliding. Moreover,

Voisin [22], [23] noted that the oscillating normal stress can trigger earthquakes


72

(stability/instability transition). Later on, Perfettini et al [24] proposed a set of the laboratory

experiments with fluctuating normal stress to study different formulations of the friction law.

In engineering systems, such oscillations (both parallel and normal to the sliding

direction) can come from the system vibrations or can be imposed artificially, as in the methods

of controlling sliding, reducing frictional energy dissipation and cutting [25], [7], [26], [27],

[28].

In this chapter, the effect of imposed vibrations (parallel and perpendicular to the

sliding direction) on frictional sliding is investigated. The chapter is organized as follows: the

effect of normal vibrations of simple spring-block models which obey constant/rate-dependent

friction law is considered in Section (4.2); the effects of horizontal vibrations with

constant/rate-dependent friction are presented in Section (4.3) which is followed by discussion

and conclusions in Section (4.4).

4.2 Effect of normal vibrations

While the normal force reduction alternates with the balancing normal force increase,

their action on the friction is asymmetric if the friction was sufficient to prevent the sliding. In

this case, the stage of increase of the normal force will keep the system stationary, while the

friction reduction associated with the reduction in the normal force can allow sliding.

Effects of changing of vibrational parameters (amplitude and frequency) on the

qualitative behaviour of the system are investigated in this section. For this purpose, vertical

vibrations which are represented either by harmonic load or random fluctuations applied to the

block. The dynamic behaviour of the motion of the slider depends on the value of the shear

force T, amplitude A of the external force (A<N) and the frequency ωF of the external force.

A block of mass with vertical spring (constant friction case)

A block mass with a vertical spring (Figure 4.1) is considered in this section. The

system is subjected to the vertical vibrations by harmonic force 0( ) c o s( )FF t F t .

All variables and constants used in equations are listed below in the Table 4.1.


73



V relative velocity of block F(t) harmonic force k single spring stiffness F0 load amplitude m block mass A amplitude N normal force P dimensionless parameter µ constant friction

coefficient T shear force

ωF forced frequency B constant ω0 eigen frequency b damping t time γ damping coefficient δ phase

Figure 4.1 – Block held by friction on a base

Vibrations of the block are governed by the following equation

2 00 cos F

Fy y y tm

(4.1)

where

20 , k b

m m (4.2)

are the eigen frequency and damping parameter respectively.


74

The steady-state solution of equation (4.1) reads

0( ) cos( ),FAFy t tm

2 2 2 2 2

0

1 ,( )F F

A

2 20

tan F

F

(4.3)

The phase of vibrations corresponding to y(t)>0 reduces the normal force by ky(t).

Thus, the sliding can be possible if

( ( ))T N ky t (4.4)

Condition of sliding stage can be written as

20 00 cos( )FN T F A t (4.5)

Condition of the existence of a sliding in each oscillating cycle is

20 ( )FP A (4.6)

where P is a dimensionless parameter.

0

0N TPF

(4.7)

During the interval [-Δt, Δt] the block is sliding and acted upon the force2

0 0 cos( )FT N F A t , which causes block acceleration in horizontal direction (in the

direction of the x-axis).

12

0

1 cos( )F F

PtA

(4.8)

The governing equation of the block movement in horizontal direction has next form

20 0( ( ) cos )

( ) 0( ) 0

F Fmx F A t Px tx t

(4.9)

Velocity of block movement is low; thus, the effect of damping can be neglected.


75

Solution of this equation reads 22

00 2[ ( )( sin cos ) ( ) ]

2 F F F FF

tmx F P A t t t P t t B

(4.10)

where B is a constant and can be find from the condition x(-Δt) =0.

The block moves the distance in time 2Δt is 2

00[ ( ) ( )] 2 ( ) sinF F

F

m x m x t x t F A t P t t

(4.11)

The resultant average velocity over the full period ( 2

F

T

) is

20 0 ( ) sin

2 /F

F FF F

FxV A t P t tm

(4.12)

Substituting equation (4.8) into (4.12) we get:

4 2 2 1 100 2 2

0 0

[ ( ) cos ]cos( ) ( )F

F F F

F P PV A P Pm A A

(4.13)

The dependence of the normalised velocity vs. the driving frequency is plotted in Figure

4.2 for different values of P and γ for the values of driving frequency that satisfy (4.6).

The first thing which is apparent is the presence of singularity when 0F . It can

only be removed by high values of P and γ when condition (4.6) cuts off the very low

frequencies. The reason for high velocities at low frequencies is clear: at low frequencies, the

condition of block sliding (4.5) is satisfied long enough for the block to gain considerable

speed. The appearance of the 1F singularity is obviously caused by neglecting the damping

in the horizontal movement.


76

a)

b)

Figure 4.2 –Dependence of the average velocity of block sliding upon the driving

frequency for the damping coefficient a) 0.1 and b) 0.5

As can be seen from Figure 4.2, at higher frequencies and low damping the velocity

shows a peak frequency (maximum of 20 ( )FA shown in the figures by dotted line) which is

close to the resonant frequency of the block. The reduction of friction coefficient is observed

under the velocity growth (frequency peak) at steady-state regime. At high values of damping

the peaks deviate from the resonant frequency or disappear altogether.

0.4 0.6 0.8 1 1.2 1.4 1.6 1.80

1

2

3

4

5

6

7

8

9

10

F/

Vm

/F 0

Amplitude responce curveP=0.1P=0.5P=1

0.2 0.4 0.6 0.8 1 1.2 1.4 1.60

0.5

1

1.5

2

2.5

3

3.5

4

F/

V

m/

F 0

Amplitude response curveP=0.1P=0.5P=1


77

Sliding of a block with horizontal spring under vertical vibrations

As shown in the previous chapter the presence of horizontal spring can qualitatively

change the process of block sliding and create a stick-slip type motion even when friction is

constant. Therefore, the next model to be considered is the block with horizontal spring. To

focus on the effect of imposed excitations and to avoid complicating effects of geometry, a

single degree of freedom block-spring model was used. A block sliding on a rigid horizontal

surface is driven by a spring whose other end is attached to a driver moving with a constant

velocity V0 (Figure 4.3).

Figure 4.3 - The single degree of freedom block-spring model

Harmonic vertical load

Two types of sliding (steady sliding and stick-slip) in both resonance ( 0 F ) and

non-resonance ( 0 F ) regimes are considered in this section. In case of non-resonance

regime, the driving frequency is different from the eigen frequency (ω0 =10) and equal to ωF=8.

The plots are built dimensionless velocity and force vs. dimensionless time. Sliding under

normal (vertical) vibrations was studied for the vertical vibration amplitudes A=0.01 and A=0.1

in the dimensionless form. The system obeys constant/rate-dependent friction law.

All constants and variables used in equations in this section are presented in Table 4.2.


78


Symbol Meaning Symbol Meaning V0 load point velocity N normal force V relative velocity of block T shear force V* arbitrary positive reference

velocity σ normal stress

DV limited velocity τ shear stress ω0 eigen frequency S unit area of surface ωF load frequency A load amplitude k spring stiffness θ state variable µ* steady-state friction Dc characteristic slip distance m block mass A0, B0 empirical constants µ friction coefficient a, b, c positive empirical constants

(dimensionless) t time

Constant friction law

Block sliding conditions: V>0 or T>µN. The system of equations representing the

motion of the block reads

0

( , ( cos ))

( )FV f T N A t

T k V V

(4.14)

where

( sin ), ( sin ) and 0( , ( sin ))

0, ( sin ) or 0F F

FF

T N A t T N A t Vf T N A t

T N A t V

(4.15)

Introduce a dimensionless time푡∗

* 20 0, kt t

m (4.16)

The dimensionless form of the system of equations (4.14)

* * *

*

( , ( sin )1

V f T N A tT V

(4.17)

where the dot stands for the derivative with respect to dimensionless time *t ; V*, T*, N* and

A* are the dimensionless velocity, shear force, normal force and load amplitude respectively.


79

*

0

VVV

, *

0 0

TT

mV , *

0 0

NN

mV , *

0 0

AmV

A

(4.18)

Figure 4.4 shows that imposed vertical excitations ( ) sin FF t A t (stick-slip regime)

with the different amplitudes do not lead to noticeable change in the behaviour of the system.

Figure 4.4 – Stick-slip sliding (V(0)=0, T(0)=0) in the presence of vertical vibrations

(A=0.01, A=0.1). The driving frequency is 0 F , 8F

Now the resonance regime is considered ( 0 F ). The oscillator produces different

responses depending on the amplitude parameter A, for which the following dimensionless

values are chosen: A=0.01, A=0.1.


(A=0.01). The driving frequency is 0 F

0 20 40 60 80 100-0.5

0

0.5

1

1.5

2

2.5

t*=to, (time)

V* =V/V

o, (ve

loci

ty)

A=0.01A=0.1

0 20 40 60 80 1000

2

4

6

8

10

t*=to, (time)

T* =T/(V

o om

ega o m

), (fo

rce)

A=0.01A=0.1

0 100 200 300 400 500-0.5

0

0.5

1

1.5

2

2.5

t*=to, (time)

V* =V/V

o, (ve

loci

ty)

0 100 200 300 400 5000

2

4

6

8

10

t*=to, (time)

T* =T/(V

o om

ega o m

), (fo

rce)

The zone of reduced vibrations The zone of reduced vibrations


80



Figures 4.5-4.6 show that the amplitude lead to formation the zone of reduced vibration.

It is seen that the zone of reduced vibrations extends with the decrease of amplitude (the smaller

the amplitude of oscillations the longer the zone of reduced vibration). Moreover, it is moves

from the origin with the decrease of amplitude.

A completely different response is observed during the steady sliding ( 0 F ). It is

seen that even small amplitude of imposed excitations breaks the steady sliding and turns it

into a stick-slip.

Figure 4.7 – Steady sliding (V(0)=V0, T(0)=T0) in the presence and absence of

vertical vibrations (A=0 and A=0.01). The driving frequency is 0 F , 8F

Figures 4.8-4.9 show the system behaviour when the driving frequency coincides with

the eigen frequency ( 0 10 ). The applied amplitudes lead to stick-slip regime. Furthermore,

in all cases the stick-slip amplitude increases to the same final values; the only difference is the

0 20 40 60 80 100-0.5

0

0.5

1

1.5

2

2.5

t*=to, (time)

V* =V/V

o, (ve

loci

ty)

0 20 40 60 80 1000

2

4

6

8

10

t*=to, (time)

T* =T/(V

o om

ega o m

), (fo

rce)

0 20 40 60 80 1000.95

1

1.05

t*=to, (time)

V* =V/V

o, (ve

loci

ty)

A=0A=0.01

0 20 40 60 80 1007.45

7.5

7.55

t*=to, (time)

T* =T/(V

o om

ega o m

), (fo

rce)

A=0A=0.01

The zone of reduced vibrations



81

time to reach the final stick-slip motion, the smaller the amplitude of oscillations the longer it

takes to reach the final regime. It can be shown that this time is inversely proportional to the

amplitude of oscillations.

Figure 4.8 – Steady sliding (V(0)=V0, T(0)=T0) in the presence of vertical

vibrations (A=0.01). The driving frequency is 0 F

Figure 4.9 – Steady sliding (V(0)=V0, T(0)=T0) regime in the presence of vertical


Rate- and state-dependent friction law

We investigate the dynamic response of the system with rate-dependent friction on

frictional sliding.

Two types of sliding (steady sliding and stick-slip) are considered in the following

sections. It is assumed that the slider has a unit area S=1 m2 and 100% contact for both types

of sliding.

0 100 200 300-0.5

0

0.5

1

1.5

2

2.5

t*=to, (time)

V* =V/V

o, (ve

loci

ty)

0 100 200 3006

6.5

7

7.5

8

8.5

9

t*=to, (time)T* =T

/(Vo o

meg

a o m),

(forc

e)

0 20 40 60 80-0.5

0

0.5

1

1.5

2

2.5

t*=to, (time)

V* =V/V

o, (ve

loci

ty)

0 20 40 60 806

6.5

7

7.5

8

8.5

9

t*=to, (time)

T* =T/(V

o om

ega o m

), (fo

rce)

Time of reaching the steady oscillations





82

Steady-sliding

In case of steady-sliding velocity V remains constant, then shear stress τ evolves toward

a steady-state value ss . In this case is a function of velocity.

( , ) ( )ssV V (4.19)

The friction at steady-state regime (V is fixed)

0*

( ) ln( )ssVa bV

(4.20)

During the steady-sliding regime, the system initially was sliding steadily at the

reference velocity *V and then the load point velocity reached a constant value V0 [29].

The block movement can be described by the following system of equations

* 0 0 *

* 0

[( ( ) ln( / ))( sin )]( )

FmV S A B V V S AS tS k S V V

(4.21)

Taking into account that slider has a unit area S=1 m2 and 100% contact, the system

(4.21) can be rewritten in next form

* *[( ( )ln( / ))( sin )]

(1 )FmV T a b V V N A t

T k V

(4.22)

Stick-slip sliding

The general formula describing the stick-slip sliding is given by

* 0 * 0 *

* 0

[( ln( / ) ln( / ))( sin )]( )

c FmV S A V V B V D S AS tS k S V V

(4.23)

and can be written in next form

* * *

0

[( ln( / ) ln( / ))( sin )]

( )c FmV T a V V b V D N A t

T k V V

(4.24)

ss


83

General friction law

According to the numerous literature publications, a block system at steady sliding and

stick-slip regimes has different governing system of equations. The general simplified friction

law was used for both types of sliding. A detailed description of the general friction law is

provided in Chapter 3, Section (3.4).

Thus, the system of equations (for the steady sliding and stick-slip sliding) has the

following form

* *

0

[ ln( / )]( sin )

( )FmV T c V V N A t

T k V V

(4.25)

where c is empirical constant and equal

*

( ) for steady sliding

ln for stick-slip

c a b

Vc a constV

(4.26)

where *lnc

Vb constD

.

The system of equations (4.25) can be written

0

( , ( sin ))

( )FmV f T N A t

T k V V

(4.27)

where the function ( , ( sin )Ff T N A t

( sin ), ( sin ) and ( , ( sin ))

0, ( sin ) or F F

FF

T N A t T N A t V DVf T N A

T N A t V DV

(4.28)

The system of equations (4.27) in dimensionless form reads * * *

*

( , ( sin ))1

V f T N A tT V

(4.29)

where the dot stands for the derivative with respect to dimensionless time *0t t ; V*, T*, N*,

and A* are the dimensionless velocity, shear force, normal force, and load amplitude

respectively.


84

*

0

VVV

, *

0 0

TT

mV , *

0 0

NN

mV , *

0 0

AmV

A

(4.30)


(A=0.01). The driving frequency is 0 F , 8F


(A=0.1). The driving frequency is 0 F , 8F

0 50 100 150 2000

1

2

3

4

5

t*=to, (time)

V* =V/V

o, (ve

loci

ty)

A=0.01

0 50 100 150 2000

1

2

3

4

5

6

t*=to, (time)

V* =V/V

o, (ve

loci

ty)

A=0.1

0 50 100 150 2000

20

40

60

80

t*=to, (time)

T* =T/(V

o om

ega o m

), (fo

rce)

A=0.1

0 50 100 150 2000

20

40

60

80

t*=to, (time)

T* =T/(V

o om

ega o m

), (fo

rce)

A=0.01


85





Figures 4.10-4.13 demonstrate that the effect of imposed vertical harmonic load on the

stick-sliding is negligible. However, at resonance regime ( 0 F ) the block oscillation

frequency is higher comparable with the non-resonance regime.

0 50 100 150 2000

1

2

3

4

5

t*=to, (time)

V* =V/V

o, (ve

loci

ty)

A=0.01

0 50 100 150 2000

1

2

3

4

5

6

t*=to, (time)

V* =V/V

o, (ve

loci

ty)

A=0.1

0 50 100 150 2000

20

40

60

80

t*=to, (time)

T* =T/(V

o om

ega o m

), (fo

rce)

A=0.01

0 50 100 150 2000

20

40

60

80

t*=to, (time)

T* =T/(V

o om

ega o m

), (fo

rce)

A=0.1


86

Figure 4.14 – Steady sliding (V(0)=V0, T(0)=T0) regime in the presence and absence

of vertical vibrations (A=0 and A=0.01). The driving frequency is 0 F , 8F




0 20 40 60 80 1000

1

2

3

4

5

t*=to, (time)

V* =V/V

o, (ve

loci

ty)

A=0A=0.01

0 20 40 60 80 10068

70

72

74

76

t*=to, (time)

T* =T/(V

o om

ega o m

), (fo

rce)

A=0A=0.01

0 20 40 60 80 1000

1

2

3

4

5

t*=to, (time)

V* =V/V

o, (ve

loci

ty)

A=0.01

0 50 100 150 20068

70

72

74

76

t*=to, (time)

T* =T/(V

o om

ega o m

), (fo

rce)

A=0.01

0 50 100 150 2000

1

2

3

4

5

6

t*=to, (time)

V* =V/V

o, (ve

loci

ty)

A=0.1

0 50 100 150 20066

68

70

72

74

76

78

t*=to, (time)

T* =T/(V

o om

ega o m

), (fo

rce)

A=0.1


87


Figures 4.14-4.16 show system instability under the action of vertical vibrations at the

steady-state sliding.

Randomly fluctuating vertical load

Geophysics is one of the main application areas of the random stress fluctuations. The

cracks creation and their growth under spatially random stress fields were studied by Dyskin

[30], [31]. In our work, we investigate the effect of random vertical fluctuations on frictional

sliding.

To reproduce the random fluctuations of vertical load we introduce a random variable

. It is a uniformly distributed random number in the interval (0,1).

2 ( ( ) 0.5)))A rand t (4.31)

The system of equations that represent motion of the block reads

0

( , ( )( )

V f T NT k V V

(4.32)

where the function f (T, µN) is defined as follows

( ), ( ) and 0( , ( )

0, ( ) or 0T N T N V

f T NT N V

(4.33)

The governing systems of equations for slip-stick motion read (dimensionless form):

* * *

0

( , ( )

( )V f T NT P V V

(4.34)

where the dot stands for the derivative with respect to dimensionless time t*; V*, A*, T*, and

N* are the dimensionless velocity, amplitude, shear force, and normal force respectively.

* 0

0

VmV VPN

, *

0

AAN

, *

0

TT

N , *

0

NN

N (4.35)


4.3.


89



V0 load point velocity T shear force

V relative velocity of

block

µ constant friction

coefficient

k single spring stiffness A load amplitude

m block mass t time

N normal force P additional parameter,

0

0

PmN

Random fluctuations are considered on two types of frictional sliding. According to

the obtained plots (Figures 4.17 - 4.18) the steady sliding is unstable under the influence of

even relatively small excitations. The “stick-slip” is stable with respect to normal oscillations

with different amplitudes.

Figure 4.17 – Steady sliding (V(0)=V0, T(0)=T0) in the presence of vertical vibrations

for A=0.01

0 50 1000.096

0.098

0.1

0.102

0.104

0.106

t*=t*0, (time)

V* =(V *

m*

0)/N0, (

velo

city

)

A=0.01no oscillations

0 50 1000.74

0.745

0.75

0.755

0.76

0.765

t*=t*0, (time)

T* =T/N

0, (fo

rce)

A=0.01no oscillations


90


for A=0.01 and A=0.1

4.3 Effect of longitudinal vibrations

In case of longitudinal vibrations, the directions of sliding and excitation are collinear.

Vibrations applied in the direction parallel to the sliding direction are sometimes used in

engineering systems (metal cutting, drilling, ultrasonic machining) involving high or low

frequency vibrations applied to the cutting tool to achieve better cutting performance [32]. A

single degree of freedom block-spring model was used for investigation the frictional sliding

in the presence of horizontal excitation (Figure 4.19). It is assumed that block is driven with

the constant velocity V0 over the base to which horizontal excitations ( ) sin FF t A t are

applied.

Figure 4.19 – Frictional sliding in the presence of horizontal vibrations

All constants and variables used in equations in this section are presented in Table 4.4.

0 50 100 150 200-0.1

0

0.1

0.2

0.3

t*=to, (time)

V* =(V

*m*o

meg

a o)/No, (

velo

city

)

A=0.01A=0.1

0 50 100 150 2000

0.2

0.4

0.6

0.8

1

t*=to, (time)

T* =T/(N

o), (fo

rce)

A=0.01A=0.1


91



V0 load point velocity N normal force

V relative velocity of block T shear force

V* arbitrary positive reference

velocity

A load amplitude

DV limited velocity c empirical constant

ω0 eigen frequency t time

ωF load frequency µ friction coefficient

k spring stiffness µ* steady-state friction

m block mass

Constant friction law

The system of equations representing the block motion reads

0

( , )

( ) cos( )F F

mV f T NT k V V A t

(4.36)

where

, and 0( , )

0, or 0T N T N V

f T NT N V

(4.37)

The governing system of equations in dimensionless form

* *

* * *

( , )

(1 ) cos( )V f T NT V A t

(4.38)

where the dot stands for the derivative with respect to dimensionless time *0t t ; V*, T*, N*

and A* are the dimensionless velocity, shear force, normal force and load amplitude

respectively.

*

0

VVV

, *

0 0

TT

mV , *

0 0

NN

mV , *

0 0

AmV

A

(4.39)


92

The imposed tangential vibrations on the system in the stick-slip sliding do not affected

the general behaviour of the system (Figure 4.20). However, the resonance regime ( 0 F ) is

more sensitive to the applied vibrations (Figures 4.21-4.22). Thus, we can observe the zone of

reduced vibrations which depends on the value of imposed amplitude.

Figure 4.20 – Stick-slip sliding (V(0)=0, T(0)=0) in the presence of horizontal

vibrations (A=0.01, A=0.1). The driving frequency is 0 F , 8F



0 20 40 60 80 100-0.5

0

0.5

1

1.5

2

2.5

t*=to, (time)

V* =V/V

o, (ve

loci

ty)

A=0.01A=0.1

0 20 40 60 80 1000

2

4

6

8

10

t*=to, (time)T* =T

/(Vo o

meg

a o m),

(forc

e)

A=0.01A=0.1

0 100 200 300 400 500-0.5

0

0.5

1

1.5

2

2.5

t*=to, (time)

V* =V/V

o, (ve

loci

ty)

0 100 200 300 400 5000

2

4

6

8

10

t*=to, (time)

T* =T/(V

o om

ega o m

), (fo

rce)

The zone of reduced vibrations The zone of reduced vibrations


93





Figure 4.23 shows that the system turns to stick-slip mode under the influence of small-

amplitude vibrations.

Figure 4.24 – Steady sliding (V(0)=V0, T(0)=T0) regime in the presence of

horizontal vibrations (A=0.01). The driving frequency is 0 F

0 20 40 60 80 100-0.5

0

0.5

1

1.5

2

2.5

t*=to, (time)

V* =V/V

o, (ve

loci

ty)

0 20 40 60 80 1000

2

4

6

8

10

t*=to, (time)

T* =T/(V

o om

ega o m

), (fo

rce)

0 20 40 60 80 1000.94

0.96

0.98

1

1.02

1.04

1.06

t*=to, (time)

V* =V/V

o, (ve

loci

ty)

A=0A=0.01

0 20 40 60 80 1007.4

7.45

7.5

7.55

7.6

7.65

t*=to, (time)

T* =T/(V

o om

ega o m

), (fo

rce)

A=0A=0.01

0 50 100 150 200 250-0.5

0

0.5

1

1.5

2

2.5

t*=to, (time)

V* =V/V

o, (ve

loci

ty)

0 50 100 150 200 2506

6.5

7

7.5

8

8.5

9

t*=to, (time)

T* =T/(V

o om

ega o m

), (fo

rce)






94

Figure 4.25 – Steady sliding (V(0)=V0, T(0)=T0) regime in the presence of horizontal


Figures 4.24-4.25 demonstrate that imposed small-amplitude horizontal vibrations

trigger stick-slip regime.Time of reaching the steady oscillations is inversely proportional to

the amplitude of vibrations.

Rate- and state-dependent friction law

We investigate the dynamic behaviour of a system that obeys a rate- and state-

dependent friction law. Two sliding modes (stick-slip and steady-sliding) have been generated

in numerical modelling. A general simplified friction law (was introduced in Chapter 3, Section

(3.4),) is used for frictional modelling.

The governing system of equations (for the steady sliding and stick-slip sliding) has the

following form

* *

0

[ ln( / )]

( ) cos( )F F

mV T c V V NT k V V A t

(4.40)

The system (4.40) can be rewritten in next form

0

( , )

( ) cos( )F F

mV f T NT k V V A t

(4.41)

where

, and ( , )

0, or T N T N V DV

f T NT N V DV

(4.42)

The system of equations (4.41) in dimensionless form reads

0 20 40 60 80-0.5

0

0.5

1

1.5

2

2.5

t*=to, (time)

V* =V/V

o, (ve

loci

ty)

0 20 40 60 806

6.5

7

7.5

8

8.5

9

t*=to, (time)

T* =T/(V

o om

ega o m

), (fo

rce)




95

* *

* *

( , )(1 ) cos( )

V f T NT V A t

(4.43)

where the dot stands for the derivative with respect to dimensionless time *0t t ; V*, T*, N*,

and A* are the dimensionless velocity, shear force, normal force, and load amplitude

respectively.

*

0

VVV

, *

0 0

TT

mV , *

0 0

NN

mV , *

0 0

AmV

A

(4.44)

Figures 4.26-4.32 show the system behaviour.


vibrations (A=0.01). The driving frequency is 0 F , 8F


vibrations (A=0.1). The driving frequency is 0 F , 8F

0 50 100 150 2000

1

2

3

4

5

t*=to, (time)

V* =V/V

o, (ve

loci

ty)

A=0.01

0 50 100 150 2000

1

2

3

4

5

6

t*=to, (time)

V* =V/V

o, (ve

loci

ty)

A=0.1

0 50 100 150 2000

20

40

60

80

t*=to, (time)

T* =T/(V

o om

ega o m

), (fo

rce)

A=0.01

0 50 100 150 2000

20

40

60

80

t*=to, (time)

T* =T/(V

o om

ega o m

), (fo

rce)

A=0.1


96





Figures 4.26-4.29 show that the longitudinal vibration has no effect on the system

behaviour at stick-slip sliding regime. However, according to Figures 4.30-4.32 the steady-

sliding regime of the system is unstable under the action of vibrations.

0 50 100 150 2000

1

2

3

4

5

t*=to, (time)

V* =V/V

o, (ve

loci

ty)

A=0.01

0 50 100 150 2000

1

2

3

4

5

6

t*=to, (time)

V* =V/V

o, (ve

loci

ty)

A=0.1

0 50 100 150 2000

20

40

60

80

t*=to, (time)

T* =T/(V

o om

ega o m

), (fo

rce)

A=0.01

0 50 100 150 2000

20

40

60

80

t*=to, (time)

T* =T/(V

o om

ega o m

), (fo

rce)

A=0.1


97



Figure 4.31– Steady sliding (V(0)=V0, T(0)=T0) regime in the presence of


Figure 4.32– Steady sliding (V(0)=V0, T(0)=T0) regime in the presence of


0 50 100 150 2000

1

2

3

4

5

t*=to, (time)

V* =V/V

o, (ve

loci

ty)

A=0A=0.01

0 50 100 150 20068

70

72

74

76

t*=to, (time)

T* =T/(V

o om

ega o m

), (fo

rce)

A=0A=0.01

0 50 100 150 2000

1

2

3

4

5

t*=to, (time)

V* =V/V

o, (ve

loci

ty)

A=0.01

0 50 100 150 20068

70

72

74

76

t*=to, (time)

T* =T/(V

o om

ega o m

), (fo

rce)

A=0.01

0 50 100 150 2000

1

2

3

4

5

6

t*=to, (time)

V* =V/V

o, (ve

loci

ty)

A=0.1

0 50 100 150 20066

68

70

72

74

76

t*=to, (time)

T* =T/(V

o om

ega o m

), (fo

rce)

A=0.1


98

4.4 Conclusions

The system with constant friction law

A single sliding block model was analysed to investigate the effect of imposed normal

(vertical) oscillations for the case when the system obeys the constant friction law. The effect

of friction reduction under the velocity growth is observed in the steady-state regime.

The effect of the imposed oscillations depends on the type of frictional sliding and

whether the system is in resonance/non-resonance regime.

Thus, for the non-resonance regime ( 0 F ), the stick-slip sliding is stable with the

respect to vertical and horizontal oscillations. However, steady sliding is unstable under the

influence of even small amplitudes for all types of vibrations.

The resonance regime for both types of sliding has a pronounced effect on the system

behaviour. The vertical and horizontal vibrations have similar effect on the system behaviour.

However, the zone of reduced vibrations (for stick-slip sliding) and time of reaching of steady

oscillations (for steady sliding) are larger for the vertical vibrations. Thus, these phenomena

open a way to control stick-slip vibrations through the application of forced vibrations on the

eigen frequency of the system with an appropriate choice of the amplitude and proper

parameters.

The system with rate- and state-dependent friction law

The effect of harmonic load acting parallel (horizontal) and perpendicular (vertical) to

the sliding direction is minor in the stick-slip sliding. At the same time, steady sliding is

unstable with respect to both types of oscillations. There is little effect in the system’s

behaviour at resonance and non-resonance regimes.

The dynamic behaviour of the system depends on the type of friction law. Furthermore,

the effect of imposed excitations is more pronounced for the system with constant friction than

for the system with rate-dependent friction under the same set of model parameters. The

simplified rate-dependent friction law leads to the removal of effects (zone of reduced

vibrations and time of reaching of steady oscillations) of the system which obeys the constant

friction law.


99

References

1. M. A. Chowdhury, D. M. Nuruzzaman, and M. L. Rahaman, Influence of external horizontal vibration on the coefficient of friction of aluminium sliding against stainless steel. Industrial Lubrication and Tribology, 2011. 63(3): p. 152-157.

2. A. M. Mitskevich, Motion of a body over a tangentially vibrating surface taking account of friction. . Soviet Phys. Acoustics, 1968. 113(3): p. 348-351.

3. T. Skare and J. E. Stahl, Static and dynamic friction processes under the influence of external vibrations. Wear, 1992. 154(1): p. 177-192.

4. K. Siegert and J. Ulmer, Superimposing ultrasonic waves on the dies in tube and wire drawing. Journal of Engineering Materials and Technology, 2001. 123(4): p. 517-523.

5. D. P. Hess and A. Soom, Normal Vibrations and Friction Under Harmonic Loads: Part I—Hertzian Contacts. Journal of tribology, 1991. 113(1): p. 80-86.

6. D. P. Hess, A. Soom, and C. H. Kim, Normal vibrations and friction at a Hertzian contact under random excitation: Theory and experiments. Journal of Sound and Vibration, 1992. 153(3): p. 491-508.

7. G. L. Chern and J. M. Liang, Study on boring and drilling with vibration cutting. International Journal of Machine Tools and Manufacture, 2007. 47(1): p. 133-140.


9. W. Littmann, H. Storck, and J. Wallaschek, Sliding friction in the presence of ultrasonic oscillations: superposition of longitudinal oscillations. Archive of Applied Mechanics, 2001. 71(8): p. 549-554.

10. V. C. Kumar and I. M. Hutchings, Reduction of the sliding friction of metals by the application of longitudinal or transverse ultrasonic vibration. Tribology International, 2004. 37(10): p. 833-840.

11. V. L. Popov, Contact Mechanics and Friction Physical Principles and Applications, ed. SpringerLink. 2010, Berlin, Heidelberg: Springer

12. P. Gutowski and M. Leuss, The effect of longitudinal tangential vibrations on friction and driving forces in sliding motion. Tribology International, 2012. 55: p. 108-118.


14. A. Helmstetter, D. Sornette, J. R. Grasso, J. V. Andersen, S. Gluzman, and V. Pisarenko, Slider block friction model for landslides: Application to Vaiont and La Clapière landslides. Journal of Geophysical Research: Solid Earth, 2004. 109(B2): p. 1974-1977.

15. J. H. Dieterich, Time-dependent friction and the mechanics of stick-slip. pure and applied geophysics, 1978. 116(4): p. 790-806.

16. J. H. Dieterich, Modeling of rock friction: 1. Experimental results and constitutive equations. Journal of Geophysical Research: Solid Earth, 1979. 84(B5): p. 2161-2168.

17. A. Ruina, Slip instability and state variable friction laws. Journal of Geophysical Research: Solid Earth, 1983. 88(B12): p. 10359-10370.

18. J. Rice, Constitutive relations for fault slip and earthquake instabilities. pure and applied geophysics, 1983. 121(3): p. 443-475.

19. D. A. Lockner and N. M. Beeler, Premonitory slip and tidal triggering of earthquakes. Journal of Geophysical Research: Solid Earth, 1999. 104(B9): p. 20133-20151.

20. N. M. Beeler and D. A. Lockner, Why earthquakes correlate weakly with the solid Earth tides: Effects of periodic stress on the rate and probability of earthquake occurrence. Journal of Geophysical Research: Solid Earth, 2003. 108(B8): p. 2391.


100

21. W. W. Tworzydlo and O. N. Hamzeh, On the importance of normal vibrations in modeling of stick slip in rock sliding. Journal of Geophysical Research: Solid Earth, 1997. 102(B7): p. 15091-15103.

22. C. Voisin, Dynamic triggering of earthquakes: The linear slip‐dependent friction case. Geophysical Research Letters, 2001. 28(17): p. 3357-3360.

23. C. Voisin, Dynamic triggering of earthquakes: The nonlinear slip‐dependent friction case. Journal of Geophysical Research: Solid Earth, 2002. 107(B12): p. ESE 10-1-ESE 10-11.

24. H. Perfettini, J. Schmittbuhl, J. R. Rice, and M. Cocco, Frictional response induced by time‐dependent fluctuations of the normal loading. Journal of Geophysical Research: Solid Earth, 2001. 106(B7): p. 13455-13472.


26. T. Jimma, Y. Kasuga, N. Iwaki, O. Miyazawa, E. Mori, K. Ito, and H. Hatano, An application of ultrasonic vibration to the deep drawing process. Journal of Materials Processing Tech., 1998. 80: p. 406-412.

27. J. C. Hung, Y. C. Tsai, and C. Hung, Frictional effect of ultrasonic-vibration on upsetting. Ultrasonics, 2007. 46(3): p. 277-284.

28. C. Nath and M. Rahman, Effect of machining parameters in ultrasonic vibration cutting. International Journal of Machine Tools and Manufacture, 2008. 48(9): p. 965-974.


30. A. V. Dyskin, On the role of stress fluctuations in brittle fracture. International Journal of Fracture, 1999. 100(1): p. 29-53.

31. A. V. Dyskin, Crack growth under spatially random stress fields., in Advances in Fracture research, Proc. 9th International Conference on Fracture. 1997. p. 2119-2126.

32. G. L. Chern and J.M. Liang, Study on boring and drilling with vibration cutting. International Journal of Machine Tools and Manufacture, 2007. 47(1): p. 133-140.

CHAPTER 5

Negative stiffness produced by particle rotations and its effect on frictional

sliding

Chapter 5. Negative stiffness produced by particle rotations and its effect on frictional sliding

102

5.1 Introduction

Cutter-rock interaction is characterized by the production of the rock fragments (gouge)

of different sizes and shapes (non-spherical) between the cutter and the rock. The purpose of

this chapter is to analyse the mechanism of cutting with the account of the influence of rock

fragments in front and under the cutter. The previous research indicated that the formation and

rotation of non-spherical particles (Figure 5.1 a) between the cutter and the rock can lead to the

appearance of negative stiffness effect [1], [2].

The effect of negative stiffness is provided by mechanisms, which are intrinsically

unstable. Assume that the length of the rod, l (Figure 5.1b) is equal to the diameter d of the

block or grain (Figure 5.1a). Then the inverted pendulum (Figure 5.1 b) can be used as a modes

of non-spherical grain systems. In other words, the inverted pendulum can be treated as a

mechanism/element with negative stiffness.

a) b)

Figure 5.1 Apparent negative stiffness produced by a) non-spherical grain (single

block in incipient stage of rotation); b) inverted pendulum

Thus, if pendulum is put in motion from the equilibrium position, the spring restoring

force (a result of two forces acting in different direction) brings the pendulum back toward its

initial state. At some stage, the system will not return to its original state and passes into an

unstable state (effect of negative stiffness occurs).


103

Negative stiffness, as a concept, represents the fact that the potential energy is negative

definite under loading and is manifested by a reversal the force-displacement relationship. The

effect of negative stiffness is provided by mechanisms, which are the systems that are

intrinsically unstable. The number of studies devoted to this topic has increased in recent years.

This interest is based on various practical applications of negative stiffness. Thus, it was

suggested that negative stiffness inclusions may effectively decrease the level of noise and

vibrations in structures [3], [4], [5].

Furthermore, structures with negative stiffness inclusions can be used in the following

applications: structural vibration control (by using magnetic negative stiffness dampers) [6];

seismic protection of structures [7], [8], [9]. Another example of practical application of such

structures is car industry (e.g. driver vibration isolation and car seats) [10], [11]. Some

manifestations of negative stiffness are observed in nature: hair-bundles in the human ear [12]

and some joins [13], [14]. Possible mechanisms of negative stiffness include post-buckling

deformation of tubes [15], columns (“S” shape configuration), shells and L-frames [16].

In nature negative stiffness characterises the post-peak softening of granular materials

[17], [18] and brittle materials (rock and concrete) [19], [20].

This chapter describes the inverted pendulum experiments (Section 5.3), the purpose of

which is to give a visual demonstration of the effect of the negative stiffness. We consider the

analytical model of loss of stability (Section 5.2) in the examples of an inverted pendulum and

compare it with the physical model results. The influence of movement and rolling of the

granular material (gouge) on friction forces is investigated in Section (5.4). Discussion and

conclusions are presented in Section (5.5).

5.2 Analytical modelling

In this section, we are going to consider the following two analytical models with

negative stiffness:

A model with negative stiffness spring;

A model of inverted pendulum.


104

A model with negative stiffness spring

A one-dimensional (1-D) chain of masses (particles) connected by normal elastic

springs is one of examples of system with negative stiffness. In general, the presence of the

negative stiffness inclusion in the system can lead to its instability. Chain of oscillators were

investigated in [21], [22], [23].

According to the Pasternak, Dyskin and Sevel [21] it was determined that a stable

system should meet the following conditions:

Constraints on the value of negative stiffness;

Limited number of negative stiffness spring inclusions (system can have no

more than one negative stiffness spring).

Pasternak, Dyskin and Sevel [21] investigated stability of the system by considering

discrete mass-springs system (Figure 5.2) with negative stiffness component and fixed chain

ends. Equations (5.1) – (5.9) are the short summary of the system stability investigation [21],

[22], [23].

Figure 5.2 – 1D mass-springs model (Pasternak et al., 2014 [21])

The equation of motion of mass-springs system can be derived from the Lagrange

equations

ii i

d L L fdt q q

, 1, ...i n (5.1)

where fi are the force, acting on masses m.

The Lagrangian is equal

2 12 2

1 1 10 1

1 ( ) ( )2 2

n ni

i i i i i ii i

muL T П k u u k u u

(5.2)

From equation (5.2) we can define a characteristic equation for the system


105

1 1 1 1( )i i i i i i i i imu k k u ku k u f (5.3)

Initial conditions of the system

0 00 00 0n n

u uu u

(5.4)

Positive definiteness of the potential energy П is the criterion of stability.

12 2

1 1 11

1 ( ) ( ) 02

n

i i i i i ii

П k u u k u u

(5.5)

Positive definiteness of equation (5.5) is equivalent to the positive definiteness of the

stiffness matrix An (equation 5.6) and occurs when all main diagonal minors of the matrix An

are positive.

1 2 2

2 2 3 3

1 1

1 1

i i i i

n n n

k k kk k k k

Ak k k k

k k k

(5.6)

1,1 1,2 1,

2,1 2,2 2,

,1 ,2 ,

det 0

i

ii

i i i i

a a aa a a

a a a

(5.7)

Substituting (5.6) into (5.7) leads to the general and sufficient condition of stability

1 1

0nn

ji j

j i

k

(5.8)

These rules are cumulative and sensitive to renumbering. The second condition of the

system stability is the minimum value of negative stiffness spring inclusion (only one stiffness

ki is allowed to be negative). For a single mass the rule is 1 2 0k k , thus only one stiffness


106

can be negative. Let’s consider a 1-D chain with n springs and suppose that there is only one

negative stiffness.

From (5.8) we have

1 2 1

2

... nn

n

k k kk

(5.9)

Therefore, if 1 2 1... 0 0n nk k k k , then 1 2 1 1( ... ) 0 0n n nk k k k k .

A model of inverted pendulum

The inverted pendulum is conventionally considered in the course of theoretical

mechanics (for example Panovko, [24]). A schematic representation of inverted pendulum is

shown in Figure 5.3.

Figure 5.3 – Schematic representation of inverted pendulum test rig

(Panovko, 1991 [24])

where m is the mass of the pendulum (including the mass of the rod), l is the height of the

spring axis location, k is the total stiffness coefficient of both springs.

The differential equation of the rotational motion of inverted pendulum (small angle

inclination case 010 ) reads:

0I b C (5.10)

where I is the moment of inertia of the moving mass about point O, φ is the angular

displacement, b is the damping coefficient, C is restoring force.


107

Two simplifications have been adopted in the analytical modelling. First, we neglect

the small damping force caused by air resistance and friction at the pivot and consider an

undamped system. Second, we consider the small oscillations of the pendulum. The

dependence of the restoring forces on generalized coordinates is nonlinear. However, in case

of small angles of deflection ( 010 ) the linear dependence is possible. These assumptions

lead to the following governing equation of motion

2( ) 0tot totI kl M (5.11)

2 2 2 2 21 1 1( ) ( )3 4 12 2tot p p p

lI m l m R r ml m l (5.12)

2 2p

tot p p

l lM M M m g mg (5.13)

where Itot is the total rotational inertia, Mtot is the total torque of pendulum, mp is the mass of

the rod of the pendulum, g is the gravity force, m is the mass of the particle (in our case is nut),

lp is the length of the rod, l is the height of the nut, R is the outer radius of nut, r is the inner

radius of nut.

5.3 Physical modelling

The apparatus used throughout experimentation was the inverted pendulum, designed

and modified at the University of Western Australia. The inverted pendulum was designed to

demonstrate the effect of negative stiffness. A photograph of the inverted pendulum is shown

in Figure 5.4. The rig consists from base (1), two frames (2), vertical rod (3), supported by two

springs of stiffness k0 (4), attached at a proper distance to the frames. The vertical rod fixed on

base fixture which has a protractor (5) for measuring the angles of inclination.


108

Figure 5.4 – Inverted pendulum test rig

There are two supports springs: the one end fixed to the vertical rod and the other to the

frame.

The rig was designed that the distance between frames and vertical rod, the height of

spring connection readily adjusted. Also, there is a possibility to vary the mass of inverted

pendulum by adding the nuts (6) which are fixed on the vertical rod. Thus, the natural frequency

of the system could be changed.

Examining the Equation 5.11, the restoring force (Equation.5.14) can be varied by

adjusting the system parameters. In particular, by increasing the mass of the system, the eigen

frequency can be drastically reduced and even equal to zero in case of instability of the system.

Stability of the system depends on the sign of restoring force C (if C<0 the system is

unstable, C>0 the system is stable). The sign of the coefficient C depends on the parameters of

the system.

The restoring force is given by

2 22( )tot

totMC kl M l kl

(5.14)

where k is the total stiffness coefficient of both springs, l is the height of the spring axis location,

Mtot is the total torque of pendulum.

(1)

(2)

(5)

(3)

(4)

(6)


109

The following equation 5.15 characterises the negative stiffness and system instability.

2totMk

l (5.15)

Spring stiffness determination

Spring stiffness is a critical parameter that controls the pendulum behaviour.

Determination of the spring stiffness starts from the experimental evaluations. Spring stiffness

was determined by conducting the simple test which obeys Hook’s law. The known force was

applied to the spring and the distance the spring stretches from its original length was measured.

The spring stiffness k0 was found to be 55.28 N/m (Figure 5.5). To verify this value,

the spring stiffness was calculated by using the standard formula based on spring geometry and

material properties.

4

0 38GdkD n

(5.16)

where d is wire diameter, G is the shear modulus of spring material (stainless steel), D is mean

spring diameter, n is number of active coils. The obtained result k0=55.38 N/m confirmed the

experimental stiffness.

Figure 5.5 – Spring stiffness determination


110

The results of applied tests (applied force vs displacement) are shown in markers. The

line F(u)=55.28u shows the linear regression, where R2=0.997 is a coefficient of determination.

Stability of the inverted pendulum

Here we investigate the stability of the inverted pendulum. Experiments were

conducted to determine the eigen frequency of inverted pendulum by increasing the total mass

of the system. In order to determine the experimental eigen frequency of the system we apply

initial displacement ( 010 ) to the inverted pendulum. Each test has been repeated three

times. We calculated the eigen frequency by using experimental data (Equation 5.17) and

compared it with the eigen frequency obtained by theoretical analysis (Equation 5.18).

exp 2Nt

(5.17)

where N is the number of vibrations for the time interval t.

2tot

partot tot

kl MCI I

(5.18)

Figure 5.6 - The eigen frequency dependence from the mass of the system

The experimental results show that variation of the parameter (in our case - increasing

the total mass) result in decreasing the eigen frequency of the system. At the point when the

eigen frequency is equal to zero, the system loses its stability and therefore demonstrates the

effect of negative stiffness. Thus, the analytical negative stiffness model has been validated by

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.54

6

8

10

12

14

16

18

20

m, [kg]

, [

1/s]

data1exp1

exp2

exp3


111

the inverted pendulum physical experiments. Now, the negative stiffness model can be used

for friction.

5.4. Friction force fluctuation associated with rotation of particles

Using the above experimentally verified model we can analyse the mechanical effect

of a set of rotating non-spherical grains/debris sitting between sliding surfaces and their

influence on friction.

Intermittent change in the force of friction between two contacting surfaces produces a

regular stick-slip motion (oscillations). However, other mechanism of irregular friction

oscillations is based on the effect of gouge rotation between the sliding surfaces.

“Fault gouge” is a layer of fragmented pieces of rock which is typical for the faults with

significant displacement. Gouge is formed by tectonic forces in a fault zones or generated

during faults sliding (due to wear of rock) [25], [26]. Sliding of gouge-filled faults involves

frictional sliding, rotation, rearrangement and even breaking of the gouge grains [27]. Effect of

grains rotation is usually investigated by modelling gouge particles as spheres (or discs in 2-

D) [28], [29]. However, the real grains have non-spherical shape. Rotation of such grains can

produce the effect of negative stiffness [1], [2], [30]. Similar effect should be associated with

rotation of rock fragments, produced by rock cutting. We will treat these fragments and the

gouge in a unified manner as a set of rotating non-spherical particles.

In this section, we compare the theoretical model results of rolling of non-spherical

gouge particles (Figure 5.1a) considered by Dyskin and Pasternak [31] with the experimental

data results. The experimental data was obtained from the rock scratching tests (Figure 5.7)

conducted in CSIRO Kensington drilling mechanics laboratory. A detailed description of the

scratching tests is provided in Chapter 2, Section 2.3.


112

Figure 5.7 – Rock scratching test (T. Richard, 2008 [32])

According to Dyskin and Pasternak [31] the shear force needed to start the rolling is

** * * *

*2 * 2

1 , , , (1 )

frfr fr

Tu u dT T u dP a ad u

(5.19)

where Tfr is the shear force, u is the displacement of upper block against the lower one, a is the

width of the particle, d is its diagonal, P is the compressive force.

However, gouge involves multiple particles. For this reason, Dyskin and Pasternak [31]

modelled a gouge consisting of multiple particles rolling independently (Figure 5.8). For

modelling the chain of rolling particles next assumptions have been made:

particles have different sizes which are randomly distributed;

the blocks are soft to keep contact with all particles;

normal force, applied to each particle is the same;

Figure 5.8 - Interaction between n identical particles

(Dyskin and Pasternak, 2014 [31])

The total friction force is the sum of friction forces associated with each particle in

dimensionless form is equal


113

* ** * * * *

*2 * * 21

, , , d , a( )

Nfri i

fr fr ii i

Ta u au dT T uP a a ad a u

(5.20)

where a is the average particle size, N is the total of rolling particles, ai is the (random) size

of each particle.

Figure 5.9 present the results of rock scratching tests. Figure 5.10 present the results

of numerical simulations for 100 particles with different shapes (d/<a> ratio) and particle size

distributions (uniform and normal distributions with the same standard deviations). Where 1/20.2*3 a is the “narrow distribution” and 1/20.495*3 a is a wide distribution.

a) b)

Figure 5.9 – Fluctuations of the friction coefficient with displacement (experimental results)


0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.080

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

L

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.080

0.2

0.4

0.6

0.8

1

1.2

1.4

L


114

The results displayed on Figures 5.9-5.10 show the fluctuation in friction forces.

Observed friction forces fluctuations (Figure 5.9) may be caused by grains rotation in front of

the cutter.

a)

b)


115

c)

Figure 5.10 - The effect of the width of the distribution and the particle shape

square particles, d/<a>=21/2, wide distribution; (b) nearly round particles, d/<a>=1.01, narrow

distribution; (c) nearly round particles, d/<a>=1.01, wide distribution.

5.5 Conclusions

Negative stiffness is a concept that provides simple expression for instabilities of

material elements such as rotating non-spherical particles. A simple model of the mechanical

effect of rotating non-spherical particle is the inverted pendulum where the role of compression

is played by gravity. Theoretical and physical models of inverted pendulum were studied to

investigate the effect of negative stiffness. The negative stiffness effect can be achieved by

geometric arrangement of mechanical components or adjusting the proper parameters (in our

case – by increasing the total mass) of a vibrating system. The obtained experimental data

supports the theoretical prediction of the negative stiffness effect. In particular it was found

that the increase of the mass reduces frequency far beyond the conventional reduction when

the frequency is proportional to the inverse square root of mass. Furthermore, there is a critical

mass at which the frequency becomes zero and the pendulum loses stability. This effect is well

described by the concept of negative stiffness: the inverted pendulum is equivalent to two

springs connected in series one of which has negative stiffness; when the value of negative

stiffness reaches the value of the stiffness of the conventional spring the frequency becomes

zero and the chain loses stability. The negative stiffness is proportional to the compressive


116

force, which in its own term is proportional to the mass. Thus, increasing the mass one can

make the value of negative stiffness equal to the critical one leading to the system instability.

The gouge material between the sliding blocks usually consists of many particles that

are non-spherical. When they rotate under block sliding they cause local effect of negative

stiffness. The combined effect of many particles exhibits itself as strong fluctuation of the

friction force, which was observed in the cutting experiments. Thus, the presence of rotating

non-spherical particles can explain the friction force variations observed in both cutting and

the models of fault sliding.

References

1. A. V. Dyskin and E. Pasternak Rock mass instability caused by incipient block rotation, in Harmonising Rock Engineering and the Environment 2012 p. 171-172.

2. A. V. Dyskin and E. Pasternak, Mechanical effect of rotating non-spherical particles on failure in compression. Philosophical Magazine, 2012. 92(28-30): p. 3451-3473.

3. Y. C. Wang and R. S. Lakes, Stable extremely-high-damping discrete viscoelastic systems due to negative stiffness elements. Applied Physics Letters, 2004. 84(22): p. 4451-4453.

4. Y. C. Wang, J. G. Swadener, and R. S. Lakes, Anomalies in stiffness and damping of a 2D discrete viscoelastic system due to negative stiffness components. Thin Solid Films, 2007. 515(6): p. 3171-3178.

5. R. S. Lakes, T. Lee, A. Bersie, and Y. C. Wang, Extreme damping in composite materials with negative-stiffness inclusions. Nature, 2001. 410(6828): p. 565.

6. S. Xiang and Z. Songye, Magnetic negative stiffness dampers. Smart Materials and Structures, 2015. 24(7): p. 072002.

7. N. Attary, M. Symans, and S. Nagarajaiah, Development of a rotation-based negative stiffness device for seismic protection of structures. Journal of Vibration and Control, 2017. 23(5): p. 853-867.

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CHAPTER 6

General conclusions and discussion

Chapter 6. General conclusions and discussion

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The thesis was concerned with the investigation of friction phenomena arising at the

contact surface during sliding and cutting.

The main aims were: a) analysis of the data of laboratory experiments conducted in

order to increase performance output of a drilling rig; b) mathematical modelling of the

oscillator with constant and nonlinear friction law; c) analysis of the influence of the harmonic

load on friction reduction; d) investigation of the effect of negative stiffness using the model

of inverted pendulum.

The study included mathematical modelling based on the theory of vibrational motion,

analysis of friction reduction effect and the theory of negative stiffness.

In this section, we summarise the outcomes of the thesis according to the main aims

and formulate the recommendations for further research.

6.1 Conclusions

Stick-slip is a phenomenon caused by continuous changing between static and rate-

dependent friction. In the domain of engineering this phenomenon is highly undesirable as it

can result in inaccurate machining operations, wear and damage of constructions, and lost

production. Oscillations of various frequencies and amplitudes are widely used in engineering

in order to influence and reduce friction. This research has been devoted to the analysis of the

influence of the external harmonic loads on friction sliding.

For this purpose, we first considered a single sliding block model with different friction

laws (constant and velocity-dependent) in the absence of vibration. The developed model

demonstrates that stick-slip behaviour can be caused by self-excited oscillations and does not

depend on whether the friction coefficient is constant or rate-dependent.

Next, we generalised this finding to the case of sliding propagating over a fault where

a stick-slip phenomenon is traditionally associated with the earthquakes. For this purpose, we

considered a simplest possible model an infinite elastic beam driven by elastic shear spring.

This model predicts that any initial sliding (impulse) moves with a p-wave velocity. This is

consistent with observed supersonic rupture propagation over faults.


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Friction is in many cases undesirable and it is important to attempt to at least reduce it.

We investigated the method of friction reduction using additional vibrations applied in normal

or tangential directions with regard to the direction of frictional sliding. To this end we

developed a block model with imposed oscillations (normal and longitudinal) which were

presented either by harmonic load or random fluctuations. The effect of applied vibrations on

frictional sliding was investigated and analysis of dynamics of system with different friction

modelling (including constant and velocity-dependent friction) was provided.

A comparative analysis of vertical and horizontal vibrations was conducted to

determine the direction in which the effect is the most pronounced. This analysis indicates that

the horizontal force oscillations imposed on the system have an effect similar to the vertical

oscillations. The computer simulations showed that the effect of the imposed excitations

depends on the type of frictional sliding (stick-slip/steady sliding), regime (resonance/non-

resonance), friction law (constant/velocity-dependent), and value of vibrational parameters

(amplitude and frequency).

It was found that the steady sliding regime is unstable under the influence of imposed

vertical/horizontal vibrations for both types of friction laws. This leads to the undesirable effect

in drilling (stick-slip motion). It was also shown that the stick-slip sliding is stable with the

respect to vertical and horizontal excitations. An important result was obtained in the case of

constant friction law at the resonance regime. It was observed that any perturbation leads to

formation of a zone of reduced stick-slip velocity and hence reduced energy loss on friction. A

special case is observed at the steady sliding regime. Then, no matter how small the amplitude

of oscillations is, the system reaches the same final stick-slip regime. The time required to

reach this limiting regime is inversely proportional to the amplitude of oscillations of the

applied harmonic force.

One important aspect of this research was the analysis of laboratory experiments

conducted at the CSIRO to investigate the frictional sliding effects. Results of rock scratching

tests with Polycrystalline Diamond Compact (PDC) single cutter indicated the presence of

considerable force fluctuations. We hypothesise that the fluctuations are caused by the rotation

of rock fragments developed in front of and under the cutter as a result of the cutting process.

Indeed, it is routinely observed in the rock cutting experiments that the rock fragments are

created and removed by the cutter. It is obvious that these grains could have different size and

shape and can have rotational degrees of freedom. The subsequent numerical simulations


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showed that the effect of rotating non-spherical particles produces shear force fluctuations

similar to those observed in the rock cutting tests.

Previous research showed that the rotation of non-spherical particles under compression

might lead to the effect of apparent negative stiffness. In order to investigate the apparent

negative stiffness further, we modelled the effect of rotating non-spherical particle and showed

that its dynamics is equivalent to the dynamics of inverted pendulum. We conducted inverted

pendulum experiments and investigated the stability of the inverted pendulum under changing

the eigen frequency. To this end we constructed a special experimental apparatus. The

experimental results showed that variation of total mass leads to the decrease of eigen

frequency of the system beyond that of an ordinary pendulum. Furthermore, there is a value

of the mass when the eigen frequency becomes equal to zero. At this point the system loses its

stability. Physical and analytical modelling of the inverted pendulum gave qualitative similar

negative stiffness results. We showed that both, the reduction in eigen frequency and the loss

of stability can be accurately modelled by introducing a negative stiffness in the equation of

coupled oscillator. This gave a convenient model to analyse the effect of rotation of non-

spherical particles.

The main conclusions of this research are: (1) stick-slip sliding can occur in the absence

of rate-dependency of the friction; (2) the propagation of sliding zone over long fault in elastic

rock occurs with the velocity of p-wave, that is supersonic; (3) dynamic behaviour of the system

depends on such parameters as type of frictional sliding (stick-slip/steady sliding), regime

(resonance/non-resonance), friction law (constant/velocity-dependent), and value of

vibrational parameters (amplitude and frequency). Moreover, the effect of imposed excitations

is more pronounced for the system with constant friction than for the system with rate-

dependent friction under the same set of model parameters. The simplified rate-dependent

friction law leads to the removal of effects of the system which obeys the constant friction law;

(4) the laboratory tests on the inverted pendulum confirm that the effect of rotating non-

spherical particles creates apparent negative stiffness. Further numerical analysis showed that

the rotation of non-spherical particles produces friction force fluctuations similar to the ones

observed in rock cutting tests.

To sum up, this work highlighted the factors that are often overlooked in studying

frictional sliding are the elasticity of the material of the sliding bodies and the non-sphericity

of the grains/cuttings can have profound effect on the sliding creating the apparent stick-slip


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behaviour and friction force fluctuations. We also clarified the effect of applied external

vibrations and identified the cases when it can affect frictional sliding. These results can

potentially be used in engineering, especially in drilling industry to control stick-slip vibrations.

6.2 Recommendation for the future work

Although a number of results have been obtained and discussed in this thesis, many

issues remain open and need to be done in future work.

Analysis of field and laboratory data

Analysis of “Thor” laboratory experiments indicates that the data is not sufficient for

the system parameters analysis. For this purpose, new experiments and data are needed for the

further research. During the “Wombat” laboratory experiments the optimal region (-5 to 5-

degree range) of the approach angle of cutter has been determined by using the discrete

simulation of a vibrational input of drill bit. However, the cutter mount is limited by 5-degree

increment. For further investigation of the effect of the back-rake angle on rock fragmentation,

a linear actuator is needed. This can allow reducing the cutter increment which is essential for

the data accuracy.

Mathematical modelling of the oscillator formed by the cutter and the rock

The complexity of the cutter-rock interaction is determined by the nonlinear nature of

rock resistance to the cutter movement towards and from the rock. The present mathematical

model of the oscillator is simplistic. The next level of complexity should include a bilinear

oscillator whereby the stiffness in compression is considerably greater than that in tension.

Moreover, the mass-spring system should include the dampers for the further examination.

Analysis of influence of harmonic load on reduction of friction force

The behaviour of the numerical model is controlled by the friction law chosen. The

general simplified nonlinear friction law was formulated for the analysis of the influence of

harmonic load on reduction of friction force. However, for further investigation of the dynamic

behaviour of the system the full rate- and state-dependent friction law with the real data

parameters is needed.


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Modelling the oscillators in the structures with negative stiffness elements

Modelling of non-spherical grains showed that the rotation of non-spherical particles

under compression might lead to the effect of the apparent negative stiffness. The effect of

negative stiffness is demonstrated in the example of an inverted pendulum experiments.

It is important to develop a combined model and analyse the system with negative

stiffness under the influence of harmonic load. This is essential for the determination of the

parameters controlling the energy distribution between cutting and friction, cutting efficiency

and tool wear.

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APPENDIX

Appendix

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Cutter 4B

Appendix

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Appendix

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Appendix

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Appendix

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Appendix

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Cutter 5B

Appendix

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Appendix

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Appendix

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Appendix

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Appendix

135

Sharp Cutter

Appendix

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Appendix

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Appendix

138

Appendix

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the UWA Profiles and Research Repository...In the middle of the nineteenth century scientists started to add the lubrication to the friction contact. The first study of fluid lubrication