Research ArticleOptimal Design Parameters of a Percussive Drilling System forEfficiency Improvement

Changheon Song12 Jintai Chung2 Jae-Sang Cho3 and Yun-Joo Nam 1

1Korea Institute of Industrial Technology Gyeongsan-si Republic of Korea2Department of Mechanical Engineering Hanyang University Ansan-si Republic of Korea3RampD Center Soosan Heavy Industries Co Ltd Hwaseong-si Republic of Korea

Correspondence should be addressed to Yun-Joo Nam yjnamkitechrekr

Received 16 June 2017 Revised 9 August 2017 Accepted 10 October 2017 Published 5 February 2018

Academic Editor Fernando Lusquintildeos

Copyright copy 2018 Changheon Song et al 0is is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

0is paper aims to determine the optimal design parameters for percussive drilling systems considering the bit-rock interactionFirst the motion dynamics of a bit impacted by a dropped piston are modeled by impact stress propagation and a rock-breakingmechanism representing the penetration resistance coefficient and unloading constant Next the damping vibration behavior ofthe bit is investigated considering the impact duration and the rock loadingunloading condition In addition the proposeddynamics are simplified by adopting two dimensionless parameters representing the bit-piston mass ratio and the rock-pistonstiffness ratio Finally the drilling efficiency defined by the energy transmitted from the piston to the rock is presented in terms ofthe proposed parameters 0e use of optimal design parameters for percussive drilling systems improved the drilling efficiency0ese results are applicable to the design and performance estimation of down-the-hole and top-hammer systems

1 Introduction

Drilling equipment is the collective term used for machinesthat apply impact and rotation forces to drill (for the mostpart) surfaces and blastholes and it is classified as top-hammer drilling (THD) down-the-hole (DTH) drillingand rotary drilling (RD) rigs depending on the operatingmethod (Figure 1) In general THD is used mostly formining and civil blasting works where the rig drills into theearth usually to a depth of 1ndash20m and at most 40m DTH isused mainly for groundwater development and can createholes to a maximum depth of 4000m and RD bores thedeepest holes most commonly for petroleum gas extractionand geothermal development being propelled by its ownweight to reach depths of up to 10000m [1]

Simply put a percussive drilling mechanism utilizes thepercussive energy resulting from the repeated impact of thedrifter (THD rigs) or the DTH hammer (DTH rigs) and thefeed force and rotation force that are transmitted to the drillbit through the drill rod 0e energy generated from the

repeated impacts is then converted into wave energy whichis transmitted to the rock via the drill bit Finally the drill bitnow with enough impact energy for drilling cuts into andcrushes the rock 0e rate at which the impact-generatedenergy in a percussive drilling system is transmitted (ie thedrilling efficiency) is determined by complex effects such asdrilling rod coupling sleeve the compressive strength of therock and interactions between the drill bit and the rock0erefore certain drilling methods are highly efficient withhigh rates of penetration when drilling soft rock (uniaxialcompressive strength UCSlt 20MPa) or medium hard rock(UCS 50ndash120MPa) but the efficiency decreases whendrilling very hard rock (UCSgt 200MPa) [2]

0ere are numerous previous studies regarding drill bitrock drilling the transmission of impact energy and drillingefficiency Hustrulid and Fairhurst [3ndash6] investigated energytransmission between the drill steel and rock and measuredthe specific energy resulting from the impact force

Chiang and Elias [7] used the finite element method(FEM) to study factors such as energy transmission

HindawiAdvances in Materials Science and EngineeringVolume 2018 Article ID 2346598 13 pageshttpsdoiorg10115520182346598

interactions between the drill bit and rock and the rockpenetration process which are important in the research andevaluation of drill equipment Song et al [1] proposedoptimal design factors and range for top-hammer drill bitbased on multiphase ow simulation and response surfacemethodology (RSM) Also Song et al [8] designed a lab-scale testing system to test drilling performance of a drill bitand investigated the e ect of bit design factors on theperformance of drilling Kwon et al [9] validated the e ect ofdrill bit button arrangement on drilling eciency througha piston percussion test using piston drop testing systemLundberg and Okrouhlik [10] conducted a study of the 3De ects on the eciency of three representative methods forpercussive rock drilling (hammer drilling DTH drilling andchurn drilling) in which the eciency of various drillingmethods was compared via 1D analysis and 3D FEMLundberg and Collet [11] investigated the optimal incidentwave that maximizes the conversion eciency of the waveenergy in the THDmethod All these prior studies essentiallyconsidered the transmission of impact energy from thepiston and drill bit However they failed to examine thee ect of the dynamic properties of the drill bit resulting fromthe piston impact Li et al [2] analyzed the piston reactionforce characteristics of percussive hammers using the stresswave propagation theory and the law of conservation ofenergy and also studied the energy transmission eciency ofa DTH hammer tool [12] e studies by Li et al [2 12]considered the dynamics of the drill bit resulting from thepiston impact However this consideration only addressedthe underdamped modes of the system response charac-teristics that result from the rotational movement of the drillbit and the e ect of the rock e work therefore neglectedthe dynamics of the drill bit generated under various otherdamped modes e study was further limited in that it

employed an excessively large range of rock-to-pistonsti ness ratios (β) in the transmission eciency of impactenergy Overall while many studies have investigated rockdrilling and the transmission of impact energy few haveaddressed the damping characteristics generated from thepenetration process the rock e ect and the movement ofthe drill parts or investigated the related drilling eciency

e analysis of behavior of drill bit should be performed forthe design and manufacturing of a percussive drilling systemwith high drilling eciency because the drilling eciency andthe range of rock fracturing depend on movement charac-teristics of drill bit during percussion and because drill bit alsopercusses the rock directly meaning the close relationship withthe delivery of percussion energy and drilling eciency

is study aimed to identify design parameters that couldoptimize the drilling eciency in a percussive drilling systemwhile taking dynamic drill bit-rock interactions resultingfrom the piston impact into consideration Drilling dynamicmodels (ie dynamic models of the drill bit) were establishedby introducing drill bit-rock interactions to determine therock-penetrating properties of the bit is was done usingimpact stress wave propagation theory the penetration re-sistance coecient and the unloading constant in relation tothe impacting piston and drill bit e drilling dynamics wasestablished considering the movement of the drill bit inducedby piston impacts and the loading and unloading conditionsresulting from the e ects of the rock

e drilling dynamics proposed here employ two di-mensionless parameters the rst (α) is related to the piston-to-drill bit mass ratio and β (dened above) and the second isβ itself is introduction is followed by a discussion of therate at which the impact energy is transmitted from thepiston to the rock during percussion (ie the drilling e-ciency) e drilling eciency is the work completed by thedrill bit relative to the incident-stress wave that causes it tocrush the rockWe also analyzed the dynamics of the drill bitwith regard to dimensionless parameters and damping ratiosand considered impact-energy transmission rates given mr(the piston-to-drill bit mass ratio) and β Combinations ofparameters tomaximize the eciency of a percussive drillingsystem are proposed for various rock strengths

To that end Section 2 of this paper proposes a drillingdynamic model for a percussive drilling system that takes intoaccount the dynamic properties of the drill bit and withconsideration to the theory of stress-wave propagation viapiston impact Drill bit behaviors and response characteristicsobserved during the percussion process (ie over critical andunderdamped modes) involving the piston the bit and therock via the parameters α and β are discussed e physicalmeaning was analyzed by vertical development of di-mensionless parameters (ie α and β) e system responsesof drill bit to the piston percussion were extended to sixconditions depending on the presence of incident stress waveand loading or unloading condition e typical motion ofdrill bits involves four conditions However this study ex-panded them into six conditions e response characteristicsresulting from the rock and themovement of the drill bit werealso analyzed to investigate the eciency or the transfer rateof the percussive energy from the piston to rock

Rock

Air flow

Rock particle + air flowDrill bit

Drifter

PercussionPiston

Rotation

Percussion

Piston

DTH hammer

Air flow

Drilling rod

Feed

Couplingsleeve

(a) (b)

Figure 1 Drilling mechanisms of two types of percussive drillingsystem (a) top-hammer drilling and (b) down-the-hole drilling(modied from Song et al [1])

2 Advances in Materials Science and Engineering

e energy transmission eciency via the impact of thepercussive drilling system is examined in Section 3 Section 4discusses the combinations of parameters that might opti-mize the drilling eciency given the dynamic drill bitcharacteristics and rock sti ness (ie β) Conclusions arepresented in Section 5

2 Dynamic Model of a Percussive DrillingSystem

21 Mechanism of Percussive Drilling Figures 1(a) and 1(b)illustrate the working principles or drilling mechanisms ofpercussive drilling systems In each case the drill bit ispropelled into the rock by percussion feed and rotationforces that are transmitted via the drifter or hammer Afterreceiving impact energy from the drill bit the rock developscracks that form a network and eventually break it ecrushed rock then undergoes further splitting via the ro-tation force and secondary impacts until it becomes powdere rock powder is discharged to the outside by compressedair supplied through a ushing duct located inside the drillbit thus enabling the drill to proceed further [1]

22 Bit Motion and Stress Wave Propagation Figure 2 il-lustrates the principles of impact and rock penetration bya percussive drilling system

e piston mass length cross-sectional area descendingspeed density and longitudinal wave velocity are designated asmp LpAp v ρ and c respectivelye underlying assumptionswere that the drill bit remains immobile on the surface and thatthe diameters of the drill bit and the piston are equal Also thee ects of buttons embedded in the bit were ignored

e following equation expresses themotion of the drill bitresulting from the force applied to it by the piston [12]

mbd2u

dt2 A σi + σr( )minusF (1)

where u is the penetration depth (displacement) of the drillbit the second term is the force applied to the drill bit whenthe impact is exerted upon it F is the force generated at therock-bit interface (ie the penetration force) and A is thecross-sectional area of the drill bit and piston

According to the relation between the dynamic stress andvelocity of a mass in an elastic body system the velocity of thecontacting end of the bit can be expressed as follows [2]

du

dtσiρcminusσrρc (2)

here σi and σr are the incident and reected stress wavesrespectively e e ect of the secondary incident stress wavewas not considered For analytical simplication the piston andthe drill bit were assumed to be of the same diameter andmaterial both were assumed to have a rectangular pulse witha duration of incident stress wave τ and an amplitude of 05 ρcv

Typically nearly all the percussion impact energy is con-verted intowave energyepiston hitting the drill bit generatesan incident stress wave (σi) at which point a portion of the

impact energy is transmitted to the rock via the drill bitcrushing the rock [11] e remaining energy is converted intoa reected stress wave (σr) and is used to increase the speed ofthe piston ascending from the drill bit top to speed ve (Figure 2)[2] e reected stress wave can in turn be transmitted to thepiston and drill bit in the form of a secondary incident stresswave at the piston-bit interface and at the drill bit-rock interface

e secondary incident stress wave in a percussivedrilling system has been reported not to contribute to thecrushing of rocks [11] therefore it would not signicantlya ect the energy transmission eciency Furthermorea study of drilling speed found the incident stress wave tohave an insignicant e ect on the impact force and pen-etration characteristics [14]

e interaction between the drill bit and rock (theforce-penetration relationship Figure 3) is expressed in (3)[2] which separately considers the loading and unloadingconditions k is the rock penetration coecient index c isthe unloading constant and τ is the duration of the in-cident stress wave

F ku(loading)ckuminus ckuf(unloading)

(3)

e duration of the incident stress wave τ generated bythe impact of the piston can be expressed with respect totime t in two scenarios as dened in (4) [2] e rstscenario involves transmission of an incident stress wavewhereas the second does not

σi(t)

12ρcv for (0lt tle τ)

0 for (tgt τ)

(4)

Bit

Piston

Rock

mb

h0

h

Crushed zone

Crack

Button

mp Lp Ap c

i

r

e

Figure 2 Schematic rock fracture mechanism by percussivedrilling (modied from Cho et al [13])

Advances in Materials Science and Engineering 3

For the loading condition we obtain the followingequations [2]

A σi + σr( ) F

du

dtσiρcminusσrρc

ku F

(5)

where the initial condition is t 0 andF 0For the unloading condition the equations can be expressed

as follows [2]

A σi + σr( ) F

du

dtσiρcminusσrρc

ckuminus cku F

(6)

where the initial condition is t τ and F(t) limtrarr τ

F(t)e response characteristics of a drill bit (ie the damped

characteristics) can be under critical or overdamped given itsmotion equation that is (1) e movement of the drill bitunder loading and unloading conditions was examined edamping characteristics of a percussive drilling system arecategorized depending on the damping ratio (ζ) and thedimensionless parameter α

When loading the equation of motion for the drill bit isdened as follows

eurou +ρcAmb

( ) _u +k

mb( )u

2Amb( )σi (7)

where the response-characteristics determinant ζ1 is

ζ1 4α(βτ)

24α(βτ)2radic

1α

radic (8)

When unloading the drill-bit equation of motion isdened by (9) e ζ2 term is given in (10)

eurou +ρcAmb

( ) _u + ck

mb( )u

2Amb( )σi + c

kufmb

(9)

ζ2 4α(βτ)

2c(4α)(βτ)2radic

1cαradic (10)

e dimensionless parameters α and β dened in (8) and(10) can be expressed as follows (extended from (2) (12))

α 4mb

ρcAmiddotk

ρcA

4mb

ρcAτmiddot β

4mb

2mpLPcLPAβ 2

mb

mp( )β

β kτρcA

k2LpcρcA

2kLhρc2A

2kLp

ρA Epρ( ) 2k

LpEpA( ) 2

k

kp( )

(11)

As explained earlier the typical motion of drill bits in-volves four conditions However this study considers sixconditions as listed in Table 1 (Appendix A) and includes theloadingunloading condition and the presence or absence ofan incident stress wave To investigate the rate at which theimpact energy is transmitted from the piston to the rockduring the percussion process (ie the drilling eciency) thisstudy rst considered the response characteristics resultingfrom the motion of the drill bit during the percussion processand the e ect of the rock e aim was to nd the responsecharacteristics that lead to the rockrsquos maximum crushingdisplacement um e study also dened the motion ofpercussive drill bits resulting from the impact of the pistonand analyzed their behavior given the system responses edrill bit-rock interaction was considered here including theforcendashpenetration relationship for the rock

Table 1 presents the relationships between loading andunloading and the duration of stress wave τ in percussivedrilling systems In Table 1 τ is the duration of the incidentstress wave being transmitted to the drill bit via the pistonduring loading it is referenced relative to the time t eloading condition can be dened in two ways 0lt tle τ orτ lt tle tm where the latter expression means that τ arrivedsooner than t that is the transmission of τ was complete butthe loading continued owing to the inertial e ect of the drillbit If τ gt tm the drill bit arrived at um before the transmissionof τ to the bit had completed and indicates the time when thesystem switches to the unloading condition

In summary in Cases 1 to 5 in Table 1 the motion of thedrill bit following the piston impact continues until theloading condition applying external force to the rock reachesτ lt tle tm After that time the stress wave arriving at the drillbit is converted into unloading at τ gt tm provided that therock displacement occurs sooner than tm that is themaximum time of arrival However the τ in Case 6 rep-resents a stress wave that is still progressing while the rockdisplacement owing to the drill bit has already reached itsmaximum that is unloading occurs during τ

F

0 u

Fm

um

k

uf

k

Figure 3 Force versus penetration relationship representing thebitndashrock interaction curve

4 Advances in Materials Science and Engineering

23 Results of BitMotionAnalysis Numerical simulations ona dynamic model of drill bit properties were run for the sixcases and used the dynamic drill bit properties during inducedpercussion (Section 2) the drill bit-rock interaction modeland the dimensionless parameters α and β

Figure 4 shows the simulation results (ζ) for differentconditions (α Table 1) and the effect of loading andunloading 0e mass ratio mr was 1 and the same con-ditions were applied to the piston mass and drill bit massFigure 4(a) shows an overview of the response characteristicsfor all six cases 0e initial black part of each curve refers tothe sustained effect of τ the red parts indicate the loadingcondition that marks the effect of drill bit inertia and theblue parts represent the transition to unloading Figure 4(b)shows Case 1 where a single-incident stress wave indicatesthe drill bit displacement at u(τ) and the drill bit arrival at umas an overdamped motion due to the loading continuingowing to external forces and inertia Later in the transitionto unloading characteristic overdamped motion appearsand there is no convergence to uf (Figure 3)

Case 2 (Figure 4(c)) appears similar to Case 1 but the bitreaches um sooner owing to τ and the effect of drill bitinertia As the system transitions to unloading and convergesto uf it shows critical damping Case 3 (Figure 4(d)) showsa longer τ compared with Case 2 because of its quickerarrival at um During unloading it converges to uf asunderdamped motion Case 4 (Figure 4(e)) has the drill bitdisplaced at u(τ) due to τ and it arrives at um as criticallydamped motion owing to continued loading resulting fromthe inertia of the drill bit 0e subsequent unloading phaseconverges to uf as underdamped motion

Case 5 (Figure 4(f)) has a longer τ than the preceding fourcases and also reaches um sooner Furthermore all its zonesshowed underdamped motion Finally in Case 6 (Figure 4(g)) the rock displacement has already reached its maximum

while τ is still in progress 0is means that unloading beginsduring τ Underdamped motion was observed in all zones

Overall the longer the duration of τ the shorter theloading time and the sooner the um is reached owing to drillbit inertia Table 2 summarizes the damping characteristics forthe loading and unloading conditions and presents the as-sociated damping mode that arrives at um Cases 1ndash3 showoverdamping Case 4 critical damping and Cases 5 and 6underdamping 0e underdamped mode showed thehighest responsiveness indicating the fastest trans-mission time of τ 0e results show that increasing theτ-transmission time decreased the loading effect resultingfrom drill bit inertia and led to faster responses 0e drillbit inertial effect is inversely proportional to τ

3 Energy Transmission in Percussive DrillingSystem

In Section 2 we examined the response characteristics ofdrill bits that can arrive at um which is themaximum drillingdisplacement of rocks Based on the examined character-istics in Section 3 we calculate the efficiency of percussivedrilling systems and discuss the combinations of parametersthat could maximize it

31 Energy Transmission Efficiency In a percussive drillingsystem the piston directly hits the drill bit at speed v (Figure 2)For a transmitted impact energy Ei at the time of the pistondescending and hitting the drill bit the final speed of the pistonimmediately preceding the impact v can be expressed asfollows [2 10]

v 2Ei

mp

1113888 1113889

12

(12)

Table 1 Expansion conditions for dynamic drill-bit properties for loading unloading and stress-wave duration τ

Case no

Initial conditions0lt tle τ τ lt tle tm tgt tm

σi 05ρcv

F ku

u0 0 _u0 0

σi 0F ku

u0 u(τ) _u0 _u(τ)

σi 0F ckuminus ckuf

u0 um _u0 0

1 αlt 1c

2 α 1c

3 1clt αlt 1

4 α 1

5 αgt 1 tm gt τ

6

0lt tle tm tm lt tle τ tgt τ

σi 05ρcv

F ku

u0 0 _u0 0

σi 05ρcv

F ckuminus ckuf

u0 um _u0 0

σi 0F F ckuminus ckuf

u0 u(τ) _u0 _u(τ)

Advances in Materials Science and Engineering 5

0000

uf

u

00

04

08

12

16

20

0001 0002 t0003 0004

Case 1Case 2Case 3Case 4Case 5Case 6

Loading by incident stress waveLoading by inertia of bitUnloading

(a)

tm

uf

um

u

t

Loading by incident stress waveLoading by inertia of bitUnloading

00040003

Overdamped motion

00020001000000

04

08

12

16

20

u()

(b)

u()

tm

uf

um

u

t

000400030002

Overdamped motion

Critically damped motion

0001000000

04

08

12

16

20

Loading by incident stress waveLoading by inertia of bitUnloading

(c)

u()

Overdamped motion

Underdamped motion

tm

uf

um

u

t

0004000300020001000000

04

08

12

16

20

Loading by incident stress waveLoading by inertia of bitUnloading

(d)

Figure 4 Continued

6 Advances in Materials Science and Engineering

e initial height h0 is v22g and the rebound eh canbe dened as the ratio between the initial height and thepistonrsquos postimpact rebound height h ese areexpressed as follows [2]

eh h

h0v2ev2 (13)

e eciency η of impact energy transmission is de-ned as the ratio between the kinetic energy generated fromthe impact of the piston and the energy transmitted to therock [12] and is given as

η kum

2

ρALhv2

Fmax( )2

ρALhv2k (14)

Which can be expressed as follows through dimensionalanalysis

η 2βu2m (15)

e value um during loading can be obtained from (7)the equation of motion for the drill bit e obtained um canbe expressed as in (16) for each damping mode where theinitial conditions are 0lt tle τ F kui u0 0 and _u0 0

u()

Critically damped motion

Underdamped motion

tm

uf

um

u

t

0004000300020001000000

04

08

12

16

20

Loading by incident stress waveLoading by inertia of bitUnloading

(e)

Loading by incident stress waveLoading by inertia of bitUnloading

u()Underdamped motion

tm

uf

um

u

t

0004000300020001000000

04

08

12

16

20

(f)

Loading by incident stress waveLoading by inertia of bitUnloading

u()Underdampedmotion

uf

u

u

t

t

000400030002

00002 00003 00004

0001000000

04

08

12

16

20

um

um

tm

(g)

Figure 4 Results of the motion response of the bit formr 1 (a) All cases (b) Case 1 (c) Case 2 (d) Case 3 (e) Case 4 (f) Case 5 (g) Case 6

Advances in Materials Science and Engineering 7

um(t)

1 +1minus

αminus 1

radic

2αminus 1

radic middot eminus(2α)(βτ)(1+

1minusα

radic)t minus

1 +1minus α

radic

21minus α

radic middot eminus(2α)(βτ)(1minus

1minusα

radic)t

1113890 1113891(overdamped)

1minus eminus2(βτ)t middot 2βτ

t + 11113890 1113891(critical damped)

1minus eminus2(βτ)t middot1

αminus 1

radic sin2

αminus 1

radic

αβτ

t1113888 1113889 + cos2

αminus 1

radic

αβτ

t1113888 11138891113890 1113891(underdamped)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(16)

Moreover for tmlt τ um is expressed as follows

um(t) 1minus eminus(2α)(βτ)

middot ⎡⎣1

αminus 1

radic sin2

αminus 1

radic

αβτ

t1113888 1113889

+ cos2

αminus 1

radic

αβτ

t1113888 1113889⎤⎦

(17)

0e initial conditions are 0lt tle τ F kui ui 0and _ui 0

Next the displacement value uf during unload-ing can be obtained via the equation of motion for a drill-bit (9) as shown in (18) 0e initial conditionsduring unloading are tgt τ F 0 u0 u(τ) and_u0 _u(τ)

uf(t)

α4

1minus α

radicτβ

middot eminus(2α)(1+

1minusα

radic)t

middot u02αβτ

(1 +1minus α

radic) minus _u0 +

4αβτu01113888 11138891113890 1113891

minusα

41minus α

radicτβ

middot eminus(2α)(1minus

1minusα

radic)t

middot u02αβτ

(1minus1minus α

radic) minus _u0 +

4αβτu01113888 11138891113890 1113891(overdamped)

eminus2(βτ)t middot _u0 + 2βτu01113888 1113889tminus u01113890 1113891(critical damped)

eminus(2α)(βτ)t middot ⎡⎣α

2αminus 1

radicτβ

1113888 1113889 _u0 middot sin2

αminus 1

radic

α1113888 1113889

βτ

t +1

αminus 1

radic u0 middot sin2

αminus 1

radic

αβτ

t1113888 1113889

+ u0 middot cos2

αminus 1

radic

αβτ

t1113888 1113889⎤⎦(underdamped)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(18)

Table 2 Dynamic drill-bit properties for loading unloading and τ and the damping modes capable of reaching um

Case no ConditionsDamping modes capable of reaching um

0lt tle τ τ lt tle tm tgt tm

1 αlt 1c

Over Overlowast Over2 α 1

cOver Overlowast Critical

3 1clt αlt 1 Over Overlowast Under

4 α 1 Critical Criticallowast Under5 αgt 1 tm gt τ Under Underlowast Under

6 αgt 1 tm lt τ0lt tle tm tm lt tle τ tgt τUnderlowast Under Under

lowastDamping modes capable of reaching um

8 Advances in Materials Science and Engineering

eeciency of the percussive drilling systems for given βcan be calculated using (14ndash17) e factor um which isimportant to the eciency calculations is discussed in depth

32 Results of Energy Transmission Eciency e results ofsimulations on the drill bit-rock interaction model shown inFigure 5 indicate the damped modes that can arrive at um (forvarying values of β) as the response by the drill bit at the time ofpiston impact e damped mode is determined by ζ and αwhich indictate the system responses Overdamped mode hasαlt 1 and can be expressed as follows

2mb

mh( )βlt 1

βlt1

2mr

(19)

e critical damped mode has α 1 and is expressed asfollows

2mb

mh( )β 1

β 1

2mr

(20)

e underdamped mode has αgt 1 and is expressed as

2mb

mh( )βgt 1

βgt1

2mr

(21)

where mbmh mr

e special underdamped mode refers to the case whereτ is faster than t indicating that the drill bit can arrive at umonly via an incident stress wave In this case the bit reachesits maximum displacement and transitions to the unloadingstate while the transmission of τ is underway

e damped mode of each mr shown in Figure 5 re-veals that the underdamped mode is dominant of thee ects Also overdamped and critical damped were ob-served where β was low is is possibly because of thee ects of the internal energy (ie compression strength)of rock and the percussive energy e overdamped andcritical damped were also observed is is possibly be-cause of internal energy (ie compression strength) ofrock and energy caused by percussion

Figure 6 shows dimensionless time tm tτ capable ofreaching Um for β and τ In Section A of tgt τ the bit reachesUm after termination of τ and in Section B of tlt τ the bitreachesUm before the termination of τe lower themr andβ the faster the termination of τ and increase in thesetended to make the duration of τ delivered to rock throughbit to increase proportionally to β is suggests that thehigher rock strength requires longer duration of τ fordrilling the rock

Figure 7 shows dimensionless Um state dened as ratioum a displacement um by β and uinfin caused by incident stresswave and loading condition e prediction of displacementof drill bit drilling displacement by characteristic of rockwas limited is study accordingly judged the di-mensionless drilling state Um only by using ratio uinfina displacement during loading and um considering inertiacaused during percussion (22)

Um umuinfin (22)

where the uinfin only by loading is dened as follows

mr = 1

mr = 2

mr = 05

0 5 10

15 20

4

3

2

Dam

ped

mod

es

1

0

mr = 1mr = 2

mr = 05

mr = 1

mr = 05

Figure 5 Damping mode capable of reaching maximum rockfracture displacement um given the piston-to-bit mass ratiomr and β1 overdamped mode 2 critically damped mode 3 underdampedmode 4 special underdamped mode

0 5 10

15 2000

02

06

10

14

18

20

04

08

12

16

Dim

ensio

nles

s pen

etra

tion

(Um

)

Section B

Section A

mr = 05mr = 1mr = 2

uinfin

Figure 6 State of reaching drill bit dimensionless penetration Umwith respect to mass ratio mr and β

Advances in Materials Science and Engineering 9

uinfin ρcAkv

τβv (23)

e e ect of initial condition on uinfin decreased with theincrease in β e um is determined by considering loadingcondition and inertia e ect is a ected by overresponse anddecreases as β increases

Section A refers to umlt uinfin with only τ and loadinge ect present (ie the condition for the normal state in theinitial mode) uinfin means only the τ e ect resulting fromloading exists Section B refers to umgt uinfin a state in whichum can be reached through τ and loading is exceeded issection is where the drill bit inertia and underdampedmotion appear e gradient of the curve for maximumdisplacement of the bit varies with β indicating that as therock strength increases the displacement allowing fracturedecreases Near value 2 β changes its gradient for maximumdisplacement is means that in Section A the rock is lesssti than the piston owing to the rock-sti ness e ect and inSection B the rock is sti er than the piston β is dened asthe sti ness ratio between the rock and the piston as in (11)Hence when βlt 2 a condition is established for rock fractureto occur in that the piston sti ness exceeds the rock sti ness

In examining um for piston-bit mass ratios of mrlt 2 therock strength that can be defeated by τ occurring in theinitial loading tends to increase is tendency is suspectedto result from the inertial e ect of the mass of the bit Forβgt 4 the um gradient tended to decrease with increasingmass of the bit Hence the eciency is considered to de-crease which is attributed to the e ect of the rockrsquos com-pressive strength e uniaxial compressive strength of veryhard rock is typically at least 200MPa while the tensilestrength of the H13 tool steels that are used mostly forpistons (ASTM A681 and DIN EN ISO 4957) is known to beat least 359ndash1170MPa [15]

Given the steels generally used to make pistons thee ective range of β is considered to be 034lt βlt 112 which

was calculated considering the strength ratio between thehard rock and an H13 steel tool Hence the e ective range ofβ could be no greater than 2 Rocks with a sti ness ratio ofβgt 2 might cause failure through the pistonrsquos plastic de-formation and fracture is study examined the e ectiveinterval of β and investigated the parameter combinationsgiving high drilling eciency in this interval

Figure 8 illustrates the motion of the bit generatedduring the percussion process and the energy transmissionrate (ie eciency) for the response characteristics τ andpercussion impact Analysis of mr found the maximumdrilling eciency in interval 1lt βlt 2 with a tendency ef-ciency decreasing after mr increased above a certain valueis result was attributed to the tendency whereby increasingmr converts the systemrsquos dynamic response to τ due to thepiston from underdamped to overdamped Furthermoreincreasing the mass of the bit can increase its internal energy(ie friction energy) to above the piston transmitted energy(ie kinetic and potential energy) Hence the bitrsquos dynamicresponse would be overdamped and could compromisedrilling eciency In other words the heavier the bit themore delayed the response to percussion

In Figure 8 the left side of red line (β18) represents thee ect of only incident stress wave meaning that the higherthe value in graph the higher the drilling eciency in in-cident stress wave e right side of red line representingunloading behavior after reaching maximum eciency inincident stress wave is a ected by both brittleness char-acteristics of rock and inertia e ect of drill bit

e piston should have twice the mass of the drill bit(iemr 05) for the most ecient drilling for βle 2 which isestablished to be the e ective interval of β and correspondsto both soft and hard rocks Drilling sti er rocks (2lt βlt 4)might be less ecient but increasing mr to 1 would helpimprove drilling eciency

00000

00002

00004

00006

00008

00010

00012

Dim

ensio

nles

s tim

e (t m

)

Section A

Section B

0 5 10

15 20

mr = 05mr = 1mr = 2

Figure 7 Dimensionless time capable of reaching Um for an in-cident stress-wave τ acting on the bit and β

0 2 4 6 8 10 12 14 16 18 20

= 1800

01

02

03

04

05

06

07

08

09

10

Effic

ienc

y

mr = 01mr = 025mr = 05mr = 1

mr = 2mr = 4mr = 10

Figure 8 Drilling eciency of percussive drilling systems withrespect to β and mr

10 Advances in Materials Science and Engineering

4 Results and Discussion

0is study defined equations for the motion of a drill bitwhen struck by a piston and expanded the dynamic prop-erties of the drill bit based on the conditions ζ and α It alsoanalyzed the relationship between the bit dynamics and theduration of the incident stress wave (τ) Underdamped drillbits displayed the greatest responsiveness and the fastesttransmission of τ Increasing the transmission of τ decreasedthe loading effect due to the drill bitrsquos inertia but led to a fastresponse 0e inertial effect of the drill bit was confirmed tobe inversely proportional to the transmission of τ

A percussive drilling system accomplishes its work viathe percussion impact of a piston In the systems studiedhere τ ended more quickly with lower mr and β values asthese values increased the duration of τ transmitted to therock via the drill bit increased in proportion to β 0e resultsalso showed that the maximum bit displacement (um) de-creased as β increased Analysis of the drilling dynamicsconfirmed the effective intervals of β With reference to therock strengths suggested by the International Society forRock Mechanics (ISRM) the effective interval of β is con-sidered to be no greater than 2

0e main purpose of this study was to examine theimpact energy transmission rate and drilling efficiency ina percussive drilling system (Figure 8) 0e results establishedamr value of 05 as the most efficient for rocks whose strengthcorresponds to the interval βlt 2 0e improved drilling ef-ficiency would lead to advantages such as reduced bit pro-duction costs Rocks stiffer than 2ltβlt 4 can be mostefficiently drilled by selecting mr 1

0emovement and response characteristics of bit duringpercussion process τ and transfer rate of energy caused bypercussion an efficiency depend on mr and β where in-crease in mr leads to overdamping of bit movement char-acteristics due to τ by piston and reduction in drillingefficiency due to internal energy effect by mass of drill bit Asthe increase in mass of drill bit for the section of 4lt βlt 6the drilling efficiency of percussive drilling system decreaseand for the very hard rock (UCSgt 200MPa) section of4lt βlt 6 it is considered that the reasonable applied mr ofdrilling tool is 1 or 2

0is study neglected the effects of the secondary incidentstress wave and also the effect of buttons embedded in thedrill bit Additionally for analytical simplification the pistonand drill bit were assumed to be of the same diameter andmaterial and a rectangular pulse with an incident stresswave of duration τ and amplitude 05ρcv was assumed 0eeffect of the flexural stress wave depending on the config-uration was not considered 0ese limitations of this papershould be addressed in further investigations that take intoaccount the configuration of the piston and drill bit theeffect of buttons and the drilling efficiency given differingshapes of the incident stress wave

5 Conclusion

0is paper aimed to identify the optimal design parametersfor percussive drilling systems by introducing a drill bit-rock

interaction model that could verify the bitrsquos motion duringpercussion and the resulting damping characteristics 0estudy analyzed drilling efficiency and drew the followingconclusions

Percussive drilling systems have six dynamic drill bitproperties that can be expandable 0is paper discussed thephysical meaning of the dimensionless parameters α and β0eir values determined the damping characteristics thatcan lead to the rockrsquos maximum fracture displacement 0efastest response of τ was observed for underdamped drill-bitmotion

Drilling was most efficient in the interval 1lt βlt 4 whereincreasing mr at a given β decreased drilling efficiency 0evalue of mr for efficient drilling was determined by the rockstrength (ie β)

0e results indicate that application of a piston-to-drill bitmass ratio of 05 (ie a piston mass twice that of the drill bitmass) to the rocks whose stiffness corresponds to βge 2 wouldlikely be most efficient and also reduce drill bit productioncosts Furthermore a selection ofmr 1 would be valid whendeveloping drill tools for boring complex rocks (1ltβlt 4) Atβgt 4 the best efficiency could be achieved when the bit massequals or exceeds the piston mass

Appendix

Theoretical Case of Bit Motion Condition(Drilling Dynamic Model)

0is paper proposed six conditions for the dynamics ofa drill bit struck by a piston impact as summarizedbelow 0e damping conditions for the damping ratiosand α from the drill-bit equations of motion (7ndash9) are asfollows

First for the overdamped condition of ζ1 gt 1 and αlt 1 τis transmitted to the drill bit indicating loading Equation(7) can be expressed as in (A1) 0e initial conditions are0lt tle τ σi 05ρcv F ku u0 0 and _u0 0

eurou +ρcA

mb

1113888 1113889 _u +k

mb

1113888 1113889u ρcA

mb

1113888 1113889v (A1)

where assuming the normal state u can be calculated asfollows

u ρcA

k1113874 1113875v (A2)

Next we examine the case where τ is complete and theloading effect is sustained owing to external forces and drill-bit inertia Applying the conditions of τ lt tle tm σi 0F ku u0 u(τ) and _u0 _u(τ) to (A1) results in thefollowing expression

eurou +ρcA

mb

1113888 1113889 _u +k

mb

1113888 1113889u 0 (A3)

where u 0 Last we examine the unloading conditions oftm lt t σi 0 F ckuminus ckuf u0 um and _u0 0 Apply-ing these to (9) when unloading gives

Advances in Materials Science and Engineering 11

eurou +ρcA

mb

1113888 1113889 _u + ck

mb

1113888 1113889u ck

mb

1113888 1113889uf (A4)

where bit displacement u uf is used to indicate the rockrsquosfracture displacement Furthermore considering the bitrsquosdamping ratio when unloading its behavior can be ex-pressed as

Damped mode

ζ2 gt 1 αlt1c

over( )

ζ2 1 α 1c

critical( )

ζ2 lt 1 αgt1c

under( )

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A5)

Second under critical damping where ζ1 1 and α 1 τis transmitted to the drill bit indicating loading Equation(7) can be expressed as in (A1) 0e initial conditions are0lt tle τ σi 05ρcv F ku u0 0 and _u0 0

In this case assuming the normal state u can be cal-culated using (A2)

0e following is the case where τ is complete and theloading effect is sustained owing to external forces and drillbit inertia Applying the conditions of τ lt tle tm σi 0F ku u0 u(τ) and _u0 _u(τ) to (A1) give (A3) whereu 0 Last we examine the unloading condition wheretm lt t σi 0 F ckuminus ckuf u0 um and _u0 0 Equation(9) when unloading can be expressed as (A4)

In this case bit displacement u uf is used to indicate therockrsquos fracture displacement Furthermore considering thebitrsquos damping ratio when unloading its behavior follows

Damped mode

ζ2 gt 1 αlt1c

(no case)

ζ2 1 α 1c

(no case)

ζ2 lt 1 αgt1c

under( )

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A6)

0e second condition of critical damping is defined asα 1When unloading drill bits c refers to the rockrsquos propertyeffect and is always less than 1 Hence when unloading noother cases but the underdamped mode are allowed In otherwords when unloading in the critically damped mode owingto the rock properties the bit is always underdamped

0e third condition is underdamping with ζ1 lt 1 andαgt 1 Here τ is transmitted to the drill bit meaning loadingEquation (7) can be expressed by (A1)0e initial conditionsare 0lt tle τ σi 05ρcv F ku u0 0 and _u0 0 Whereassuming the normal state u can be calculated as (A2)

0e following is the case where τ is complete and theloading effect is sustained owing to external forces and drillbit inertia Applying the conditions of τ lt tle tm σi 0F ku u0 u(τ) and _u0 _u(τ) to (A1) results in (A3)

where u 0 Last we examine the unloading conditionwhere tm lt t σi 0 F ckuminus ckuf u0 um and _u0 0Equation (9) when unloading can be expressed as (A4)

In this case the drill bit displacement u uf is used toindicate the rockrsquos fracture displacement Furthermoreconsidering the damping ratio in the drill bitrsquos unloadingthe bitrsquos behavior can be expressed as

Damped mode

ζ2 gt 1 αlt1c

(no case)

ζ2 1 α 1c

(no case)

ζ2 lt 1 αgt1c

under( )

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A7)

0e third condition is the critically damped modedefined as αgt 1 During drill bit unloading c refers to therock property effect and is always less than 1 Hencewhen unloading no other cases but underdamping areallowed

Next the underdamped condition is where ζ1 lt 1 andαgt 1 and τ is slower than tm (the time at which the rockrsquosmaximum fracture displacement is reached) that is tm lt τIn this case τ is transmitted to the drill bit meaningloading Equation (7) can be expressed by (A1) 0e initialconditions are 0lt tle tm σi 05ρcv F ku u0 0 and_u0 0 Here assuming the normal state u can be calculatedas (A2)

0e following case is where conversion is made tounloading after the drill bit reaches its maximum dis-placement but the incident stress wave is sustained Ap-plying the conditions of tm lt tle τ σi 05ρcvF ckuminus ckuf u0 um and _u0 0 to (9) results in

eurou +ρcA

mb

1113888 1113889 _u + ck

mb

1113888 1113889u ρcA

mb

1113888 1113889v + ck

mb

1113888 1113889uf (A8)

where the definition of u 1c(ρcAk)v + uf is possibleLast this unloading condition is where the incident

stress wave is resolved When unloading the drill bit usingthe conditions tlt τ σi 0 F ckuminus ckuf u0 u(τ) and_u0 _u(τ) allows (9) to be expressed as (A4)

In this case drill bit displacement u uf is used to in-dicate the rockrsquos fracture displacement 0e analysis of drillbit dynamics showed that when loading transitions tounloading in the underdamped condition the dynamicproperties of the bit exist only in the underdamping mode

Conflicts of Interest

The authors declare that there are no conflicts of interest

References

[1] C H Song K B Kwon J Y Park et al ldquoOptimum design ofthe internal flushing channel of a drill bit using RSM and CFDsimulationrdquo International Journal of Precision Engineeringand Manufacturing vol 15 no 6 pp 1041ndash1050 2014

12 Advances in Materials Science and Engineering

[2] X Li G Rupert D A Summers P Santi and D Liu ldquoAnalysisof impact hammer rebound to estimate rock drillabilityrdquo RockMechanics and Rock Engineering vol 33 no 1 pp 1ndash13 2000

[3] W A Hustrulid and C Fairhurst ldquoA theoretical and ex-perimental study of the percussive drilling of rock part Itheory of percussive drillingrdquo International Journal of RockMechanics and Mining Sciences amp Geomechanics Abstractsvol 8 no 4 pp 311ndash333 1971

[4] W A Hustrulid and C Fairhurst ldquoA theoretical and ex-perimental study of the percussive drilling of rock part IIforce-penetration and specific energy determinationrdquo In-ternational Journal of Rock Mechanics and Mining Sciences ampGeomechanics Abstracts vol 8 no 4 pp 335ndash356 1971

[5] W A Hustrulid and C Fairhurst ldquoA theoretical and ex-perimental study of the percussive drilling of rock part IIIexperimental verification of the mathematical theoryrdquo In-ternational Journal of Rock Mechanics and Mining Sciences ampGeomechanics Abstracts vol 9 no 3 pp 417-418 1972

[6] W A Hustrulid and C Fairhurst ldquoA theoretical and ex-perimental study of the percussive drilling of rock part IVapplication of the model to actual percussive drillingrdquo In-ternational Journal of Rock Mechanics and Mining Sciencesvol 9 no 3 pp 431ndash442 1972

[7] L E Chiang and D A Elias ldquoA 3D FEM methodology forsimulating the impact in rock-drilling hammersrdquo In-ternational Journal of Rock Mechanics and Mining Sciencesvol 45 no 5 pp 701ndash711 2008

[8] C H Song K B Kwon M G Cho J Y Oh D Y Shin andJ W Cho ldquoDevelopment of lab-scale rock drill apparatus fortesting performance of a drill bitrdquo International Journal ofPrecision Engineering and Manufacturing vol 16 no 7pp 1402ndash1414 2015

[9] K B Kwon C H Song J Y Park J Y Oh J W Lee andJ W Cho ldquoEvaluation of drilling efficiency by percussiontesting of a drill bit with new button arrangementrdquo In-ternational Journal of Precision Engineering andManufacturing vol 15 no 6 pp 1063ndash1068 2014

[10] B Lundberg andM Okrouhlik ldquoInfluence of 3D effects on theefficiency of percussive rock drillingrdquo International Journal ofimpact Engineering vol 25 no 4 pp 345ndash360 2001

[11] B Lundberg and P Collet ldquoOptimal wave shape with respectto efficiency in percussive drilling with detachable drill bitrdquoInternational Journal of impact Engineering vol 86 pp 179ndash187 2015

[12] X B Li G Rupert and D A Summers ldquoEnergy transmissionof down-hole hammer tool and its conditionalityrdquo Trans-actions of Nonferrous Metals Society of China vol 10 no 1pp 109ndash111 2000

[13] J W Cho S Jeon S H Yu and S H Chang ldquoOptimumspacing of TBM disc cutters a numerical simulation using thethree-dimensional dynamic fracturing methodrdquo Tunnellingand Underground Space Technology vol 25 no 3 pp 230ndash244 2010

[14] W Changming ldquoAn analytical study of percussive energytransfer in hydraulic rock drillsrdquo Mining Science and Tech-nology vol 13 no 1 pp 57ndash68 1991

[15] Online Materials Information Resource httpwwwmatwebcomsearchDataSheetaspxMatGUID8bc5d558f4174e6082ddf4966e382bd6ampckck1

Advances in Materials Science and Engineering 13

e energy transmission eciency via the impact of thepercussive drilling system is examined in Section 3 Section 4discusses the combinations of parameters that might opti-mize the drilling eciency given the dynamic drill bitcharacteristics and rock sti ness (ie β) Conclusions arepresented in Section 5

2 Dynamic Model of a Percussive DrillingSystem

21 Mechanism of Percussive Drilling Figures 1(a) and 1(b)illustrate the working principles or drilling mechanisms ofpercussive drilling systems In each case the drill bit ispropelled into the rock by percussion feed and rotationforces that are transmitted via the drifter or hammer Afterreceiving impact energy from the drill bit the rock developscracks that form a network and eventually break it ecrushed rock then undergoes further splitting via the ro-tation force and secondary impacts until it becomes powdere rock powder is discharged to the outside by compressedair supplied through a ushing duct located inside the drillbit thus enabling the drill to proceed further [1]

22 Bit Motion and Stress Wave Propagation Figure 2 il-lustrates the principles of impact and rock penetration bya percussive drilling system

e piston mass length cross-sectional area descendingspeed density and longitudinal wave velocity are designated asmp LpAp v ρ and c respectivelye underlying assumptionswere that the drill bit remains immobile on the surface and thatthe diameters of the drill bit and the piston are equal Also thee ects of buttons embedded in the bit were ignored

e following equation expresses themotion of the drill bitresulting from the force applied to it by the piston [12]

mbd2u

dt2 A σi + σr( )minusF (1)

where u is the penetration depth (displacement) of the drillbit the second term is the force applied to the drill bit whenthe impact is exerted upon it F is the force generated at therock-bit interface (ie the penetration force) and A is thecross-sectional area of the drill bit and piston

According to the relation between the dynamic stress andvelocity of a mass in an elastic body system the velocity of thecontacting end of the bit can be expressed as follows [2]

du

dtσiρcminusσrρc (2)

here σi and σr are the incident and reected stress wavesrespectively e e ect of the secondary incident stress wavewas not considered For analytical simplication the piston andthe drill bit were assumed to be of the same diameter andmaterial both were assumed to have a rectangular pulse witha duration of incident stress wave τ and an amplitude of 05 ρcv

Typically nearly all the percussion impact energy is con-verted intowave energyepiston hitting the drill bit generatesan incident stress wave (σi) at which point a portion of the

impact energy is transmitted to the rock via the drill bitcrushing the rock [11] e remaining energy is converted intoa reected stress wave (σr) and is used to increase the speed ofthe piston ascending from the drill bit top to speed ve (Figure 2)[2] e reected stress wave can in turn be transmitted to thepiston and drill bit in the form of a secondary incident stresswave at the piston-bit interface and at the drill bit-rock interface

e secondary incident stress wave in a percussivedrilling system has been reported not to contribute to thecrushing of rocks [11] therefore it would not signicantlya ect the energy transmission eciency Furthermorea study of drilling speed found the incident stress wave tohave an insignicant e ect on the impact force and pen-etration characteristics [14]

e interaction between the drill bit and rock (theforce-penetration relationship Figure 3) is expressed in (3)[2] which separately considers the loading and unloadingconditions k is the rock penetration coecient index c isthe unloading constant and τ is the duration of the in-cident stress wave

F ku(loading)ckuminus ckuf(unloading)

(3)

e duration of the incident stress wave τ generated bythe impact of the piston can be expressed with respect totime t in two scenarios as dened in (4) [2] e rstscenario involves transmission of an incident stress wavewhereas the second does not

σi(t)

12ρcv for (0lt tle τ)

0 for (tgt τ)

(4)

Bit

Piston

Rock

mb

h0

h

Crushed zone

Crack

Button

mp Lp Ap c

i

r

e

Figure 2 Schematic rock fracture mechanism by percussivedrilling (modied from Cho et al [13])

Advances in Materials Science and Engineering 3

For the loading condition we obtain the followingequations [2]

A σi + σr( ) F

du

dtσiρcminusσrρc

ku F

(5)

where the initial condition is t 0 andF 0For the unloading condition the equations can be expressed

as follows [2]

A σi + σr( ) F

du

dtσiρcminusσrρc

ckuminus cku F

(6)

where the initial condition is t τ and F(t) limtrarr τ

F(t)e response characteristics of a drill bit (ie the damped

characteristics) can be under critical or overdamped given itsmotion equation that is (1) e movement of the drill bitunder loading and unloading conditions was examined edamping characteristics of a percussive drilling system arecategorized depending on the damping ratio (ζ) and thedimensionless parameter α

When loading the equation of motion for the drill bit isdened as follows

eurou +ρcAmb

( ) _u +k

mb( )u

2Amb( )σi (7)

where the response-characteristics determinant ζ1 is

ζ1 4α(βτ)

24α(βτ)2radic

1α

radic (8)

When unloading the drill-bit equation of motion isdened by (9) e ζ2 term is given in (10)

eurou +ρcAmb

( ) _u + ck

mb( )u

2Amb( )σi + c

kufmb

(9)

ζ2 4α(βτ)

2c(4α)(βτ)2radic

1cαradic (10)

e dimensionless parameters α and β dened in (8) and(10) can be expressed as follows (extended from (2) (12))

α 4mb

ρcAmiddotk

ρcA

4mb

ρcAτmiddot β

4mb

2mpLPcLPAβ 2

mb

mp( )β

β kτρcA

k2LpcρcA

2kLhρc2A

2kLp

ρA Epρ( ) 2k

LpEpA( ) 2

k

kp( )

(11)

As explained earlier the typical motion of drill bits in-volves four conditions However this study considers sixconditions as listed in Table 1 (Appendix A) and includes theloadingunloading condition and the presence or absence ofan incident stress wave To investigate the rate at which theimpact energy is transmitted from the piston to the rockduring the percussion process (ie the drilling eciency) thisstudy rst considered the response characteristics resultingfrom the motion of the drill bit during the percussion processand the e ect of the rock e aim was to nd the responsecharacteristics that lead to the rockrsquos maximum crushingdisplacement um e study also dened the motion ofpercussive drill bits resulting from the impact of the pistonand analyzed their behavior given the system responses edrill bit-rock interaction was considered here including theforcendashpenetration relationship for the rock

Table 1 presents the relationships between loading andunloading and the duration of stress wave τ in percussivedrilling systems In Table 1 τ is the duration of the incidentstress wave being transmitted to the drill bit via the pistonduring loading it is referenced relative to the time t eloading condition can be dened in two ways 0lt tle τ orτ lt tle tm where the latter expression means that τ arrivedsooner than t that is the transmission of τ was complete butthe loading continued owing to the inertial e ect of the drillbit If τ gt tm the drill bit arrived at um before the transmissionof τ to the bit had completed and indicates the time when thesystem switches to the unloading condition

In summary in Cases 1 to 5 in Table 1 the motion of thedrill bit following the piston impact continues until theloading condition applying external force to the rock reachesτ lt tle tm After that time the stress wave arriving at the drillbit is converted into unloading at τ gt tm provided that therock displacement occurs sooner than tm that is themaximum time of arrival However the τ in Case 6 rep-resents a stress wave that is still progressing while the rockdisplacement owing to the drill bit has already reached itsmaximum that is unloading occurs during τ

F

0 u

Fm

um

k

uf

k

Figure 3 Force versus penetration relationship representing thebitndashrock interaction curve

4 Advances in Materials Science and Engineering

23 Results of BitMotionAnalysis Numerical simulations ona dynamic model of drill bit properties were run for the sixcases and used the dynamic drill bit properties during inducedpercussion (Section 2) the drill bit-rock interaction modeland the dimensionless parameters α and β

Figure 4 shows the simulation results (ζ) for differentconditions (α Table 1) and the effect of loading andunloading 0e mass ratio mr was 1 and the same con-ditions were applied to the piston mass and drill bit massFigure 4(a) shows an overview of the response characteristicsfor all six cases 0e initial black part of each curve refers tothe sustained effect of τ the red parts indicate the loadingcondition that marks the effect of drill bit inertia and theblue parts represent the transition to unloading Figure 4(b)shows Case 1 where a single-incident stress wave indicatesthe drill bit displacement at u(τ) and the drill bit arrival at umas an overdamped motion due to the loading continuingowing to external forces and inertia Later in the transitionto unloading characteristic overdamped motion appearsand there is no convergence to uf (Figure 3)

Case 2 (Figure 4(c)) appears similar to Case 1 but the bitreaches um sooner owing to τ and the effect of drill bitinertia As the system transitions to unloading and convergesto uf it shows critical damping Case 3 (Figure 4(d)) showsa longer τ compared with Case 2 because of its quickerarrival at um During unloading it converges to uf asunderdamped motion Case 4 (Figure 4(e)) has the drill bitdisplaced at u(τ) due to τ and it arrives at um as criticallydamped motion owing to continued loading resulting fromthe inertia of the drill bit 0e subsequent unloading phaseconverges to uf as underdamped motion

Case 5 (Figure 4(f)) has a longer τ than the preceding fourcases and also reaches um sooner Furthermore all its zonesshowed underdamped motion Finally in Case 6 (Figure 4(g)) the rock displacement has already reached its maximum

while τ is still in progress 0is means that unloading beginsduring τ Underdamped motion was observed in all zones

Overall the longer the duration of τ the shorter theloading time and the sooner the um is reached owing to drillbit inertia Table 2 summarizes the damping characteristics forthe loading and unloading conditions and presents the as-sociated damping mode that arrives at um Cases 1ndash3 showoverdamping Case 4 critical damping and Cases 5 and 6underdamping 0e underdamped mode showed thehighest responsiveness indicating the fastest trans-mission time of τ 0e results show that increasing theτ-transmission time decreased the loading effect resultingfrom drill bit inertia and led to faster responses 0e drillbit inertial effect is inversely proportional to τ

3 Energy Transmission in Percussive DrillingSystem

In Section 2 we examined the response characteristics ofdrill bits that can arrive at um which is themaximum drillingdisplacement of rocks Based on the examined character-istics in Section 3 we calculate the efficiency of percussivedrilling systems and discuss the combinations of parametersthat could maximize it

31 Energy Transmission Efficiency In a percussive drillingsystem the piston directly hits the drill bit at speed v (Figure 2)For a transmitted impact energy Ei at the time of the pistondescending and hitting the drill bit the final speed of the pistonimmediately preceding the impact v can be expressed asfollows [2 10]

v 2Ei

mp

1113888 1113889

12

(12)

Table 1 Expansion conditions for dynamic drill-bit properties for loading unloading and stress-wave duration τ

Case no

Initial conditions0lt tle τ τ lt tle tm tgt tm

σi 05ρcv

F ku

u0 0 _u0 0

σi 0F ku

u0 u(τ) _u0 _u(τ)

σi 0F ckuminus ckuf

u0 um _u0 0

1 αlt 1c

2 α 1c

3 1clt αlt 1

4 α 1

5 αgt 1 tm gt τ

6

0lt tle tm tm lt tle τ tgt τ

σi 05ρcv

F ku

u0 0 _u0 0

σi 05ρcv

F ckuminus ckuf

u0 um _u0 0

σi 0F F ckuminus ckuf

u0 u(τ) _u0 _u(τ)

Advances in Materials Science and Engineering 5

0000

uf

u

00

04

08

12

16

20

0001 0002 t0003 0004

Case 1Case 2Case 3Case 4Case 5Case 6

Loading by incident stress waveLoading by inertia of bitUnloading

(a)

tm

uf

um

u

t

Loading by incident stress waveLoading by inertia of bitUnloading

00040003

Overdamped motion

00020001000000

04

08

12

16

20

u()

(b)

u()

tm

uf

um

u

t

000400030002

Overdamped motion

Critically damped motion

0001000000

04

08

12

16

20

Loading by incident stress waveLoading by inertia of bitUnloading

(c)

u()

Overdamped motion

Underdamped motion

tm

uf

um

u

t

0004000300020001000000

04

08

12

16

20

Loading by incident stress waveLoading by inertia of bitUnloading

(d)

Figure 4 Continued

6 Advances in Materials Science and Engineering

e initial height h0 is v22g and the rebound eh canbe dened as the ratio between the initial height and thepistonrsquos postimpact rebound height h ese areexpressed as follows [2]

eh h

h0v2ev2 (13)

e eciency η of impact energy transmission is de-ned as the ratio between the kinetic energy generated fromthe impact of the piston and the energy transmitted to therock [12] and is given as

η kum

2

ρALhv2

Fmax( )2

ρALhv2k (14)

Which can be expressed as follows through dimensionalanalysis

η 2βu2m (15)

e value um during loading can be obtained from (7)the equation of motion for the drill bit e obtained um canbe expressed as in (16) for each damping mode where theinitial conditions are 0lt tle τ F kui u0 0 and _u0 0

u()

Critically damped motion

Underdamped motion

tm

uf

um

u

t

0004000300020001000000

04

08

12

16

20

Loading by incident stress waveLoading by inertia of bitUnloading

(e)

Loading by incident stress waveLoading by inertia of bitUnloading

u()Underdamped motion

tm

uf

um

u

t

0004000300020001000000

04

08

12

16

20

(f)

Loading by incident stress waveLoading by inertia of bitUnloading

u()Underdampedmotion

uf

u

u

t

t

000400030002

00002 00003 00004

0001000000

04

08

12

16

20

um

um

tm

(g)

Figure 4 Results of the motion response of the bit formr 1 (a) All cases (b) Case 1 (c) Case 2 (d) Case 3 (e) Case 4 (f) Case 5 (g) Case 6

Advances in Materials Science and Engineering 7

um(t)

1 +1minus

αminus 1

radic

2αminus 1

radic middot eminus(2α)(βτ)(1+

1minusα

radic)t minus

1 +1minus α

radic

21minus α

radic middot eminus(2α)(βτ)(1minus

1minusα

radic)t

1113890 1113891(overdamped)

1minus eminus2(βτ)t middot 2βτ

t + 11113890 1113891(critical damped)

1minus eminus2(βτ)t middot1

αminus 1

radic sin2

αminus 1

radic

αβτ

t1113888 1113889 + cos2

αminus 1

radic

αβτ

t1113888 11138891113890 1113891(underdamped)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(16)

Moreover for tmlt τ um is expressed as follows

um(t) 1minus eminus(2α)(βτ)

middot ⎡⎣1

αminus 1

radic sin2

αminus 1

radic

αβτ

t1113888 1113889

+ cos2

αminus 1

radic

αβτ

t1113888 1113889⎤⎦

(17)

0e initial conditions are 0lt tle τ F kui ui 0and _ui 0

Next the displacement value uf during unload-ing can be obtained via the equation of motion for a drill-bit (9) as shown in (18) 0e initial conditionsduring unloading are tgt τ F 0 u0 u(τ) and_u0 _u(τ)

uf(t)

α4

1minus α

radicτβ

middot eminus(2α)(1+

1minusα

radic)t

middot u02αβτ

(1 +1minus α

radic) minus _u0 +

4αβτu01113888 11138891113890 1113891

minusα

41minus α

radicτβ

middot eminus(2α)(1minus

1minusα

radic)t

middot u02αβτ

(1minus1minus α

radic) minus _u0 +

4αβτu01113888 11138891113890 1113891(overdamped)

eminus2(βτ)t middot _u0 + 2βτu01113888 1113889tminus u01113890 1113891(critical damped)

eminus(2α)(βτ)t middot ⎡⎣α

2αminus 1

radicτβ

1113888 1113889 _u0 middot sin2

αminus 1

radic

α1113888 1113889

βτ

t +1

αminus 1

radic u0 middot sin2

αminus 1

radic

αβτ

t1113888 1113889

+ u0 middot cos2

αminus 1

radic

αβτ

t1113888 1113889⎤⎦(underdamped)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(18)

Table 2 Dynamic drill-bit properties for loading unloading and τ and the damping modes capable of reaching um

Case no ConditionsDamping modes capable of reaching um

0lt tle τ τ lt tle tm tgt tm

1 αlt 1c

Over Overlowast Over2 α 1

cOver Overlowast Critical

3 1clt αlt 1 Over Overlowast Under

4 α 1 Critical Criticallowast Under5 αgt 1 tm gt τ Under Underlowast Under

6 αgt 1 tm lt τ0lt tle tm tm lt tle τ tgt τUnderlowast Under Under

lowastDamping modes capable of reaching um

8 Advances in Materials Science and Engineering

eeciency of the percussive drilling systems for given βcan be calculated using (14ndash17) e factor um which isimportant to the eciency calculations is discussed in depth

32 Results of Energy Transmission Eciency e results ofsimulations on the drill bit-rock interaction model shown inFigure 5 indicate the damped modes that can arrive at um (forvarying values of β) as the response by the drill bit at the time ofpiston impact e damped mode is determined by ζ and αwhich indictate the system responses Overdamped mode hasαlt 1 and can be expressed as follows

2mb

mh( )βlt 1

βlt1

2mr

(19)

e critical damped mode has α 1 and is expressed asfollows

2mb

mh( )β 1

β 1

2mr

(20)

e underdamped mode has αgt 1 and is expressed as

2mb

mh( )βgt 1

βgt1

2mr

(21)

where mbmh mr

e special underdamped mode refers to the case whereτ is faster than t indicating that the drill bit can arrive at umonly via an incident stress wave In this case the bit reachesits maximum displacement and transitions to the unloadingstate while the transmission of τ is underway

e damped mode of each mr shown in Figure 5 re-veals that the underdamped mode is dominant of thee ects Also overdamped and critical damped were ob-served where β was low is is possibly because of thee ects of the internal energy (ie compression strength)of rock and the percussive energy e overdamped andcritical damped were also observed is is possibly be-cause of internal energy (ie compression strength) ofrock and energy caused by percussion

Figure 6 shows dimensionless time tm tτ capable ofreaching Um for β and τ In Section A of tgt τ the bit reachesUm after termination of τ and in Section B of tlt τ the bitreachesUm before the termination of τe lower themr andβ the faster the termination of τ and increase in thesetended to make the duration of τ delivered to rock throughbit to increase proportionally to β is suggests that thehigher rock strength requires longer duration of τ fordrilling the rock

Figure 7 shows dimensionless Um state dened as ratioum a displacement um by β and uinfin caused by incident stresswave and loading condition e prediction of displacementof drill bit drilling displacement by characteristic of rockwas limited is study accordingly judged the di-mensionless drilling state Um only by using ratio uinfina displacement during loading and um considering inertiacaused during percussion (22)

Um umuinfin (22)

where the uinfin only by loading is dened as follows

mr = 1

mr = 2

mr = 05

0 5 10

15 20

4

3

2

Dam

ped

mod

es

1

0

mr = 1mr = 2

mr = 05

mr = 1

mr = 05

Figure 5 Damping mode capable of reaching maximum rockfracture displacement um given the piston-to-bit mass ratiomr and β1 overdamped mode 2 critically damped mode 3 underdampedmode 4 special underdamped mode

0 5 10

15 2000

02

06

10

14

18

20

04

08

12

16

Dim

ensio

nles

s pen

etra

tion

(Um

)

Section B

Section A

mr = 05mr = 1mr = 2

uinfin

Figure 6 State of reaching drill bit dimensionless penetration Umwith respect to mass ratio mr and β

Advances in Materials Science and Engineering 9

uinfin ρcAkv

τβv (23)

e e ect of initial condition on uinfin decreased with theincrease in β e um is determined by considering loadingcondition and inertia e ect is a ected by overresponse anddecreases as β increases

Section A refers to umlt uinfin with only τ and loadinge ect present (ie the condition for the normal state in theinitial mode) uinfin means only the τ e ect resulting fromloading exists Section B refers to umgt uinfin a state in whichum can be reached through τ and loading is exceeded issection is where the drill bit inertia and underdampedmotion appear e gradient of the curve for maximumdisplacement of the bit varies with β indicating that as therock strength increases the displacement allowing fracturedecreases Near value 2 β changes its gradient for maximumdisplacement is means that in Section A the rock is lesssti than the piston owing to the rock-sti ness e ect and inSection B the rock is sti er than the piston β is dened asthe sti ness ratio between the rock and the piston as in (11)Hence when βlt 2 a condition is established for rock fractureto occur in that the piston sti ness exceeds the rock sti ness

In examining um for piston-bit mass ratios of mrlt 2 therock strength that can be defeated by τ occurring in theinitial loading tends to increase is tendency is suspectedto result from the inertial e ect of the mass of the bit Forβgt 4 the um gradient tended to decrease with increasingmass of the bit Hence the eciency is considered to de-crease which is attributed to the e ect of the rockrsquos com-pressive strength e uniaxial compressive strength of veryhard rock is typically at least 200MPa while the tensilestrength of the H13 tool steels that are used mostly forpistons (ASTM A681 and DIN EN ISO 4957) is known to beat least 359ndash1170MPa [15]

Given the steels generally used to make pistons thee ective range of β is considered to be 034lt βlt 112 which

was calculated considering the strength ratio between thehard rock and an H13 steel tool Hence the e ective range ofβ could be no greater than 2 Rocks with a sti ness ratio ofβgt 2 might cause failure through the pistonrsquos plastic de-formation and fracture is study examined the e ectiveinterval of β and investigated the parameter combinationsgiving high drilling eciency in this interval

Figure 8 illustrates the motion of the bit generatedduring the percussion process and the energy transmissionrate (ie eciency) for the response characteristics τ andpercussion impact Analysis of mr found the maximumdrilling eciency in interval 1lt βlt 2 with a tendency ef-ciency decreasing after mr increased above a certain valueis result was attributed to the tendency whereby increasingmr converts the systemrsquos dynamic response to τ due to thepiston from underdamped to overdamped Furthermoreincreasing the mass of the bit can increase its internal energy(ie friction energy) to above the piston transmitted energy(ie kinetic and potential energy) Hence the bitrsquos dynamicresponse would be overdamped and could compromisedrilling eciency In other words the heavier the bit themore delayed the response to percussion

In Figure 8 the left side of red line (β18) represents thee ect of only incident stress wave meaning that the higherthe value in graph the higher the drilling eciency in in-cident stress wave e right side of red line representingunloading behavior after reaching maximum eciency inincident stress wave is a ected by both brittleness char-acteristics of rock and inertia e ect of drill bit

e piston should have twice the mass of the drill bit(iemr 05) for the most ecient drilling for βle 2 which isestablished to be the e ective interval of β and correspondsto both soft and hard rocks Drilling sti er rocks (2lt βlt 4)might be less ecient but increasing mr to 1 would helpimprove drilling eciency

00000

00002

00004

00006

00008

00010

00012

Dim

ensio

nles

s tim

e (t m

)

Section A

Section B

0 5 10

15 20

mr = 05mr = 1mr = 2

Figure 7 Dimensionless time capable of reaching Um for an in-cident stress-wave τ acting on the bit and β

0 2 4 6 8 10 12 14 16 18 20

= 1800

01

02

03

04

05

06

07

08

09

10

Effic

ienc

y

mr = 01mr = 025mr = 05mr = 1

mr = 2mr = 4mr = 10

Figure 8 Drilling eciency of percussive drilling systems withrespect to β and mr

10 Advances in Materials Science and Engineering

4 Results and Discussion

0is study defined equations for the motion of a drill bitwhen struck by a piston and expanded the dynamic prop-erties of the drill bit based on the conditions ζ and α It alsoanalyzed the relationship between the bit dynamics and theduration of the incident stress wave (τ) Underdamped drillbits displayed the greatest responsiveness and the fastesttransmission of τ Increasing the transmission of τ decreasedthe loading effect due to the drill bitrsquos inertia but led to a fastresponse 0e inertial effect of the drill bit was confirmed tobe inversely proportional to the transmission of τ

A percussive drilling system accomplishes its work viathe percussion impact of a piston In the systems studiedhere τ ended more quickly with lower mr and β values asthese values increased the duration of τ transmitted to therock via the drill bit increased in proportion to β 0e resultsalso showed that the maximum bit displacement (um) de-creased as β increased Analysis of the drilling dynamicsconfirmed the effective intervals of β With reference to therock strengths suggested by the International Society forRock Mechanics (ISRM) the effective interval of β is con-sidered to be no greater than 2

0e main purpose of this study was to examine theimpact energy transmission rate and drilling efficiency ina percussive drilling system (Figure 8) 0e results establishedamr value of 05 as the most efficient for rocks whose strengthcorresponds to the interval βlt 2 0e improved drilling ef-ficiency would lead to advantages such as reduced bit pro-duction costs Rocks stiffer than 2ltβlt 4 can be mostefficiently drilled by selecting mr 1

0emovement and response characteristics of bit duringpercussion process τ and transfer rate of energy caused bypercussion an efficiency depend on mr and β where in-crease in mr leads to overdamping of bit movement char-acteristics due to τ by piston and reduction in drillingefficiency due to internal energy effect by mass of drill bit Asthe increase in mass of drill bit for the section of 4lt βlt 6the drilling efficiency of percussive drilling system decreaseand for the very hard rock (UCSgt 200MPa) section of4lt βlt 6 it is considered that the reasonable applied mr ofdrilling tool is 1 or 2

0is study neglected the effects of the secondary incidentstress wave and also the effect of buttons embedded in thedrill bit Additionally for analytical simplification the pistonand drill bit were assumed to be of the same diameter andmaterial and a rectangular pulse with an incident stresswave of duration τ and amplitude 05ρcv was assumed 0eeffect of the flexural stress wave depending on the config-uration was not considered 0ese limitations of this papershould be addressed in further investigations that take intoaccount the configuration of the piston and drill bit theeffect of buttons and the drilling efficiency given differingshapes of the incident stress wave

5 Conclusion

0is paper aimed to identify the optimal design parametersfor percussive drilling systems by introducing a drill bit-rock

interaction model that could verify the bitrsquos motion duringpercussion and the resulting damping characteristics 0estudy analyzed drilling efficiency and drew the followingconclusions

Percussive drilling systems have six dynamic drill bitproperties that can be expandable 0is paper discussed thephysical meaning of the dimensionless parameters α and β0eir values determined the damping characteristics thatcan lead to the rockrsquos maximum fracture displacement 0efastest response of τ was observed for underdamped drill-bitmotion

Drilling was most efficient in the interval 1lt βlt 4 whereincreasing mr at a given β decreased drilling efficiency 0evalue of mr for efficient drilling was determined by the rockstrength (ie β)

0e results indicate that application of a piston-to-drill bitmass ratio of 05 (ie a piston mass twice that of the drill bitmass) to the rocks whose stiffness corresponds to βge 2 wouldlikely be most efficient and also reduce drill bit productioncosts Furthermore a selection ofmr 1 would be valid whendeveloping drill tools for boring complex rocks (1ltβlt 4) Atβgt 4 the best efficiency could be achieved when the bit massequals or exceeds the piston mass

Appendix

Theoretical Case of Bit Motion Condition(Drilling Dynamic Model)

0is paper proposed six conditions for the dynamics ofa drill bit struck by a piston impact as summarizedbelow 0e damping conditions for the damping ratiosand α from the drill-bit equations of motion (7ndash9) are asfollows

First for the overdamped condition of ζ1 gt 1 and αlt 1 τis transmitted to the drill bit indicating loading Equation(7) can be expressed as in (A1) 0e initial conditions are0lt tle τ σi 05ρcv F ku u0 0 and _u0 0

eurou +ρcA

mb

1113888 1113889 _u +k

mb

1113888 1113889u ρcA

mb

1113888 1113889v (A1)

where assuming the normal state u can be calculated asfollows

u ρcA

k1113874 1113875v (A2)

Next we examine the case where τ is complete and theloading effect is sustained owing to external forces and drill-bit inertia Applying the conditions of τ lt tle tm σi 0F ku u0 u(τ) and _u0 _u(τ) to (A1) results in thefollowing expression

eurou +ρcA

mb

1113888 1113889 _u +k

mb

1113888 1113889u 0 (A3)

where u 0 Last we examine the unloading conditions oftm lt t σi 0 F ckuminus ckuf u0 um and _u0 0 Apply-ing these to (9) when unloading gives

Advances in Materials Science and Engineering 11

eurou +ρcA

mb

1113888 1113889 _u + ck

mb

1113888 1113889u ck

mb

1113888 1113889uf (A4)

where bit displacement u uf is used to indicate the rockrsquosfracture displacement Furthermore considering the bitrsquosdamping ratio when unloading its behavior can be ex-pressed as

Damped mode

ζ2 gt 1 αlt1c

over( )

ζ2 1 α 1c

critical( )

ζ2 lt 1 αgt1c

under( )

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A5)

Second under critical damping where ζ1 1 and α 1 τis transmitted to the drill bit indicating loading Equation(7) can be expressed as in (A1) 0e initial conditions are0lt tle τ σi 05ρcv F ku u0 0 and _u0 0

In this case assuming the normal state u can be cal-culated using (A2)

0e following is the case where τ is complete and theloading effect is sustained owing to external forces and drillbit inertia Applying the conditions of τ lt tle tm σi 0F ku u0 u(τ) and _u0 _u(τ) to (A1) give (A3) whereu 0 Last we examine the unloading condition wheretm lt t σi 0 F ckuminus ckuf u0 um and _u0 0 Equation(9) when unloading can be expressed as (A4)

In this case bit displacement u uf is used to indicate therockrsquos fracture displacement Furthermore considering thebitrsquos damping ratio when unloading its behavior follows

Damped mode

ζ2 gt 1 αlt1c

(no case)

ζ2 1 α 1c

(no case)

ζ2 lt 1 αgt1c

under( )

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A6)

0e second condition of critical damping is defined asα 1When unloading drill bits c refers to the rockrsquos propertyeffect and is always less than 1 Hence when unloading noother cases but the underdamped mode are allowed In otherwords when unloading in the critically damped mode owingto the rock properties the bit is always underdamped

0e third condition is underdamping with ζ1 lt 1 andαgt 1 Here τ is transmitted to the drill bit meaning loadingEquation (7) can be expressed by (A1)0e initial conditionsare 0lt tle τ σi 05ρcv F ku u0 0 and _u0 0 Whereassuming the normal state u can be calculated as (A2)

0e following is the case where τ is complete and theloading effect is sustained owing to external forces and drillbit inertia Applying the conditions of τ lt tle tm σi 0F ku u0 u(τ) and _u0 _u(τ) to (A1) results in (A3)

where u 0 Last we examine the unloading conditionwhere tm lt t σi 0 F ckuminus ckuf u0 um and _u0 0Equation (9) when unloading can be expressed as (A4)

In this case the drill bit displacement u uf is used toindicate the rockrsquos fracture displacement Furthermoreconsidering the damping ratio in the drill bitrsquos unloadingthe bitrsquos behavior can be expressed as

Damped mode

ζ2 gt 1 αlt1c

(no case)

ζ2 1 α 1c

(no case)

ζ2 lt 1 αgt1c

under( )

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A7)

0e third condition is the critically damped modedefined as αgt 1 During drill bit unloading c refers to therock property effect and is always less than 1 Hencewhen unloading no other cases but underdamping areallowed

Next the underdamped condition is where ζ1 lt 1 andαgt 1 and τ is slower than tm (the time at which the rockrsquosmaximum fracture displacement is reached) that is tm lt τIn this case τ is transmitted to the drill bit meaningloading Equation (7) can be expressed by (A1) 0e initialconditions are 0lt tle tm σi 05ρcv F ku u0 0 and_u0 0 Here assuming the normal state u can be calculatedas (A2)

0e following case is where conversion is made tounloading after the drill bit reaches its maximum dis-placement but the incident stress wave is sustained Ap-plying the conditions of tm lt tle τ σi 05ρcvF ckuminus ckuf u0 um and _u0 0 to (9) results in

eurou +ρcA

mb

1113888 1113889 _u + ck

mb

1113888 1113889u ρcA

mb

1113888 1113889v + ck

mb

1113888 1113889uf (A8)

where the definition of u 1c(ρcAk)v + uf is possibleLast this unloading condition is where the incident

stress wave is resolved When unloading the drill bit usingthe conditions tlt τ σi 0 F ckuminus ckuf u0 u(τ) and_u0 _u(τ) allows (9) to be expressed as (A4)

In this case drill bit displacement u uf is used to in-dicate the rockrsquos fracture displacement 0e analysis of drillbit dynamics showed that when loading transitions tounloading in the underdamped condition the dynamicproperties of the bit exist only in the underdamping mode

Conflicts of Interest

The authors declare that there are no conflicts of interest

References

[1] C H Song K B Kwon J Y Park et al ldquoOptimum design ofthe internal flushing channel of a drill bit using RSM and CFDsimulationrdquo International Journal of Precision Engineeringand Manufacturing vol 15 no 6 pp 1041ndash1050 2014

12 Advances in Materials Science and Engineering

[2] X Li G Rupert D A Summers P Santi and D Liu ldquoAnalysisof impact hammer rebound to estimate rock drillabilityrdquo RockMechanics and Rock Engineering vol 33 no 1 pp 1ndash13 2000

[3] W A Hustrulid and C Fairhurst ldquoA theoretical and ex-perimental study of the percussive drilling of rock part Itheory of percussive drillingrdquo International Journal of RockMechanics and Mining Sciences amp Geomechanics Abstractsvol 8 no 4 pp 311ndash333 1971

[4] W A Hustrulid and C Fairhurst ldquoA theoretical and ex-perimental study of the percussive drilling of rock part IIforce-penetration and specific energy determinationrdquo In-ternational Journal of Rock Mechanics and Mining Sciences ampGeomechanics Abstracts vol 8 no 4 pp 335ndash356 1971

[5] W A Hustrulid and C Fairhurst ldquoA theoretical and ex-perimental study of the percussive drilling of rock part IIIexperimental verification of the mathematical theoryrdquo In-ternational Journal of Rock Mechanics and Mining Sciences ampGeomechanics Abstracts vol 9 no 3 pp 417-418 1972

[6] W A Hustrulid and C Fairhurst ldquoA theoretical and ex-perimental study of the percussive drilling of rock part IVapplication of the model to actual percussive drillingrdquo In-ternational Journal of Rock Mechanics and Mining Sciencesvol 9 no 3 pp 431ndash442 1972

[7] L E Chiang and D A Elias ldquoA 3D FEM methodology forsimulating the impact in rock-drilling hammersrdquo In-ternational Journal of Rock Mechanics and Mining Sciencesvol 45 no 5 pp 701ndash711 2008

[8] C H Song K B Kwon M G Cho J Y Oh D Y Shin andJ W Cho ldquoDevelopment of lab-scale rock drill apparatus fortesting performance of a drill bitrdquo International Journal ofPrecision Engineering and Manufacturing vol 16 no 7pp 1402ndash1414 2015

[9] K B Kwon C H Song J Y Park J Y Oh J W Lee andJ W Cho ldquoEvaluation of drilling efficiency by percussiontesting of a drill bit with new button arrangementrdquo In-ternational Journal of Precision Engineering andManufacturing vol 15 no 6 pp 1063ndash1068 2014

[10] B Lundberg andM Okrouhlik ldquoInfluence of 3D effects on theefficiency of percussive rock drillingrdquo International Journal ofimpact Engineering vol 25 no 4 pp 345ndash360 2001

[11] B Lundberg and P Collet ldquoOptimal wave shape with respectto efficiency in percussive drilling with detachable drill bitrdquoInternational Journal of impact Engineering vol 86 pp 179ndash187 2015

[12] X B Li G Rupert and D A Summers ldquoEnergy transmissionof down-hole hammer tool and its conditionalityrdquo Trans-actions of Nonferrous Metals Society of China vol 10 no 1pp 109ndash111 2000

[13] J W Cho S Jeon S H Yu and S H Chang ldquoOptimumspacing of TBM disc cutters a numerical simulation using thethree-dimensional dynamic fracturing methodrdquo Tunnellingand Underground Space Technology vol 25 no 3 pp 230ndash244 2010

[14] W Changming ldquoAn analytical study of percussive energytransfer in hydraulic rock drillsrdquo Mining Science and Tech-nology vol 13 no 1 pp 57ndash68 1991

[15] Online Materials Information Resource httpwwwmatwebcomsearchDataSheetaspxMatGUID8bc5d558f4174e6082ddf4966e382bd6ampckck1

Advances in Materials Science and Engineering 13

For the loading condition we obtain the followingequations [2]

A σi + σr( ) F

du

dtσiρcminusσrρc

ku F

(5)

where the initial condition is t 0 andF 0For the unloading condition the equations can be expressed

as follows [2]

A σi + σr( ) F

du

dtσiρcminusσrρc

ckuminus cku F

(6)

where the initial condition is t τ and F(t) limtrarr τ

F(t)e response characteristics of a drill bit (ie the damped

characteristics) can be under critical or overdamped given itsmotion equation that is (1) e movement of the drill bitunder loading and unloading conditions was examined edamping characteristics of a percussive drilling system arecategorized depending on the damping ratio (ζ) and thedimensionless parameter α

When loading the equation of motion for the drill bit isdened as follows

eurou +ρcAmb

( ) _u +k

mb( )u

2Amb( )σi (7)

where the response-characteristics determinant ζ1 is

ζ1 4α(βτ)

24α(βτ)2radic

1α

radic (8)

When unloading the drill-bit equation of motion isdened by (9) e ζ2 term is given in (10)

eurou +ρcAmb

( ) _u + ck

mb( )u

2Amb( )σi + c

kufmb

(9)

ζ2 4α(βτ)

2c(4α)(βτ)2radic

1cαradic (10)

e dimensionless parameters α and β dened in (8) and(10) can be expressed as follows (extended from (2) (12))

α 4mb

ρcAmiddotk

ρcA

4mb

ρcAτmiddot β

4mb

2mpLPcLPAβ 2

mb

mp( )β

β kτρcA

k2LpcρcA

2kLhρc2A

2kLp

ρA Epρ( ) 2k

LpEpA( ) 2

k

kp( )

(11)

As explained earlier the typical motion of drill bits in-volves four conditions However this study considers sixconditions as listed in Table 1 (Appendix A) and includes theloadingunloading condition and the presence or absence ofan incident stress wave To investigate the rate at which theimpact energy is transmitted from the piston to the rockduring the percussion process (ie the drilling eciency) thisstudy rst considered the response characteristics resultingfrom the motion of the drill bit during the percussion processand the e ect of the rock e aim was to nd the responsecharacteristics that lead to the rockrsquos maximum crushingdisplacement um e study also dened the motion ofpercussive drill bits resulting from the impact of the pistonand analyzed their behavior given the system responses edrill bit-rock interaction was considered here including theforcendashpenetration relationship for the rock

Table 1 presents the relationships between loading andunloading and the duration of stress wave τ in percussivedrilling systems In Table 1 τ is the duration of the incidentstress wave being transmitted to the drill bit via the pistonduring loading it is referenced relative to the time t eloading condition can be dened in two ways 0lt tle τ orτ lt tle tm where the latter expression means that τ arrivedsooner than t that is the transmission of τ was complete butthe loading continued owing to the inertial e ect of the drillbit If τ gt tm the drill bit arrived at um before the transmissionof τ to the bit had completed and indicates the time when thesystem switches to the unloading condition

In summary in Cases 1 to 5 in Table 1 the motion of thedrill bit following the piston impact continues until theloading condition applying external force to the rock reachesτ lt tle tm After that time the stress wave arriving at the drillbit is converted into unloading at τ gt tm provided that therock displacement occurs sooner than tm that is themaximum time of arrival However the τ in Case 6 rep-resents a stress wave that is still progressing while the rockdisplacement owing to the drill bit has already reached itsmaximum that is unloading occurs during τ

F

0 u

Fm

um

k

uf

k

Figure 3 Force versus penetration relationship representing thebitndashrock interaction curve

4 Advances in Materials Science and Engineering

23 Results of BitMotionAnalysis Numerical simulations ona dynamic model of drill bit properties were run for the sixcases and used the dynamic drill bit properties during inducedpercussion (Section 2) the drill bit-rock interaction modeland the dimensionless parameters α and β

Figure 4 shows the simulation results (ζ) for differentconditions (α Table 1) and the effect of loading andunloading 0e mass ratio mr was 1 and the same con-ditions were applied to the piston mass and drill bit massFigure 4(a) shows an overview of the response characteristicsfor all six cases 0e initial black part of each curve refers tothe sustained effect of τ the red parts indicate the loadingcondition that marks the effect of drill bit inertia and theblue parts represent the transition to unloading Figure 4(b)shows Case 1 where a single-incident stress wave indicatesthe drill bit displacement at u(τ) and the drill bit arrival at umas an overdamped motion due to the loading continuingowing to external forces and inertia Later in the transitionto unloading characteristic overdamped motion appearsand there is no convergence to uf (Figure 3)

Case 2 (Figure 4(c)) appears similar to Case 1 but the bitreaches um sooner owing to τ and the effect of drill bitinertia As the system transitions to unloading and convergesto uf it shows critical damping Case 3 (Figure 4(d)) showsa longer τ compared with Case 2 because of its quickerarrival at um During unloading it converges to uf asunderdamped motion Case 4 (Figure 4(e)) has the drill bitdisplaced at u(τ) due to τ and it arrives at um as criticallydamped motion owing to continued loading resulting fromthe inertia of the drill bit 0e subsequent unloading phaseconverges to uf as underdamped motion

Case 5 (Figure 4(f)) has a longer τ than the preceding fourcases and also reaches um sooner Furthermore all its zonesshowed underdamped motion Finally in Case 6 (Figure 4(g)) the rock displacement has already reached its maximum

while τ is still in progress 0is means that unloading beginsduring τ Underdamped motion was observed in all zones

Overall the longer the duration of τ the shorter theloading time and the sooner the um is reached owing to drillbit inertia Table 2 summarizes the damping characteristics forthe loading and unloading conditions and presents the as-sociated damping mode that arrives at um Cases 1ndash3 showoverdamping Case 4 critical damping and Cases 5 and 6underdamping 0e underdamped mode showed thehighest responsiveness indicating the fastest trans-mission time of τ 0e results show that increasing theτ-transmission time decreased the loading effect resultingfrom drill bit inertia and led to faster responses 0e drillbit inertial effect is inversely proportional to τ

3 Energy Transmission in Percussive DrillingSystem

In Section 2 we examined the response characteristics ofdrill bits that can arrive at um which is themaximum drillingdisplacement of rocks Based on the examined character-istics in Section 3 we calculate the efficiency of percussivedrilling systems and discuss the combinations of parametersthat could maximize it

31 Energy Transmission Efficiency In a percussive drillingsystem the piston directly hits the drill bit at speed v (Figure 2)For a transmitted impact energy Ei at the time of the pistondescending and hitting the drill bit the final speed of the pistonimmediately preceding the impact v can be expressed asfollows [2 10]

v 2Ei

mp

1113888 1113889

12

(12)

Table 1 Expansion conditions for dynamic drill-bit properties for loading unloading and stress-wave duration τ

Case no

Initial conditions0lt tle τ τ lt tle tm tgt tm

σi 05ρcv

F ku

u0 0 _u0 0

σi 0F ku

u0 u(τ) _u0 _u(τ)

σi 0F ckuminus ckuf

u0 um _u0 0

1 αlt 1c

2 α 1c

3 1clt αlt 1

4 α 1

5 αgt 1 tm gt τ

6

0lt tle tm tm lt tle τ tgt τ

σi 05ρcv

F ku

u0 0 _u0 0

σi 05ρcv

F ckuminus ckuf

u0 um _u0 0

σi 0F F ckuminus ckuf

u0 u(τ) _u0 _u(τ)

Advances in Materials Science and Engineering 5

0000

uf

u

00

04

08

12

16

20

0001 0002 t0003 0004

Case 1Case 2Case 3Case 4Case 5Case 6

Loading by incident stress waveLoading by inertia of bitUnloading

(a)

tm

uf

um

u

t

Loading by incident stress waveLoading by inertia of bitUnloading

00040003

Overdamped motion

00020001000000

04

08

12

16

20

u()

(b)

u()

tm

uf

um

u

t

000400030002

Overdamped motion

Critically damped motion

0001000000

04

08

12

16

20

Loading by incident stress waveLoading by inertia of bitUnloading

(c)

u()

Overdamped motion

Underdamped motion

tm

uf

um

u

t

0004000300020001000000

04

08

12

16

20

Loading by incident stress waveLoading by inertia of bitUnloading

(d)

Figure 4 Continued

6 Advances in Materials Science and Engineering

e initial height h0 is v22g and the rebound eh canbe dened as the ratio between the initial height and thepistonrsquos postimpact rebound height h ese areexpressed as follows [2]

eh h

h0v2ev2 (13)

e eciency η of impact energy transmission is de-ned as the ratio between the kinetic energy generated fromthe impact of the piston and the energy transmitted to therock [12] and is given as

η kum

2

ρALhv2

Fmax( )2

ρALhv2k (14)

Which can be expressed as follows through dimensionalanalysis

η 2βu2m (15)

e value um during loading can be obtained from (7)the equation of motion for the drill bit e obtained um canbe expressed as in (16) for each damping mode where theinitial conditions are 0lt tle τ F kui u0 0 and _u0 0

u()

Critically damped motion

Underdamped motion

tm

uf

um

u

t

0004000300020001000000

04

08

12

16

20

Loading by incident stress waveLoading by inertia of bitUnloading

(e)

Loading by incident stress waveLoading by inertia of bitUnloading

u()Underdamped motion

tm

uf

um

u

t

0004000300020001000000

04

08

12

16

20

(f)

Loading by incident stress waveLoading by inertia of bitUnloading

u()Underdampedmotion

uf

u

u

t

t

000400030002

00002 00003 00004

0001000000

04

08

12

16

20

um

um

tm

(g)

Figure 4 Results of the motion response of the bit formr 1 (a) All cases (b) Case 1 (c) Case 2 (d) Case 3 (e) Case 4 (f) Case 5 (g) Case 6

Advances in Materials Science and Engineering 7

um(t)

1 +1minus

αminus 1

radic

2αminus 1

radic middot eminus(2α)(βτ)(1+

1minusα

radic)t minus

1 +1minus α

radic

21minus α

radic middot eminus(2α)(βτ)(1minus

1minusα

radic)t

1113890 1113891(overdamped)

1minus eminus2(βτ)t middot 2βτ

t + 11113890 1113891(critical damped)

1minus eminus2(βτ)t middot1

αminus 1

radic sin2

αminus 1

radic

αβτ

t1113888 1113889 + cos2

αminus 1

radic

αβτ

t1113888 11138891113890 1113891(underdamped)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(16)

Moreover for tmlt τ um is expressed as follows

um(t) 1minus eminus(2α)(βτ)

middot ⎡⎣1

αminus 1

radic sin2

αminus 1

radic

αβτ

t1113888 1113889

+ cos2

αminus 1

radic

αβτ

t1113888 1113889⎤⎦

(17)

0e initial conditions are 0lt tle τ F kui ui 0and _ui 0

Next the displacement value uf during unload-ing can be obtained via the equation of motion for a drill-bit (9) as shown in (18) 0e initial conditionsduring unloading are tgt τ F 0 u0 u(τ) and_u0 _u(τ)

uf(t)

α4

1minus α

radicτβ

middot eminus(2α)(1+

1minusα

radic)t

middot u02αβτ

(1 +1minus α

radic) minus _u0 +

4αβτu01113888 11138891113890 1113891

minusα

41minus α

radicτβ

middot eminus(2α)(1minus

1minusα

radic)t

middot u02αβτ

(1minus1minus α

radic) minus _u0 +

4αβτu01113888 11138891113890 1113891(overdamped)

eminus2(βτ)t middot _u0 + 2βτu01113888 1113889tminus u01113890 1113891(critical damped)

eminus(2α)(βτ)t middot ⎡⎣α

2αminus 1

radicτβ

1113888 1113889 _u0 middot sin2

αminus 1

radic

α1113888 1113889

βτ

t +1

αminus 1

radic u0 middot sin2

αminus 1

radic

αβτ

t1113888 1113889

+ u0 middot cos2

αminus 1

radic

αβτ

t1113888 1113889⎤⎦(underdamped)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(18)

Table 2 Dynamic drill-bit properties for loading unloading and τ and the damping modes capable of reaching um

Case no ConditionsDamping modes capable of reaching um

0lt tle τ τ lt tle tm tgt tm

1 αlt 1c

Over Overlowast Over2 α 1

cOver Overlowast Critical

3 1clt αlt 1 Over Overlowast Under

4 α 1 Critical Criticallowast Under5 αgt 1 tm gt τ Under Underlowast Under

6 αgt 1 tm lt τ0lt tle tm tm lt tle τ tgt τUnderlowast Under Under

lowastDamping modes capable of reaching um

8 Advances in Materials Science and Engineering

eeciency of the percussive drilling systems for given βcan be calculated using (14ndash17) e factor um which isimportant to the eciency calculations is discussed in depth

32 Results of Energy Transmission Eciency e results ofsimulations on the drill bit-rock interaction model shown inFigure 5 indicate the damped modes that can arrive at um (forvarying values of β) as the response by the drill bit at the time ofpiston impact e damped mode is determined by ζ and αwhich indictate the system responses Overdamped mode hasαlt 1 and can be expressed as follows

2mb

mh( )βlt 1

βlt1

2mr

(19)

e critical damped mode has α 1 and is expressed asfollows

2mb

mh( )β 1

β 1

2mr

(20)

e underdamped mode has αgt 1 and is expressed as

2mb

mh( )βgt 1

βgt1

2mr

(21)

where mbmh mr

e special underdamped mode refers to the case whereτ is faster than t indicating that the drill bit can arrive at umonly via an incident stress wave In this case the bit reachesits maximum displacement and transitions to the unloadingstate while the transmission of τ is underway

e damped mode of each mr shown in Figure 5 re-veals that the underdamped mode is dominant of thee ects Also overdamped and critical damped were ob-served where β was low is is possibly because of thee ects of the internal energy (ie compression strength)of rock and the percussive energy e overdamped andcritical damped were also observed is is possibly be-cause of internal energy (ie compression strength) ofrock and energy caused by percussion

Figure 6 shows dimensionless time tm tτ capable ofreaching Um for β and τ In Section A of tgt τ the bit reachesUm after termination of τ and in Section B of tlt τ the bitreachesUm before the termination of τe lower themr andβ the faster the termination of τ and increase in thesetended to make the duration of τ delivered to rock throughbit to increase proportionally to β is suggests that thehigher rock strength requires longer duration of τ fordrilling the rock

Figure 7 shows dimensionless Um state dened as ratioum a displacement um by β and uinfin caused by incident stresswave and loading condition e prediction of displacementof drill bit drilling displacement by characteristic of rockwas limited is study accordingly judged the di-mensionless drilling state Um only by using ratio uinfina displacement during loading and um considering inertiacaused during percussion (22)

Um umuinfin (22)

where the uinfin only by loading is dened as follows

mr = 1

mr = 2

mr = 05

0 5 10

15 20

4

3

2

Dam

ped

mod

es

1

0

mr = 1mr = 2

mr = 05

mr = 1

mr = 05

Figure 5 Damping mode capable of reaching maximum rockfracture displacement um given the piston-to-bit mass ratiomr and β1 overdamped mode 2 critically damped mode 3 underdampedmode 4 special underdamped mode

0 5 10

15 2000

02

06

10

14

18

20

04

08

12

16

Dim

ensio

nles

s pen

etra

tion

(Um

)

Section B

Section A

mr = 05mr = 1mr = 2

uinfin

Figure 6 State of reaching drill bit dimensionless penetration Umwith respect to mass ratio mr and β

Advances in Materials Science and Engineering 9

uinfin ρcAkv

τβv (23)

e e ect of initial condition on uinfin decreased with theincrease in β e um is determined by considering loadingcondition and inertia e ect is a ected by overresponse anddecreases as β increases

Section A refers to umlt uinfin with only τ and loadinge ect present (ie the condition for the normal state in theinitial mode) uinfin means only the τ e ect resulting fromloading exists Section B refers to umgt uinfin a state in whichum can be reached through τ and loading is exceeded issection is where the drill bit inertia and underdampedmotion appear e gradient of the curve for maximumdisplacement of the bit varies with β indicating that as therock strength increases the displacement allowing fracturedecreases Near value 2 β changes its gradient for maximumdisplacement is means that in Section A the rock is lesssti than the piston owing to the rock-sti ness e ect and inSection B the rock is sti er than the piston β is dened asthe sti ness ratio between the rock and the piston as in (11)Hence when βlt 2 a condition is established for rock fractureto occur in that the piston sti ness exceeds the rock sti ness

In examining um for piston-bit mass ratios of mrlt 2 therock strength that can be defeated by τ occurring in theinitial loading tends to increase is tendency is suspectedto result from the inertial e ect of the mass of the bit Forβgt 4 the um gradient tended to decrease with increasingmass of the bit Hence the eciency is considered to de-crease which is attributed to the e ect of the rockrsquos com-pressive strength e uniaxial compressive strength of veryhard rock is typically at least 200MPa while the tensilestrength of the H13 tool steels that are used mostly forpistons (ASTM A681 and DIN EN ISO 4957) is known to beat least 359ndash1170MPa [15]

Given the steels generally used to make pistons thee ective range of β is considered to be 034lt βlt 112 which

was calculated considering the strength ratio between thehard rock and an H13 steel tool Hence the e ective range ofβ could be no greater than 2 Rocks with a sti ness ratio ofβgt 2 might cause failure through the pistonrsquos plastic de-formation and fracture is study examined the e ectiveinterval of β and investigated the parameter combinationsgiving high drilling eciency in this interval

Figure 8 illustrates the motion of the bit generatedduring the percussion process and the energy transmissionrate (ie eciency) for the response characteristics τ andpercussion impact Analysis of mr found the maximumdrilling eciency in interval 1lt βlt 2 with a tendency ef-ciency decreasing after mr increased above a certain valueis result was attributed to the tendency whereby increasingmr converts the systemrsquos dynamic response to τ due to thepiston from underdamped to overdamped Furthermoreincreasing the mass of the bit can increase its internal energy(ie friction energy) to above the piston transmitted energy(ie kinetic and potential energy) Hence the bitrsquos dynamicresponse would be overdamped and could compromisedrilling eciency In other words the heavier the bit themore delayed the response to percussion

In Figure 8 the left side of red line (β18) represents thee ect of only incident stress wave meaning that the higherthe value in graph the higher the drilling eciency in in-cident stress wave e right side of red line representingunloading behavior after reaching maximum eciency inincident stress wave is a ected by both brittleness char-acteristics of rock and inertia e ect of drill bit

e piston should have twice the mass of the drill bit(iemr 05) for the most ecient drilling for βle 2 which isestablished to be the e ective interval of β and correspondsto both soft and hard rocks Drilling sti er rocks (2lt βlt 4)might be less ecient but increasing mr to 1 would helpimprove drilling eciency

00000

00002

00004

00006

00008

00010

00012

Dim

ensio

nles

s tim

e (t m

)

Section A

Section B

0 5 10

15 20

mr = 05mr = 1mr = 2

Figure 7 Dimensionless time capable of reaching Um for an in-cident stress-wave τ acting on the bit and β

0 2 4 6 8 10 12 14 16 18 20

= 1800

01

02

03

04

05

06

07

08

09

10

Effic

ienc

y

mr = 01mr = 025mr = 05mr = 1

mr = 2mr = 4mr = 10

Figure 8 Drilling eciency of percussive drilling systems withrespect to β and mr

10 Advances in Materials Science and Engineering

4 Results and Discussion

0is study defined equations for the motion of a drill bitwhen struck by a piston and expanded the dynamic prop-erties of the drill bit based on the conditions ζ and α It alsoanalyzed the relationship between the bit dynamics and theduration of the incident stress wave (τ) Underdamped drillbits displayed the greatest responsiveness and the fastesttransmission of τ Increasing the transmission of τ decreasedthe loading effect due to the drill bitrsquos inertia but led to a fastresponse 0e inertial effect of the drill bit was confirmed tobe inversely proportional to the transmission of τ

A percussive drilling system accomplishes its work viathe percussion impact of a piston In the systems studiedhere τ ended more quickly with lower mr and β values asthese values increased the duration of τ transmitted to therock via the drill bit increased in proportion to β 0e resultsalso showed that the maximum bit displacement (um) de-creased as β increased Analysis of the drilling dynamicsconfirmed the effective intervals of β With reference to therock strengths suggested by the International Society forRock Mechanics (ISRM) the effective interval of β is con-sidered to be no greater than 2

0e main purpose of this study was to examine theimpact energy transmission rate and drilling efficiency ina percussive drilling system (Figure 8) 0e results establishedamr value of 05 as the most efficient for rocks whose strengthcorresponds to the interval βlt 2 0e improved drilling ef-ficiency would lead to advantages such as reduced bit pro-duction costs Rocks stiffer than 2ltβlt 4 can be mostefficiently drilled by selecting mr 1

0emovement and response characteristics of bit duringpercussion process τ and transfer rate of energy caused bypercussion an efficiency depend on mr and β where in-crease in mr leads to overdamping of bit movement char-acteristics due to τ by piston and reduction in drillingefficiency due to internal energy effect by mass of drill bit Asthe increase in mass of drill bit for the section of 4lt βlt 6the drilling efficiency of percussive drilling system decreaseand for the very hard rock (UCSgt 200MPa) section of4lt βlt 6 it is considered that the reasonable applied mr ofdrilling tool is 1 or 2

0is study neglected the effects of the secondary incidentstress wave and also the effect of buttons embedded in thedrill bit Additionally for analytical simplification the pistonand drill bit were assumed to be of the same diameter andmaterial and a rectangular pulse with an incident stresswave of duration τ and amplitude 05ρcv was assumed 0eeffect of the flexural stress wave depending on the config-uration was not considered 0ese limitations of this papershould be addressed in further investigations that take intoaccount the configuration of the piston and drill bit theeffect of buttons and the drilling efficiency given differingshapes of the incident stress wave

5 Conclusion

0is paper aimed to identify the optimal design parametersfor percussive drilling systems by introducing a drill bit-rock

interaction model that could verify the bitrsquos motion duringpercussion and the resulting damping characteristics 0estudy analyzed drilling efficiency and drew the followingconclusions

Percussive drilling systems have six dynamic drill bitproperties that can be expandable 0is paper discussed thephysical meaning of the dimensionless parameters α and β0eir values determined the damping characteristics thatcan lead to the rockrsquos maximum fracture displacement 0efastest response of τ was observed for underdamped drill-bitmotion

Drilling was most efficient in the interval 1lt βlt 4 whereincreasing mr at a given β decreased drilling efficiency 0evalue of mr for efficient drilling was determined by the rockstrength (ie β)

0e results indicate that application of a piston-to-drill bitmass ratio of 05 (ie a piston mass twice that of the drill bitmass) to the rocks whose stiffness corresponds to βge 2 wouldlikely be most efficient and also reduce drill bit productioncosts Furthermore a selection ofmr 1 would be valid whendeveloping drill tools for boring complex rocks (1ltβlt 4) Atβgt 4 the best efficiency could be achieved when the bit massequals or exceeds the piston mass

Appendix

Theoretical Case of Bit Motion Condition(Drilling Dynamic Model)

0is paper proposed six conditions for the dynamics ofa drill bit struck by a piston impact as summarizedbelow 0e damping conditions for the damping ratiosand α from the drill-bit equations of motion (7ndash9) are asfollows

First for the overdamped condition of ζ1 gt 1 and αlt 1 τis transmitted to the drill bit indicating loading Equation(7) can be expressed as in (A1) 0e initial conditions are0lt tle τ σi 05ρcv F ku u0 0 and _u0 0

eurou +ρcA

mb

1113888 1113889 _u +k

mb

1113888 1113889u ρcA

mb

1113888 1113889v (A1)

where assuming the normal state u can be calculated asfollows

u ρcA

k1113874 1113875v (A2)

Next we examine the case where τ is complete and theloading effect is sustained owing to external forces and drill-bit inertia Applying the conditions of τ lt tle tm σi 0F ku u0 u(τ) and _u0 _u(τ) to (A1) results in thefollowing expression

eurou +ρcA

mb

1113888 1113889 _u +k

mb

1113888 1113889u 0 (A3)

where u 0 Last we examine the unloading conditions oftm lt t σi 0 F ckuminus ckuf u0 um and _u0 0 Apply-ing these to (9) when unloading gives

Advances in Materials Science and Engineering 11

eurou +ρcA

mb

1113888 1113889 _u + ck

mb

1113888 1113889u ck

mb

1113888 1113889uf (A4)

where bit displacement u uf is used to indicate the rockrsquosfracture displacement Furthermore considering the bitrsquosdamping ratio when unloading its behavior can be ex-pressed as

Damped mode

ζ2 gt 1 αlt1c

over( )

ζ2 1 α 1c

critical( )

ζ2 lt 1 αgt1c

under( )

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A5)

Second under critical damping where ζ1 1 and α 1 τis transmitted to the drill bit indicating loading Equation(7) can be expressed as in (A1) 0e initial conditions are0lt tle τ σi 05ρcv F ku u0 0 and _u0 0

In this case assuming the normal state u can be cal-culated using (A2)

0e following is the case where τ is complete and theloading effect is sustained owing to external forces and drillbit inertia Applying the conditions of τ lt tle tm σi 0F ku u0 u(τ) and _u0 _u(τ) to (A1) give (A3) whereu 0 Last we examine the unloading condition wheretm lt t σi 0 F ckuminus ckuf u0 um and _u0 0 Equation(9) when unloading can be expressed as (A4)

In this case bit displacement u uf is used to indicate therockrsquos fracture displacement Furthermore considering thebitrsquos damping ratio when unloading its behavior follows

Damped mode

ζ2 gt 1 αlt1c

(no case)

ζ2 1 α 1c

(no case)

ζ2 lt 1 αgt1c

under( )

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A6)

0e second condition of critical damping is defined asα 1When unloading drill bits c refers to the rockrsquos propertyeffect and is always less than 1 Hence when unloading noother cases but the underdamped mode are allowed In otherwords when unloading in the critically damped mode owingto the rock properties the bit is always underdamped

0e third condition is underdamping with ζ1 lt 1 andαgt 1 Here τ is transmitted to the drill bit meaning loadingEquation (7) can be expressed by (A1)0e initial conditionsare 0lt tle τ σi 05ρcv F ku u0 0 and _u0 0 Whereassuming the normal state u can be calculated as (A2)

0e following is the case where τ is complete and theloading effect is sustained owing to external forces and drillbit inertia Applying the conditions of τ lt tle tm σi 0F ku u0 u(τ) and _u0 _u(τ) to (A1) results in (A3)

where u 0 Last we examine the unloading conditionwhere tm lt t σi 0 F ckuminus ckuf u0 um and _u0 0Equation (9) when unloading can be expressed as (A4)

In this case the drill bit displacement u uf is used toindicate the rockrsquos fracture displacement Furthermoreconsidering the damping ratio in the drill bitrsquos unloadingthe bitrsquos behavior can be expressed as

Damped mode

ζ2 gt 1 αlt1c

(no case)

ζ2 1 α 1c

(no case)

ζ2 lt 1 αgt1c

under( )

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A7)

0e third condition is the critically damped modedefined as αgt 1 During drill bit unloading c refers to therock property effect and is always less than 1 Hencewhen unloading no other cases but underdamping areallowed

Next the underdamped condition is where ζ1 lt 1 andαgt 1 and τ is slower than tm (the time at which the rockrsquosmaximum fracture displacement is reached) that is tm lt τIn this case τ is transmitted to the drill bit meaningloading Equation (7) can be expressed by (A1) 0e initialconditions are 0lt tle tm σi 05ρcv F ku u0 0 and_u0 0 Here assuming the normal state u can be calculatedas (A2)

0e following case is where conversion is made tounloading after the drill bit reaches its maximum dis-placement but the incident stress wave is sustained Ap-plying the conditions of tm lt tle τ σi 05ρcvF ckuminus ckuf u0 um and _u0 0 to (9) results in

eurou +ρcA

mb

1113888 1113889 _u + ck

mb

1113888 1113889u ρcA

mb

1113888 1113889v + ck

mb

1113888 1113889uf (A8)

where the definition of u 1c(ρcAk)v + uf is possibleLast this unloading condition is where the incident

stress wave is resolved When unloading the drill bit usingthe conditions tlt τ σi 0 F ckuminus ckuf u0 u(τ) and_u0 _u(τ) allows (9) to be expressed as (A4)

In this case drill bit displacement u uf is used to in-dicate the rockrsquos fracture displacement 0e analysis of drillbit dynamics showed that when loading transitions tounloading in the underdamped condition the dynamicproperties of the bit exist only in the underdamping mode

Conflicts of Interest

The authors declare that there are no conflicts of interest

References

[1] C H Song K B Kwon J Y Park et al ldquoOptimum design ofthe internal flushing channel of a drill bit using RSM and CFDsimulationrdquo International Journal of Precision Engineeringand Manufacturing vol 15 no 6 pp 1041ndash1050 2014

12 Advances in Materials Science and Engineering

[2] X Li G Rupert D A Summers P Santi and D Liu ldquoAnalysisof impact hammer rebound to estimate rock drillabilityrdquo RockMechanics and Rock Engineering vol 33 no 1 pp 1ndash13 2000

[3] W A Hustrulid and C Fairhurst ldquoA theoretical and ex-perimental study of the percussive drilling of rock part Itheory of percussive drillingrdquo International Journal of RockMechanics and Mining Sciences amp Geomechanics Abstractsvol 8 no 4 pp 311ndash333 1971

[4] W A Hustrulid and C Fairhurst ldquoA theoretical and ex-perimental study of the percussive drilling of rock part IIforce-penetration and specific energy determinationrdquo In-ternational Journal of Rock Mechanics and Mining Sciences ampGeomechanics Abstracts vol 8 no 4 pp 335ndash356 1971

[5] W A Hustrulid and C Fairhurst ldquoA theoretical and ex-perimental study of the percussive drilling of rock part IIIexperimental verification of the mathematical theoryrdquo In-ternational Journal of Rock Mechanics and Mining Sciences ampGeomechanics Abstracts vol 9 no 3 pp 417-418 1972

[6] W A Hustrulid and C Fairhurst ldquoA theoretical and ex-perimental study of the percussive drilling of rock part IVapplication of the model to actual percussive drillingrdquo In-ternational Journal of Rock Mechanics and Mining Sciencesvol 9 no 3 pp 431ndash442 1972

[7] L E Chiang and D A Elias ldquoA 3D FEM methodology forsimulating the impact in rock-drilling hammersrdquo In-ternational Journal of Rock Mechanics and Mining Sciencesvol 45 no 5 pp 701ndash711 2008

[8] C H Song K B Kwon M G Cho J Y Oh D Y Shin andJ W Cho ldquoDevelopment of lab-scale rock drill apparatus fortesting performance of a drill bitrdquo International Journal ofPrecision Engineering and Manufacturing vol 16 no 7pp 1402ndash1414 2015

[9] K B Kwon C H Song J Y Park J Y Oh J W Lee andJ W Cho ldquoEvaluation of drilling efficiency by percussiontesting of a drill bit with new button arrangementrdquo In-ternational Journal of Precision Engineering andManufacturing vol 15 no 6 pp 1063ndash1068 2014

[10] B Lundberg andM Okrouhlik ldquoInfluence of 3D effects on theefficiency of percussive rock drillingrdquo International Journal ofimpact Engineering vol 25 no 4 pp 345ndash360 2001

[11] B Lundberg and P Collet ldquoOptimal wave shape with respectto efficiency in percussive drilling with detachable drill bitrdquoInternational Journal of impact Engineering vol 86 pp 179ndash187 2015

[12] X B Li G Rupert and D A Summers ldquoEnergy transmissionof down-hole hammer tool and its conditionalityrdquo Trans-actions of Nonferrous Metals Society of China vol 10 no 1pp 109ndash111 2000

[13] J W Cho S Jeon S H Yu and S H Chang ldquoOptimumspacing of TBM disc cutters a numerical simulation using thethree-dimensional dynamic fracturing methodrdquo Tunnellingand Underground Space Technology vol 25 no 3 pp 230ndash244 2010

[14] W Changming ldquoAn analytical study of percussive energytransfer in hydraulic rock drillsrdquo Mining Science and Tech-nology vol 13 no 1 pp 57ndash68 1991

[15] Online Materials Information Resource httpwwwmatwebcomsearchDataSheetaspxMatGUID8bc5d558f4174e6082ddf4966e382bd6ampckck1

Advances in Materials Science and Engineering 13

23 Results of BitMotionAnalysis Numerical simulations ona dynamic model of drill bit properties were run for the sixcases and used the dynamic drill bit properties during inducedpercussion (Section 2) the drill bit-rock interaction modeland the dimensionless parameters α and β

Figure 4 shows the simulation results (ζ) for differentconditions (α Table 1) and the effect of loading andunloading 0e mass ratio mr was 1 and the same con-ditions were applied to the piston mass and drill bit massFigure 4(a) shows an overview of the response characteristicsfor all six cases 0e initial black part of each curve refers tothe sustained effect of τ the red parts indicate the loadingcondition that marks the effect of drill bit inertia and theblue parts represent the transition to unloading Figure 4(b)shows Case 1 where a single-incident stress wave indicatesthe drill bit displacement at u(τ) and the drill bit arrival at umas an overdamped motion due to the loading continuingowing to external forces and inertia Later in the transitionto unloading characteristic overdamped motion appearsand there is no convergence to uf (Figure 3)

Case 2 (Figure 4(c)) appears similar to Case 1 but the bitreaches um sooner owing to τ and the effect of drill bitinertia As the system transitions to unloading and convergesto uf it shows critical damping Case 3 (Figure 4(d)) showsa longer τ compared with Case 2 because of its quickerarrival at um During unloading it converges to uf asunderdamped motion Case 4 (Figure 4(e)) has the drill bitdisplaced at u(τ) due to τ and it arrives at um as criticallydamped motion owing to continued loading resulting fromthe inertia of the drill bit 0e subsequent unloading phaseconverges to uf as underdamped motion

Case 5 (Figure 4(f)) has a longer τ than the preceding fourcases and also reaches um sooner Furthermore all its zonesshowed underdamped motion Finally in Case 6 (Figure 4(g)) the rock displacement has already reached its maximum

while τ is still in progress 0is means that unloading beginsduring τ Underdamped motion was observed in all zones

Overall the longer the duration of τ the shorter theloading time and the sooner the um is reached owing to drillbit inertia Table 2 summarizes the damping characteristics forthe loading and unloading conditions and presents the as-sociated damping mode that arrives at um Cases 1ndash3 showoverdamping Case 4 critical damping and Cases 5 and 6underdamping 0e underdamped mode showed thehighest responsiveness indicating the fastest trans-mission time of τ 0e results show that increasing theτ-transmission time decreased the loading effect resultingfrom drill bit inertia and led to faster responses 0e drillbit inertial effect is inversely proportional to τ

3 Energy Transmission in Percussive DrillingSystem

In Section 2 we examined the response characteristics ofdrill bits that can arrive at um which is themaximum drillingdisplacement of rocks Based on the examined character-istics in Section 3 we calculate the efficiency of percussivedrilling systems and discuss the combinations of parametersthat could maximize it

31 Energy Transmission Efficiency In a percussive drillingsystem the piston directly hits the drill bit at speed v (Figure 2)For a transmitted impact energy Ei at the time of the pistondescending and hitting the drill bit the final speed of the pistonimmediately preceding the impact v can be expressed asfollows [2 10]

v 2Ei

mp

1113888 1113889

12

(12)

Table 1 Expansion conditions for dynamic drill-bit properties for loading unloading and stress-wave duration τ

Case no

Initial conditions0lt tle τ τ lt tle tm tgt tm

σi 05ρcv

F ku

u0 0 _u0 0

σi 0F ku

u0 u(τ) _u0 _u(τ)

σi 0F ckuminus ckuf

u0 um _u0 0

1 αlt 1c

2 α 1c

3 1clt αlt 1

4 α 1

5 αgt 1 tm gt τ

6

0lt tle tm tm lt tle τ tgt τ

σi 05ρcv

F ku

u0 0 _u0 0

σi 05ρcv

F ckuminus ckuf

u0 um _u0 0

σi 0F F ckuminus ckuf

u0 u(τ) _u0 _u(τ)

Advances in Materials Science and Engineering 5

0000

uf

u

00

04

08

12

16

20

0001 0002 t0003 0004

Case 1Case 2Case 3Case 4Case 5Case 6

Loading by incident stress waveLoading by inertia of bitUnloading

(a)

tm

uf

um

u

t

Loading by incident stress waveLoading by inertia of bitUnloading

00040003

Overdamped motion

00020001000000

04

08

12

16

20

u()

(b)

u()

tm

uf

um

u

t

000400030002

Overdamped motion

Critically damped motion

0001000000

04

08

12

16

20

Loading by incident stress waveLoading by inertia of bitUnloading

(c)

u()

Overdamped motion

Underdamped motion

tm

uf

um

u

t

0004000300020001000000

04

08

12

16

20

Loading by incident stress waveLoading by inertia of bitUnloading

(d)

Figure 4 Continued

6 Advances in Materials Science and Engineering

e initial height h0 is v22g and the rebound eh canbe dened as the ratio between the initial height and thepistonrsquos postimpact rebound height h ese areexpressed as follows [2]

eh h

h0v2ev2 (13)

e eciency η of impact energy transmission is de-ned as the ratio between the kinetic energy generated fromthe impact of the piston and the energy transmitted to therock [12] and is given as

η kum

2

ρALhv2

Fmax( )2

ρALhv2k (14)

Which can be expressed as follows through dimensionalanalysis

η 2βu2m (15)

e value um during loading can be obtained from (7)the equation of motion for the drill bit e obtained um canbe expressed as in (16) for each damping mode where theinitial conditions are 0lt tle τ F kui u0 0 and _u0 0

u()

Critically damped motion

Underdamped motion

tm

uf

um

u

t

0004000300020001000000

04

08

12

16

20

Loading by incident stress waveLoading by inertia of bitUnloading

(e)

Loading by incident stress waveLoading by inertia of bitUnloading

u()Underdamped motion

tm

uf

um

u

t

0004000300020001000000

04

08

12

16

20

(f)

Loading by incident stress waveLoading by inertia of bitUnloading

u()Underdampedmotion

uf

u

u

t

t

000400030002

00002 00003 00004

0001000000

04

08

12

16

20

um

um

tm

(g)

Figure 4 Results of the motion response of the bit formr 1 (a) All cases (b) Case 1 (c) Case 2 (d) Case 3 (e) Case 4 (f) Case 5 (g) Case 6

Advances in Materials Science and Engineering 7

um(t)

1 +1minus

αminus 1

radic

2αminus 1

radic middot eminus(2α)(βτ)(1+

1minusα

radic)t minus

1 +1minus α

radic

21minus α

radic middot eminus(2α)(βτ)(1minus

1minusα

radic)t

1113890 1113891(overdamped)

1minus eminus2(βτ)t middot 2βτ

t + 11113890 1113891(critical damped)

1minus eminus2(βτ)t middot1

αminus 1

radic sin2

αminus 1

radic

αβτ

t1113888 1113889 + cos2

αminus 1

radic

αβτ

t1113888 11138891113890 1113891(underdamped)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(16)

Moreover for tmlt τ um is expressed as follows

um(t) 1minus eminus(2α)(βτ)

middot ⎡⎣1

αminus 1

radic sin2

αminus 1

radic

αβτ

t1113888 1113889

+ cos2

αminus 1

radic

αβτ

t1113888 1113889⎤⎦

(17)

0e initial conditions are 0lt tle τ F kui ui 0and _ui 0

Next the displacement value uf during unload-ing can be obtained via the equation of motion for a drill-bit (9) as shown in (18) 0e initial conditionsduring unloading are tgt τ F 0 u0 u(τ) and_u0 _u(τ)

uf(t)

α4

1minus α

radicτβ

middot eminus(2α)(1+

1minusα

radic)t

middot u02αβτ

(1 +1minus α

radic) minus _u0 +

4αβτu01113888 11138891113890 1113891

minusα

41minus α

radicτβ

middot eminus(2α)(1minus

1minusα

radic)t

middot u02αβτ

(1minus1minus α

radic) minus _u0 +

4αβτu01113888 11138891113890 1113891(overdamped)

eminus2(βτ)t middot _u0 + 2βτu01113888 1113889tminus u01113890 1113891(critical damped)

eminus(2α)(βτ)t middot ⎡⎣α

2αminus 1

radicτβ

1113888 1113889 _u0 middot sin2

αminus 1

radic

α1113888 1113889

βτ

t +1

αminus 1

radic u0 middot sin2

αminus 1

radic

αβτ

t1113888 1113889

+ u0 middot cos2

αminus 1

radic

αβτ

t1113888 1113889⎤⎦(underdamped)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(18)

Table 2 Dynamic drill-bit properties for loading unloading and τ and the damping modes capable of reaching um

Case no ConditionsDamping modes capable of reaching um

0lt tle τ τ lt tle tm tgt tm

1 αlt 1c

Over Overlowast Over2 α 1

cOver Overlowast Critical

3 1clt αlt 1 Over Overlowast Under

4 α 1 Critical Criticallowast Under5 αgt 1 tm gt τ Under Underlowast Under

6 αgt 1 tm lt τ0lt tle tm tm lt tle τ tgt τUnderlowast Under Under

lowastDamping modes capable of reaching um

8 Advances in Materials Science and Engineering

eeciency of the percussive drilling systems for given βcan be calculated using (14ndash17) e factor um which isimportant to the eciency calculations is discussed in depth

32 Results of Energy Transmission Eciency e results ofsimulations on the drill bit-rock interaction model shown inFigure 5 indicate the damped modes that can arrive at um (forvarying values of β) as the response by the drill bit at the time ofpiston impact e damped mode is determined by ζ and αwhich indictate the system responses Overdamped mode hasαlt 1 and can be expressed as follows

2mb

mh( )βlt 1

βlt1

2mr

(19)

e critical damped mode has α 1 and is expressed asfollows

2mb

mh( )β 1

β 1

2mr

(20)

e underdamped mode has αgt 1 and is expressed as

2mb

mh( )βgt 1

βgt1

2mr

(21)

where mbmh mr

e special underdamped mode refers to the case whereτ is faster than t indicating that the drill bit can arrive at umonly via an incident stress wave In this case the bit reachesits maximum displacement and transitions to the unloadingstate while the transmission of τ is underway

e damped mode of each mr shown in Figure 5 re-veals that the underdamped mode is dominant of thee ects Also overdamped and critical damped were ob-served where β was low is is possibly because of thee ects of the internal energy (ie compression strength)of rock and the percussive energy e overdamped andcritical damped were also observed is is possibly be-cause of internal energy (ie compression strength) ofrock and energy caused by percussion

Figure 6 shows dimensionless time tm tτ capable ofreaching Um for β and τ In Section A of tgt τ the bit reachesUm after termination of τ and in Section B of tlt τ the bitreachesUm before the termination of τe lower themr andβ the faster the termination of τ and increase in thesetended to make the duration of τ delivered to rock throughbit to increase proportionally to β is suggests that thehigher rock strength requires longer duration of τ fordrilling the rock

Figure 7 shows dimensionless Um state dened as ratioum a displacement um by β and uinfin caused by incident stresswave and loading condition e prediction of displacementof drill bit drilling displacement by characteristic of rockwas limited is study accordingly judged the di-mensionless drilling state Um only by using ratio uinfina displacement during loading and um considering inertiacaused during percussion (22)

Um umuinfin (22)

where the uinfin only by loading is dened as follows

mr = 1

mr = 2

mr = 05

0 5 10

15 20

4

3

2

Dam

ped

mod

es

1

0

mr = 1mr = 2

mr = 05

mr = 1

mr = 05

Figure 5 Damping mode capable of reaching maximum rockfracture displacement um given the piston-to-bit mass ratiomr and β1 overdamped mode 2 critically damped mode 3 underdampedmode 4 special underdamped mode

0 5 10

15 2000

02

06

10

14

18

20

04

08

12

16

Dim

ensio

nles

s pen

etra

tion

(Um

)

Section B

Section A

mr = 05mr = 1mr = 2

uinfin

Figure 6 State of reaching drill bit dimensionless penetration Umwith respect to mass ratio mr and β

Advances in Materials Science and Engineering 9

uinfin ρcAkv

τβv (23)

e e ect of initial condition on uinfin decreased with theincrease in β e um is determined by considering loadingcondition and inertia e ect is a ected by overresponse anddecreases as β increases

Section A refers to umlt uinfin with only τ and loadinge ect present (ie the condition for the normal state in theinitial mode) uinfin means only the τ e ect resulting fromloading exists Section B refers to umgt uinfin a state in whichum can be reached through τ and loading is exceeded issection is where the drill bit inertia and underdampedmotion appear e gradient of the curve for maximumdisplacement of the bit varies with β indicating that as therock strength increases the displacement allowing fracturedecreases Near value 2 β changes its gradient for maximumdisplacement is means that in Section A the rock is lesssti than the piston owing to the rock-sti ness e ect and inSection B the rock is sti er than the piston β is dened asthe sti ness ratio between the rock and the piston as in (11)Hence when βlt 2 a condition is established for rock fractureto occur in that the piston sti ness exceeds the rock sti ness

In examining um for piston-bit mass ratios of mrlt 2 therock strength that can be defeated by τ occurring in theinitial loading tends to increase is tendency is suspectedto result from the inertial e ect of the mass of the bit Forβgt 4 the um gradient tended to decrease with increasingmass of the bit Hence the eciency is considered to de-crease which is attributed to the e ect of the rockrsquos com-pressive strength e uniaxial compressive strength of veryhard rock is typically at least 200MPa while the tensilestrength of the H13 tool steels that are used mostly forpistons (ASTM A681 and DIN EN ISO 4957) is known to beat least 359ndash1170MPa [15]

Given the steels generally used to make pistons thee ective range of β is considered to be 034lt βlt 112 which

was calculated considering the strength ratio between thehard rock and an H13 steel tool Hence the e ective range ofβ could be no greater than 2 Rocks with a sti ness ratio ofβgt 2 might cause failure through the pistonrsquos plastic de-formation and fracture is study examined the e ectiveinterval of β and investigated the parameter combinationsgiving high drilling eciency in this interval

Figure 8 illustrates the motion of the bit generatedduring the percussion process and the energy transmissionrate (ie eciency) for the response characteristics τ andpercussion impact Analysis of mr found the maximumdrilling eciency in interval 1lt βlt 2 with a tendency ef-ciency decreasing after mr increased above a certain valueis result was attributed to the tendency whereby increasingmr converts the systemrsquos dynamic response to τ due to thepiston from underdamped to overdamped Furthermoreincreasing the mass of the bit can increase its internal energy(ie friction energy) to above the piston transmitted energy(ie kinetic and potential energy) Hence the bitrsquos dynamicresponse would be overdamped and could compromisedrilling eciency In other words the heavier the bit themore delayed the response to percussion

In Figure 8 the left side of red line (β18) represents thee ect of only incident stress wave meaning that the higherthe value in graph the higher the drilling eciency in in-cident stress wave e right side of red line representingunloading behavior after reaching maximum eciency inincident stress wave is a ected by both brittleness char-acteristics of rock and inertia e ect of drill bit

e piston should have twice the mass of the drill bit(iemr 05) for the most ecient drilling for βle 2 which isestablished to be the e ective interval of β and correspondsto both soft and hard rocks Drilling sti er rocks (2lt βlt 4)might be less ecient but increasing mr to 1 would helpimprove drilling eciency

00000

00002

00004

00006

00008

00010

00012

Dim

ensio

nles

s tim

e (t m

)

Section A

Section B

0 5 10

15 20

mr = 05mr = 1mr = 2

Figure 7 Dimensionless time capable of reaching Um for an in-cident stress-wave τ acting on the bit and β

0 2 4 6 8 10 12 14 16 18 20

= 1800

01

02

03

04

05

06

07

08

09

10

Effic

ienc

y

mr = 01mr = 025mr = 05mr = 1

mr = 2mr = 4mr = 10

Figure 8 Drilling eciency of percussive drilling systems withrespect to β and mr

10 Advances in Materials Science and Engineering

4 Results and Discussion

0is study defined equations for the motion of a drill bitwhen struck by a piston and expanded the dynamic prop-erties of the drill bit based on the conditions ζ and α It alsoanalyzed the relationship between the bit dynamics and theduration of the incident stress wave (τ) Underdamped drillbits displayed the greatest responsiveness and the fastesttransmission of τ Increasing the transmission of τ decreasedthe loading effect due to the drill bitrsquos inertia but led to a fastresponse 0e inertial effect of the drill bit was confirmed tobe inversely proportional to the transmission of τ

A percussive drilling system accomplishes its work viathe percussion impact of a piston In the systems studiedhere τ ended more quickly with lower mr and β values asthese values increased the duration of τ transmitted to therock via the drill bit increased in proportion to β 0e resultsalso showed that the maximum bit displacement (um) de-creased as β increased Analysis of the drilling dynamicsconfirmed the effective intervals of β With reference to therock strengths suggested by the International Society forRock Mechanics (ISRM) the effective interval of β is con-sidered to be no greater than 2

0e main purpose of this study was to examine theimpact energy transmission rate and drilling efficiency ina percussive drilling system (Figure 8) 0e results establishedamr value of 05 as the most efficient for rocks whose strengthcorresponds to the interval βlt 2 0e improved drilling ef-ficiency would lead to advantages such as reduced bit pro-duction costs Rocks stiffer than 2ltβlt 4 can be mostefficiently drilled by selecting mr 1

0emovement and response characteristics of bit duringpercussion process τ and transfer rate of energy caused bypercussion an efficiency depend on mr and β where in-crease in mr leads to overdamping of bit movement char-acteristics due to τ by piston and reduction in drillingefficiency due to internal energy effect by mass of drill bit Asthe increase in mass of drill bit for the section of 4lt βlt 6the drilling efficiency of percussive drilling system decreaseand for the very hard rock (UCSgt 200MPa) section of4lt βlt 6 it is considered that the reasonable applied mr ofdrilling tool is 1 or 2

0is study neglected the effects of the secondary incidentstress wave and also the effect of buttons embedded in thedrill bit Additionally for analytical simplification the pistonand drill bit were assumed to be of the same diameter andmaterial and a rectangular pulse with an incident stresswave of duration τ and amplitude 05ρcv was assumed 0eeffect of the flexural stress wave depending on the config-uration was not considered 0ese limitations of this papershould be addressed in further investigations that take intoaccount the configuration of the piston and drill bit theeffect of buttons and the drilling efficiency given differingshapes of the incident stress wave

5 Conclusion

0is paper aimed to identify the optimal design parametersfor percussive drilling systems by introducing a drill bit-rock

interaction model that could verify the bitrsquos motion duringpercussion and the resulting damping characteristics 0estudy analyzed drilling efficiency and drew the followingconclusions

Percussive drilling systems have six dynamic drill bitproperties that can be expandable 0is paper discussed thephysical meaning of the dimensionless parameters α and β0eir values determined the damping characteristics thatcan lead to the rockrsquos maximum fracture displacement 0efastest response of τ was observed for underdamped drill-bitmotion

Drilling was most efficient in the interval 1lt βlt 4 whereincreasing mr at a given β decreased drilling efficiency 0evalue of mr for efficient drilling was determined by the rockstrength (ie β)

0e results indicate that application of a piston-to-drill bitmass ratio of 05 (ie a piston mass twice that of the drill bitmass) to the rocks whose stiffness corresponds to βge 2 wouldlikely be most efficient and also reduce drill bit productioncosts Furthermore a selection ofmr 1 would be valid whendeveloping drill tools for boring complex rocks (1ltβlt 4) Atβgt 4 the best efficiency could be achieved when the bit massequals or exceeds the piston mass

Appendix

Theoretical Case of Bit Motion Condition(Drilling Dynamic Model)

0is paper proposed six conditions for the dynamics ofa drill bit struck by a piston impact as summarizedbelow 0e damping conditions for the damping ratiosand α from the drill-bit equations of motion (7ndash9) are asfollows

First for the overdamped condition of ζ1 gt 1 and αlt 1 τis transmitted to the drill bit indicating loading Equation(7) can be expressed as in (A1) 0e initial conditions are0lt tle τ σi 05ρcv F ku u0 0 and _u0 0

eurou +ρcA

mb

1113888 1113889 _u +k

mb

1113888 1113889u ρcA

mb

1113888 1113889v (A1)

where assuming the normal state u can be calculated asfollows

u ρcA

k1113874 1113875v (A2)

Next we examine the case where τ is complete and theloading effect is sustained owing to external forces and drill-bit inertia Applying the conditions of τ lt tle tm σi 0F ku u0 u(τ) and _u0 _u(τ) to (A1) results in thefollowing expression

eurou +ρcA

mb

1113888 1113889 _u +k

mb

1113888 1113889u 0 (A3)

where u 0 Last we examine the unloading conditions oftm lt t σi 0 F ckuminus ckuf u0 um and _u0 0 Apply-ing these to (9) when unloading gives

Advances in Materials Science and Engineering 11

eurou +ρcA

mb

1113888 1113889 _u + ck

mb

1113888 1113889u ck

mb

1113888 1113889uf (A4)

where bit displacement u uf is used to indicate the rockrsquosfracture displacement Furthermore considering the bitrsquosdamping ratio when unloading its behavior can be ex-pressed as

Damped mode

ζ2 gt 1 αlt1c

over( )

ζ2 1 α 1c

critical( )

ζ2 lt 1 αgt1c

under( )

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A5)

Second under critical damping where ζ1 1 and α 1 τis transmitted to the drill bit indicating loading Equation(7) can be expressed as in (A1) 0e initial conditions are0lt tle τ σi 05ρcv F ku u0 0 and _u0 0

In this case assuming the normal state u can be cal-culated using (A2)

0e following is the case where τ is complete and theloading effect is sustained owing to external forces and drillbit inertia Applying the conditions of τ lt tle tm σi 0F ku u0 u(τ) and _u0 _u(τ) to (A1) give (A3) whereu 0 Last we examine the unloading condition wheretm lt t σi 0 F ckuminus ckuf u0 um and _u0 0 Equation(9) when unloading can be expressed as (A4)

In this case bit displacement u uf is used to indicate therockrsquos fracture displacement Furthermore considering thebitrsquos damping ratio when unloading its behavior follows

Damped mode

ζ2 gt 1 αlt1c

(no case)

ζ2 1 α 1c

(no case)

ζ2 lt 1 αgt1c

under( )

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A6)

0e second condition of critical damping is defined asα 1When unloading drill bits c refers to the rockrsquos propertyeffect and is always less than 1 Hence when unloading noother cases but the underdamped mode are allowed In otherwords when unloading in the critically damped mode owingto the rock properties the bit is always underdamped

0e third condition is underdamping with ζ1 lt 1 andαgt 1 Here τ is transmitted to the drill bit meaning loadingEquation (7) can be expressed by (A1)0e initial conditionsare 0lt tle τ σi 05ρcv F ku u0 0 and _u0 0 Whereassuming the normal state u can be calculated as (A2)

0e following is the case where τ is complete and theloading effect is sustained owing to external forces and drillbit inertia Applying the conditions of τ lt tle tm σi 0F ku u0 u(τ) and _u0 _u(τ) to (A1) results in (A3)

where u 0 Last we examine the unloading conditionwhere tm lt t σi 0 F ckuminus ckuf u0 um and _u0 0Equation (9) when unloading can be expressed as (A4)

In this case the drill bit displacement u uf is used toindicate the rockrsquos fracture displacement Furthermoreconsidering the damping ratio in the drill bitrsquos unloadingthe bitrsquos behavior can be expressed as

Damped mode

ζ2 gt 1 αlt1c

(no case)

ζ2 1 α 1c

(no case)

ζ2 lt 1 αgt1c

under( )

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A7)

0e third condition is the critically damped modedefined as αgt 1 During drill bit unloading c refers to therock property effect and is always less than 1 Hencewhen unloading no other cases but underdamping areallowed

Next the underdamped condition is where ζ1 lt 1 andαgt 1 and τ is slower than tm (the time at which the rockrsquosmaximum fracture displacement is reached) that is tm lt τIn this case τ is transmitted to the drill bit meaningloading Equation (7) can be expressed by (A1) 0e initialconditions are 0lt tle tm σi 05ρcv F ku u0 0 and_u0 0 Here assuming the normal state u can be calculatedas (A2)

0e following case is where conversion is made tounloading after the drill bit reaches its maximum dis-placement but the incident stress wave is sustained Ap-plying the conditions of tm lt tle τ σi 05ρcvF ckuminus ckuf u0 um and _u0 0 to (9) results in

eurou +ρcA

mb

1113888 1113889 _u + ck

mb

1113888 1113889u ρcA

mb

1113888 1113889v + ck

mb

1113888 1113889uf (A8)

where the definition of u 1c(ρcAk)v + uf is possibleLast this unloading condition is where the incident

stress wave is resolved When unloading the drill bit usingthe conditions tlt τ σi 0 F ckuminus ckuf u0 u(τ) and_u0 _u(τ) allows (9) to be expressed as (A4)

In this case drill bit displacement u uf is used to in-dicate the rockrsquos fracture displacement 0e analysis of drillbit dynamics showed that when loading transitions tounloading in the underdamped condition the dynamicproperties of the bit exist only in the underdamping mode

Conflicts of Interest

The authors declare that there are no conflicts of interest

References

[1] C H Song K B Kwon J Y Park et al ldquoOptimum design ofthe internal flushing channel of a drill bit using RSM and CFDsimulationrdquo International Journal of Precision Engineeringand Manufacturing vol 15 no 6 pp 1041ndash1050 2014

12 Advances in Materials Science and Engineering

[2] X Li G Rupert D A Summers P Santi and D Liu ldquoAnalysisof impact hammer rebound to estimate rock drillabilityrdquo RockMechanics and Rock Engineering vol 33 no 1 pp 1ndash13 2000

[3] W A Hustrulid and C Fairhurst ldquoA theoretical and ex-perimental study of the percussive drilling of rock part Itheory of percussive drillingrdquo International Journal of RockMechanics and Mining Sciences amp Geomechanics Abstractsvol 8 no 4 pp 311ndash333 1971

[4] W A Hustrulid and C Fairhurst ldquoA theoretical and ex-perimental study of the percussive drilling of rock part IIforce-penetration and specific energy determinationrdquo In-ternational Journal of Rock Mechanics and Mining Sciences ampGeomechanics Abstracts vol 8 no 4 pp 335ndash356 1971

[5] W A Hustrulid and C Fairhurst ldquoA theoretical and ex-perimental study of the percussive drilling of rock part IIIexperimental verification of the mathematical theoryrdquo In-ternational Journal of Rock Mechanics and Mining Sciences ampGeomechanics Abstracts vol 9 no 3 pp 417-418 1972

[6] W A Hustrulid and C Fairhurst ldquoA theoretical and ex-perimental study of the percussive drilling of rock part IVapplication of the model to actual percussive drillingrdquo In-ternational Journal of Rock Mechanics and Mining Sciencesvol 9 no 3 pp 431ndash442 1972

[7] L E Chiang and D A Elias ldquoA 3D FEM methodology forsimulating the impact in rock-drilling hammersrdquo In-ternational Journal of Rock Mechanics and Mining Sciencesvol 45 no 5 pp 701ndash711 2008

[8] C H Song K B Kwon M G Cho J Y Oh D Y Shin andJ W Cho ldquoDevelopment of lab-scale rock drill apparatus fortesting performance of a drill bitrdquo International Journal ofPrecision Engineering and Manufacturing vol 16 no 7pp 1402ndash1414 2015

[9] K B Kwon C H Song J Y Park J Y Oh J W Lee andJ W Cho ldquoEvaluation of drilling efficiency by percussiontesting of a drill bit with new button arrangementrdquo In-ternational Journal of Precision Engineering andManufacturing vol 15 no 6 pp 1063ndash1068 2014

[10] B Lundberg andM Okrouhlik ldquoInfluence of 3D effects on theefficiency of percussive rock drillingrdquo International Journal ofimpact Engineering vol 25 no 4 pp 345ndash360 2001

[11] B Lundberg and P Collet ldquoOptimal wave shape with respectto efficiency in percussive drilling with detachable drill bitrdquoInternational Journal of impact Engineering vol 86 pp 179ndash187 2015

[12] X B Li G Rupert and D A Summers ldquoEnergy transmissionof down-hole hammer tool and its conditionalityrdquo Trans-actions of Nonferrous Metals Society of China vol 10 no 1pp 109ndash111 2000

[13] J W Cho S Jeon S H Yu and S H Chang ldquoOptimumspacing of TBM disc cutters a numerical simulation using thethree-dimensional dynamic fracturing methodrdquo Tunnellingand Underground Space Technology vol 25 no 3 pp 230ndash244 2010

[14] W Changming ldquoAn analytical study of percussive energytransfer in hydraulic rock drillsrdquo Mining Science and Tech-nology vol 13 no 1 pp 57ndash68 1991

[15] Online Materials Information Resource httpwwwmatwebcomsearchDataSheetaspxMatGUID8bc5d558f4174e6082ddf4966e382bd6ampckck1

Advances in Materials Science and Engineering 13

0000

uf

u

00

04

08

12

16

20

0001 0002 t0003 0004

Case 1Case 2Case 3Case 4Case 5Case 6

Loading by incident stress waveLoading by inertia of bitUnloading

(a)

tm

uf

um

u

t

Loading by incident stress waveLoading by inertia of bitUnloading

00040003

Overdamped motion

00020001000000

04

08

12

16

20

u()

(b)

u()

tm

uf

um

u

t

000400030002

Overdamped motion

Critically damped motion

0001000000

04

08

12

16

20

Loading by incident stress waveLoading by inertia of bitUnloading

(c)

u()

Overdamped motion

Underdamped motion

tm

uf

um

u

t

0004000300020001000000

04

08

12

16

20

Loading by incident stress waveLoading by inertia of bitUnloading

(d)

Figure 4 Continued

6 Advances in Materials Science and Engineering

e initial height h0 is v22g and the rebound eh canbe dened as the ratio between the initial height and thepistonrsquos postimpact rebound height h ese areexpressed as follows [2]

eh h

h0v2ev2 (13)

e eciency η of impact energy transmission is de-ned as the ratio between the kinetic energy generated fromthe impact of the piston and the energy transmitted to therock [12] and is given as

η kum

2

ρALhv2

Fmax( )2

ρALhv2k (14)

Which can be expressed as follows through dimensionalanalysis

η 2βu2m (15)

e value um during loading can be obtained from (7)the equation of motion for the drill bit e obtained um canbe expressed as in (16) for each damping mode where theinitial conditions are 0lt tle τ F kui u0 0 and _u0 0

u()

Critically damped motion

Underdamped motion

tm

uf

um

u

t

0004000300020001000000

04

08

12

16

20

Loading by incident stress waveLoading by inertia of bitUnloading

(e)

Loading by incident stress waveLoading by inertia of bitUnloading

u()Underdamped motion

tm

uf

um

u

t

0004000300020001000000

04

08

12

16

20

(f)

Loading by incident stress waveLoading by inertia of bitUnloading

u()Underdampedmotion

uf

u

u

t

t

000400030002

00002 00003 00004

0001000000

04

08

12

16

20

um

um

tm

(g)

Figure 4 Results of the motion response of the bit formr 1 (a) All cases (b) Case 1 (c) Case 2 (d) Case 3 (e) Case 4 (f) Case 5 (g) Case 6

Advances in Materials Science and Engineering 7

um(t)

1 +1minus

αminus 1

radic

2αminus 1

radic middot eminus(2α)(βτ)(1+

1minusα

radic)t minus

1 +1minus α

radic

21minus α

radic middot eminus(2α)(βτ)(1minus

1minusα

radic)t

1113890 1113891(overdamped)

1minus eminus2(βτ)t middot 2βτ

t + 11113890 1113891(critical damped)

1minus eminus2(βτ)t middot1

αminus 1

radic sin2

αminus 1

radic

αβτ

t1113888 1113889 + cos2

αminus 1

radic

αβτ

t1113888 11138891113890 1113891(underdamped)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(16)

Moreover for tmlt τ um is expressed as follows

um(t) 1minus eminus(2α)(βτ)

middot ⎡⎣1

αminus 1

radic sin2

αminus 1

radic

αβτ

t1113888 1113889

+ cos2

αminus 1

radic

αβτ

t1113888 1113889⎤⎦

(17)

0e initial conditions are 0lt tle τ F kui ui 0and _ui 0

Next the displacement value uf during unload-ing can be obtained via the equation of motion for a drill-bit (9) as shown in (18) 0e initial conditionsduring unloading are tgt τ F 0 u0 u(τ) and_u0 _u(τ)

uf(t)

α4

1minus α

radicτβ

middot eminus(2α)(1+

1minusα

radic)t

middot u02αβτ

(1 +1minus α

radic) minus _u0 +

4αβτu01113888 11138891113890 1113891

minusα

41minus α

radicτβ

middot eminus(2α)(1minus

1minusα

radic)t

middot u02αβτ

(1minus1minus α

radic) minus _u0 +

4αβτu01113888 11138891113890 1113891(overdamped)

eminus2(βτ)t middot _u0 + 2βτu01113888 1113889tminus u01113890 1113891(critical damped)

eminus(2α)(βτ)t middot ⎡⎣α

2αminus 1

radicτβ

1113888 1113889 _u0 middot sin2

αminus 1

radic

α1113888 1113889

βτ

t +1

αminus 1

radic u0 middot sin2

αminus 1

radic

αβτ

t1113888 1113889

+ u0 middot cos2

αminus 1

radic

αβτ

t1113888 1113889⎤⎦(underdamped)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(18)

Table 2 Dynamic drill-bit properties for loading unloading and τ and the damping modes capable of reaching um

Case no ConditionsDamping modes capable of reaching um

0lt tle τ τ lt tle tm tgt tm

1 αlt 1c

Over Overlowast Over2 α 1

cOver Overlowast Critical

3 1clt αlt 1 Over Overlowast Under

4 α 1 Critical Criticallowast Under5 αgt 1 tm gt τ Under Underlowast Under

6 αgt 1 tm lt τ0lt tle tm tm lt tle τ tgt τUnderlowast Under Under

lowastDamping modes capable of reaching um

8 Advances in Materials Science and Engineering

eeciency of the percussive drilling systems for given βcan be calculated using (14ndash17) e factor um which isimportant to the eciency calculations is discussed in depth

32 Results of Energy Transmission Eciency e results ofsimulations on the drill bit-rock interaction model shown inFigure 5 indicate the damped modes that can arrive at um (forvarying values of β) as the response by the drill bit at the time ofpiston impact e damped mode is determined by ζ and αwhich indictate the system responses Overdamped mode hasαlt 1 and can be expressed as follows

2mb

mh( )βlt 1

βlt1

2mr

(19)

e critical damped mode has α 1 and is expressed asfollows

2mb

mh( )β 1

β 1

2mr

(20)

e underdamped mode has αgt 1 and is expressed as

2mb

mh( )βgt 1

βgt1

2mr

(21)

where mbmh mr

e special underdamped mode refers to the case whereτ is faster than t indicating that the drill bit can arrive at umonly via an incident stress wave In this case the bit reachesits maximum displacement and transitions to the unloadingstate while the transmission of τ is underway

e damped mode of each mr shown in Figure 5 re-veals that the underdamped mode is dominant of thee ects Also overdamped and critical damped were ob-served where β was low is is possibly because of thee ects of the internal energy (ie compression strength)of rock and the percussive energy e overdamped andcritical damped were also observed is is possibly be-cause of internal energy (ie compression strength) ofrock and energy caused by percussion

Figure 6 shows dimensionless time tm tτ capable ofreaching Um for β and τ In Section A of tgt τ the bit reachesUm after termination of τ and in Section B of tlt τ the bitreachesUm before the termination of τe lower themr andβ the faster the termination of τ and increase in thesetended to make the duration of τ delivered to rock throughbit to increase proportionally to β is suggests that thehigher rock strength requires longer duration of τ fordrilling the rock

Figure 7 shows dimensionless Um state dened as ratioum a displacement um by β and uinfin caused by incident stresswave and loading condition e prediction of displacementof drill bit drilling displacement by characteristic of rockwas limited is study accordingly judged the di-mensionless drilling state Um only by using ratio uinfina displacement during loading and um considering inertiacaused during percussion (22)

Um umuinfin (22)

where the uinfin only by loading is dened as follows

mr = 1

mr = 2

mr = 05

0 5 10

15 20

4

3

2

Dam

ped

mod

es

1

0

mr = 1mr = 2

mr = 05

mr = 1

mr = 05

Figure 5 Damping mode capable of reaching maximum rockfracture displacement um given the piston-to-bit mass ratiomr and β1 overdamped mode 2 critically damped mode 3 underdampedmode 4 special underdamped mode

0 5 10

15 2000

02

06

10

14

18

20

04

08

12

16

Dim

ensio

nles

s pen

etra

tion

(Um

)

Section B

Section A

mr = 05mr = 1mr = 2

uinfin

Figure 6 State of reaching drill bit dimensionless penetration Umwith respect to mass ratio mr and β

Advances in Materials Science and Engineering 9

uinfin ρcAkv

τβv (23)

e e ect of initial condition on uinfin decreased with theincrease in β e um is determined by considering loadingcondition and inertia e ect is a ected by overresponse anddecreases as β increases

Section A refers to umlt uinfin with only τ and loadinge ect present (ie the condition for the normal state in theinitial mode) uinfin means only the τ e ect resulting fromloading exists Section B refers to umgt uinfin a state in whichum can be reached through τ and loading is exceeded issection is where the drill bit inertia and underdampedmotion appear e gradient of the curve for maximumdisplacement of the bit varies with β indicating that as therock strength increases the displacement allowing fracturedecreases Near value 2 β changes its gradient for maximumdisplacement is means that in Section A the rock is lesssti than the piston owing to the rock-sti ness e ect and inSection B the rock is sti er than the piston β is dened asthe sti ness ratio between the rock and the piston as in (11)Hence when βlt 2 a condition is established for rock fractureto occur in that the piston sti ness exceeds the rock sti ness

In examining um for piston-bit mass ratios of mrlt 2 therock strength that can be defeated by τ occurring in theinitial loading tends to increase is tendency is suspectedto result from the inertial e ect of the mass of the bit Forβgt 4 the um gradient tended to decrease with increasingmass of the bit Hence the eciency is considered to de-crease which is attributed to the e ect of the rockrsquos com-pressive strength e uniaxial compressive strength of veryhard rock is typically at least 200MPa while the tensilestrength of the H13 tool steels that are used mostly forpistons (ASTM A681 and DIN EN ISO 4957) is known to beat least 359ndash1170MPa [15]

Given the steels generally used to make pistons thee ective range of β is considered to be 034lt βlt 112 which

was calculated considering the strength ratio between thehard rock and an H13 steel tool Hence the e ective range ofβ could be no greater than 2 Rocks with a sti ness ratio ofβgt 2 might cause failure through the pistonrsquos plastic de-formation and fracture is study examined the e ectiveinterval of β and investigated the parameter combinationsgiving high drilling eciency in this interval

Figure 8 illustrates the motion of the bit generatedduring the percussion process and the energy transmissionrate (ie eciency) for the response characteristics τ andpercussion impact Analysis of mr found the maximumdrilling eciency in interval 1lt βlt 2 with a tendency ef-ciency decreasing after mr increased above a certain valueis result was attributed to the tendency whereby increasingmr converts the systemrsquos dynamic response to τ due to thepiston from underdamped to overdamped Furthermoreincreasing the mass of the bit can increase its internal energy(ie friction energy) to above the piston transmitted energy(ie kinetic and potential energy) Hence the bitrsquos dynamicresponse would be overdamped and could compromisedrilling eciency In other words the heavier the bit themore delayed the response to percussion

In Figure 8 the left side of red line (β18) represents thee ect of only incident stress wave meaning that the higherthe value in graph the higher the drilling eciency in in-cident stress wave e right side of red line representingunloading behavior after reaching maximum eciency inincident stress wave is a ected by both brittleness char-acteristics of rock and inertia e ect of drill bit

e piston should have twice the mass of the drill bit(iemr 05) for the most ecient drilling for βle 2 which isestablished to be the e ective interval of β and correspondsto both soft and hard rocks Drilling sti er rocks (2lt βlt 4)might be less ecient but increasing mr to 1 would helpimprove drilling eciency

00000

00002

00004

00006

00008

00010

00012

Dim

ensio

nles

s tim

e (t m

)

Section A

Section B

0 5 10

15 20

mr = 05mr = 1mr = 2

Figure 7 Dimensionless time capable of reaching Um for an in-cident stress-wave τ acting on the bit and β

0 2 4 6 8 10 12 14 16 18 20

= 1800

01

02

03

04

05

06

07

08

09

10

Effic

ienc

y

mr = 01mr = 025mr = 05mr = 1

mr = 2mr = 4mr = 10

Figure 8 Drilling eciency of percussive drilling systems withrespect to β and mr

10 Advances in Materials Science and Engineering

4 Results and Discussion

0is study defined equations for the motion of a drill bitwhen struck by a piston and expanded the dynamic prop-erties of the drill bit based on the conditions ζ and α It alsoanalyzed the relationship between the bit dynamics and theduration of the incident stress wave (τ) Underdamped drillbits displayed the greatest responsiveness and the fastesttransmission of τ Increasing the transmission of τ decreasedthe loading effect due to the drill bitrsquos inertia but led to a fastresponse 0e inertial effect of the drill bit was confirmed tobe inversely proportional to the transmission of τ

A percussive drilling system accomplishes its work viathe percussion impact of a piston In the systems studiedhere τ ended more quickly with lower mr and β values asthese values increased the duration of τ transmitted to therock via the drill bit increased in proportion to β 0e resultsalso showed that the maximum bit displacement (um) de-creased as β increased Analysis of the drilling dynamicsconfirmed the effective intervals of β With reference to therock strengths suggested by the International Society forRock Mechanics (ISRM) the effective interval of β is con-sidered to be no greater than 2

0e main purpose of this study was to examine theimpact energy transmission rate and drilling efficiency ina percussive drilling system (Figure 8) 0e results establishedamr value of 05 as the most efficient for rocks whose strengthcorresponds to the interval βlt 2 0e improved drilling ef-ficiency would lead to advantages such as reduced bit pro-duction costs Rocks stiffer than 2ltβlt 4 can be mostefficiently drilled by selecting mr 1

0emovement and response characteristics of bit duringpercussion process τ and transfer rate of energy caused bypercussion an efficiency depend on mr and β where in-crease in mr leads to overdamping of bit movement char-acteristics due to τ by piston and reduction in drillingefficiency due to internal energy effect by mass of drill bit Asthe increase in mass of drill bit for the section of 4lt βlt 6the drilling efficiency of percussive drilling system decreaseand for the very hard rock (UCSgt 200MPa) section of4lt βlt 6 it is considered that the reasonable applied mr ofdrilling tool is 1 or 2

0is study neglected the effects of the secondary incidentstress wave and also the effect of buttons embedded in thedrill bit Additionally for analytical simplification the pistonand drill bit were assumed to be of the same diameter andmaterial and a rectangular pulse with an incident stresswave of duration τ and amplitude 05ρcv was assumed 0eeffect of the flexural stress wave depending on the config-uration was not considered 0ese limitations of this papershould be addressed in further investigations that take intoaccount the configuration of the piston and drill bit theeffect of buttons and the drilling efficiency given differingshapes of the incident stress wave

5 Conclusion

0is paper aimed to identify the optimal design parametersfor percussive drilling systems by introducing a drill bit-rock

interaction model that could verify the bitrsquos motion duringpercussion and the resulting damping characteristics 0estudy analyzed drilling efficiency and drew the followingconclusions

Percussive drilling systems have six dynamic drill bitproperties that can be expandable 0is paper discussed thephysical meaning of the dimensionless parameters α and β0eir values determined the damping characteristics thatcan lead to the rockrsquos maximum fracture displacement 0efastest response of τ was observed for underdamped drill-bitmotion

Drilling was most efficient in the interval 1lt βlt 4 whereincreasing mr at a given β decreased drilling efficiency 0evalue of mr for efficient drilling was determined by the rockstrength (ie β)

0e results indicate that application of a piston-to-drill bitmass ratio of 05 (ie a piston mass twice that of the drill bitmass) to the rocks whose stiffness corresponds to βge 2 wouldlikely be most efficient and also reduce drill bit productioncosts Furthermore a selection ofmr 1 would be valid whendeveloping drill tools for boring complex rocks (1ltβlt 4) Atβgt 4 the best efficiency could be achieved when the bit massequals or exceeds the piston mass

Appendix

Theoretical Case of Bit Motion Condition(Drilling Dynamic Model)

0is paper proposed six conditions for the dynamics ofa drill bit struck by a piston impact as summarizedbelow 0e damping conditions for the damping ratiosand α from the drill-bit equations of motion (7ndash9) are asfollows

First for the overdamped condition of ζ1 gt 1 and αlt 1 τis transmitted to the drill bit indicating loading Equation(7) can be expressed as in (A1) 0e initial conditions are0lt tle τ σi 05ρcv F ku u0 0 and _u0 0

eurou +ρcA

mb

1113888 1113889 _u +k

mb

1113888 1113889u ρcA

mb

1113888 1113889v (A1)

where assuming the normal state u can be calculated asfollows

u ρcA

k1113874 1113875v (A2)

Next we examine the case where τ is complete and theloading effect is sustained owing to external forces and drill-bit inertia Applying the conditions of τ lt tle tm σi 0F ku u0 u(τ) and _u0 _u(τ) to (A1) results in thefollowing expression

eurou +ρcA

mb

1113888 1113889 _u +k

mb

1113888 1113889u 0 (A3)

where u 0 Last we examine the unloading conditions oftm lt t σi 0 F ckuminus ckuf u0 um and _u0 0 Apply-ing these to (9) when unloading gives

Advances in Materials Science and Engineering 11

eurou +ρcA

mb

1113888 1113889 _u + ck

mb

1113888 1113889u ck

mb

1113888 1113889uf (A4)

where bit displacement u uf is used to indicate the rockrsquosfracture displacement Furthermore considering the bitrsquosdamping ratio when unloading its behavior can be ex-pressed as

Damped mode

ζ2 gt 1 αlt1c

over( )

ζ2 1 α 1c

critical( )

ζ2 lt 1 αgt1c

under( )

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A5)

Second under critical damping where ζ1 1 and α 1 τis transmitted to the drill bit indicating loading Equation(7) can be expressed as in (A1) 0e initial conditions are0lt tle τ σi 05ρcv F ku u0 0 and _u0 0

In this case assuming the normal state u can be cal-culated using (A2)

0e following is the case where τ is complete and theloading effect is sustained owing to external forces and drillbit inertia Applying the conditions of τ lt tle tm σi 0F ku u0 u(τ) and _u0 _u(τ) to (A1) give (A3) whereu 0 Last we examine the unloading condition wheretm lt t σi 0 F ckuminus ckuf u0 um and _u0 0 Equation(9) when unloading can be expressed as (A4)

In this case bit displacement u uf is used to indicate therockrsquos fracture displacement Furthermore considering thebitrsquos damping ratio when unloading its behavior follows

Damped mode

ζ2 gt 1 αlt1c

(no case)

ζ2 1 α 1c

(no case)

ζ2 lt 1 αgt1c

under( )

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A6)

0e second condition of critical damping is defined asα 1When unloading drill bits c refers to the rockrsquos propertyeffect and is always less than 1 Hence when unloading noother cases but the underdamped mode are allowed In otherwords when unloading in the critically damped mode owingto the rock properties the bit is always underdamped

0e third condition is underdamping with ζ1 lt 1 andαgt 1 Here τ is transmitted to the drill bit meaning loadingEquation (7) can be expressed by (A1)0e initial conditionsare 0lt tle τ σi 05ρcv F ku u0 0 and _u0 0 Whereassuming the normal state u can be calculated as (A2)

0e following is the case where τ is complete and theloading effect is sustained owing to external forces and drillbit inertia Applying the conditions of τ lt tle tm σi 0F ku u0 u(τ) and _u0 _u(τ) to (A1) results in (A3)

where u 0 Last we examine the unloading conditionwhere tm lt t σi 0 F ckuminus ckuf u0 um and _u0 0Equation (9) when unloading can be expressed as (A4)

In this case the drill bit displacement u uf is used toindicate the rockrsquos fracture displacement Furthermoreconsidering the damping ratio in the drill bitrsquos unloadingthe bitrsquos behavior can be expressed as

Damped mode

ζ2 gt 1 αlt1c

(no case)

ζ2 1 α 1c

(no case)

ζ2 lt 1 αgt1c

under( )

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A7)

0e third condition is the critically damped modedefined as αgt 1 During drill bit unloading c refers to therock property effect and is always less than 1 Hencewhen unloading no other cases but underdamping areallowed

Next the underdamped condition is where ζ1 lt 1 andαgt 1 and τ is slower than tm (the time at which the rockrsquosmaximum fracture displacement is reached) that is tm lt τIn this case τ is transmitted to the drill bit meaningloading Equation (7) can be expressed by (A1) 0e initialconditions are 0lt tle tm σi 05ρcv F ku u0 0 and_u0 0 Here assuming the normal state u can be calculatedas (A2)

0e following case is where conversion is made tounloading after the drill bit reaches its maximum dis-placement but the incident stress wave is sustained Ap-plying the conditions of tm lt tle τ σi 05ρcvF ckuminus ckuf u0 um and _u0 0 to (9) results in

eurou +ρcA

mb

1113888 1113889 _u + ck

mb

1113888 1113889u ρcA

mb

1113888 1113889v + ck

mb

1113888 1113889uf (A8)

where the definition of u 1c(ρcAk)v + uf is possibleLast this unloading condition is where the incident

stress wave is resolved When unloading the drill bit usingthe conditions tlt τ σi 0 F ckuminus ckuf u0 u(τ) and_u0 _u(τ) allows (9) to be expressed as (A4)

In this case drill bit displacement u uf is used to in-dicate the rockrsquos fracture displacement 0e analysis of drillbit dynamics showed that when loading transitions tounloading in the underdamped condition the dynamicproperties of the bit exist only in the underdamping mode

Conflicts of Interest

The authors declare that there are no conflicts of interest

References

[1] C H Song K B Kwon J Y Park et al ldquoOptimum design ofthe internal flushing channel of a drill bit using RSM and CFDsimulationrdquo International Journal of Precision Engineeringand Manufacturing vol 15 no 6 pp 1041ndash1050 2014

12 Advances in Materials Science and Engineering

[2] X Li G Rupert D A Summers P Santi and D Liu ldquoAnalysisof impact hammer rebound to estimate rock drillabilityrdquo RockMechanics and Rock Engineering vol 33 no 1 pp 1ndash13 2000

[3] W A Hustrulid and C Fairhurst ldquoA theoretical and ex-perimental study of the percussive drilling of rock part Itheory of percussive drillingrdquo International Journal of RockMechanics and Mining Sciences amp Geomechanics Abstractsvol 8 no 4 pp 311ndash333 1971

[4] W A Hustrulid and C Fairhurst ldquoA theoretical and ex-perimental study of the percussive drilling of rock part IIforce-penetration and specific energy determinationrdquo In-ternational Journal of Rock Mechanics and Mining Sciences ampGeomechanics Abstracts vol 8 no 4 pp 335ndash356 1971

[5] W A Hustrulid and C Fairhurst ldquoA theoretical and ex-perimental study of the percussive drilling of rock part IIIexperimental verification of the mathematical theoryrdquo In-ternational Journal of Rock Mechanics and Mining Sciences ampGeomechanics Abstracts vol 9 no 3 pp 417-418 1972

[6] W A Hustrulid and C Fairhurst ldquoA theoretical and ex-perimental study of the percussive drilling of rock part IVapplication of the model to actual percussive drillingrdquo In-ternational Journal of Rock Mechanics and Mining Sciencesvol 9 no 3 pp 431ndash442 1972

[7] L E Chiang and D A Elias ldquoA 3D FEM methodology forsimulating the impact in rock-drilling hammersrdquo In-ternational Journal of Rock Mechanics and Mining Sciencesvol 45 no 5 pp 701ndash711 2008

[8] C H Song K B Kwon M G Cho J Y Oh D Y Shin andJ W Cho ldquoDevelopment of lab-scale rock drill apparatus fortesting performance of a drill bitrdquo International Journal ofPrecision Engineering and Manufacturing vol 16 no 7pp 1402ndash1414 2015

[9] K B Kwon C H Song J Y Park J Y Oh J W Lee andJ W Cho ldquoEvaluation of drilling efficiency by percussiontesting of a drill bit with new button arrangementrdquo In-ternational Journal of Precision Engineering andManufacturing vol 15 no 6 pp 1063ndash1068 2014

[10] B Lundberg andM Okrouhlik ldquoInfluence of 3D effects on theefficiency of percussive rock drillingrdquo International Journal ofimpact Engineering vol 25 no 4 pp 345ndash360 2001

[11] B Lundberg and P Collet ldquoOptimal wave shape with respectto efficiency in percussive drilling with detachable drill bitrdquoInternational Journal of impact Engineering vol 86 pp 179ndash187 2015

[12] X B Li G Rupert and D A Summers ldquoEnergy transmissionof down-hole hammer tool and its conditionalityrdquo Trans-actions of Nonferrous Metals Society of China vol 10 no 1pp 109ndash111 2000

[13] J W Cho S Jeon S H Yu and S H Chang ldquoOptimumspacing of TBM disc cutters a numerical simulation using thethree-dimensional dynamic fracturing methodrdquo Tunnellingand Underground Space Technology vol 25 no 3 pp 230ndash244 2010

[14] W Changming ldquoAn analytical study of percussive energytransfer in hydraulic rock drillsrdquo Mining Science and Tech-nology vol 13 no 1 pp 57ndash68 1991

[15] Online Materials Information Resource httpwwwmatwebcomsearchDataSheetaspxMatGUID8bc5d558f4174e6082ddf4966e382bd6ampckck1

Advances in Materials Science and Engineering 13

e initial height h0 is v22g and the rebound eh canbe dened as the ratio between the initial height and thepistonrsquos postimpact rebound height h ese areexpressed as follows [2]

eh h

h0v2ev2 (13)

e eciency η of impact energy transmission is de-ned as the ratio between the kinetic energy generated fromthe impact of the piston and the energy transmitted to therock [12] and is given as

η kum

2

ρALhv2

Fmax( )2

ρALhv2k (14)

Which can be expressed as follows through dimensionalanalysis

η 2βu2m (15)

e value um during loading can be obtained from (7)the equation of motion for the drill bit e obtained um canbe expressed as in (16) for each damping mode where theinitial conditions are 0lt tle τ F kui u0 0 and _u0 0

u()

Critically damped motion

Underdamped motion

tm

uf

um

u

t

0004000300020001000000

04

08

12

16

20

Loading by incident stress waveLoading by inertia of bitUnloading

(e)

Loading by incident stress waveLoading by inertia of bitUnloading

u()Underdamped motion

tm

uf

um

u

t

0004000300020001000000

04

08

12

16

20

(f)

Loading by incident stress waveLoading by inertia of bitUnloading

u()Underdampedmotion

uf

u

u

t

t

000400030002

00002 00003 00004

0001000000

04

08

12

16

20

um

um

tm

(g)

Figure 4 Results of the motion response of the bit formr 1 (a) All cases (b) Case 1 (c) Case 2 (d) Case 3 (e) Case 4 (f) Case 5 (g) Case 6

Advances in Materials Science and Engineering 7

um(t)

1 +1minus

αminus 1

radic

2αminus 1

radic middot eminus(2α)(βτ)(1+

1minusα

radic)t minus

1 +1minus α

radic

21minus α

radic middot eminus(2α)(βτ)(1minus

1minusα

radic)t

1113890 1113891(overdamped)

1minus eminus2(βτ)t middot 2βτ

t + 11113890 1113891(critical damped)

1minus eminus2(βτ)t middot1

αminus 1

radic sin2

αminus 1

radic

αβτ

t1113888 1113889 + cos2

αminus 1

radic

αβτ

t1113888 11138891113890 1113891(underdamped)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(16)

Moreover for tmlt τ um is expressed as follows

um(t) 1minus eminus(2α)(βτ)

middot ⎡⎣1

αminus 1

radic sin2

αminus 1

radic

αβτ

t1113888 1113889

+ cos2

αminus 1

radic

αβτ

t1113888 1113889⎤⎦

(17)

0e initial conditions are 0lt tle τ F kui ui 0and _ui 0

Next the displacement value uf during unload-ing can be obtained via the equation of motion for a drill-bit (9) as shown in (18) 0e initial conditionsduring unloading are tgt τ F 0 u0 u(τ) and_u0 _u(τ)

uf(t)

α4

1minus α

radicτβ

middot eminus(2α)(1+

1minusα

radic)t

middot u02αβτ

(1 +1minus α

radic) minus _u0 +

4αβτu01113888 11138891113890 1113891

minusα

41minus α

radicτβ

middot eminus(2α)(1minus

1minusα

radic)t

middot u02αβτ

(1minus1minus α

radic) minus _u0 +

4αβτu01113888 11138891113890 1113891(overdamped)

eminus2(βτ)t middot _u0 + 2βτu01113888 1113889tminus u01113890 1113891(critical damped)

eminus(2α)(βτ)t middot ⎡⎣α

2αminus 1

radicτβ

1113888 1113889 _u0 middot sin2

αminus 1

radic

α1113888 1113889

βτ

t +1

αminus 1

radic u0 middot sin2

αminus 1

radic

αβτ

t1113888 1113889

+ u0 middot cos2

αminus 1

radic

αβτ

t1113888 1113889⎤⎦(underdamped)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(18)

Table 2 Dynamic drill-bit properties for loading unloading and τ and the damping modes capable of reaching um

Case no ConditionsDamping modes capable of reaching um

0lt tle τ τ lt tle tm tgt tm

1 αlt 1c

Over Overlowast Over2 α 1

cOver Overlowast Critical

3 1clt αlt 1 Over Overlowast Under

4 α 1 Critical Criticallowast Under5 αgt 1 tm gt τ Under Underlowast Under

6 αgt 1 tm lt τ0lt tle tm tm lt tle τ tgt τUnderlowast Under Under

lowastDamping modes capable of reaching um

8 Advances in Materials Science and Engineering

eeciency of the percussive drilling systems for given βcan be calculated using (14ndash17) e factor um which isimportant to the eciency calculations is discussed in depth

32 Results of Energy Transmission Eciency e results ofsimulations on the drill bit-rock interaction model shown inFigure 5 indicate the damped modes that can arrive at um (forvarying values of β) as the response by the drill bit at the time ofpiston impact e damped mode is determined by ζ and αwhich indictate the system responses Overdamped mode hasαlt 1 and can be expressed as follows

2mb

mh( )βlt 1

βlt1

2mr

(19)

e critical damped mode has α 1 and is expressed asfollows

2mb

mh( )β 1

β 1

2mr

(20)

e underdamped mode has αgt 1 and is expressed as

2mb

mh( )βgt 1

βgt1

2mr

(21)

where mbmh mr

e special underdamped mode refers to the case whereτ is faster than t indicating that the drill bit can arrive at umonly via an incident stress wave In this case the bit reachesits maximum displacement and transitions to the unloadingstate while the transmission of τ is underway

e damped mode of each mr shown in Figure 5 re-veals that the underdamped mode is dominant of thee ects Also overdamped and critical damped were ob-served where β was low is is possibly because of thee ects of the internal energy (ie compression strength)of rock and the percussive energy e overdamped andcritical damped were also observed is is possibly be-cause of internal energy (ie compression strength) ofrock and energy caused by percussion

Figure 6 shows dimensionless time tm tτ capable ofreaching Um for β and τ In Section A of tgt τ the bit reachesUm after termination of τ and in Section B of tlt τ the bitreachesUm before the termination of τe lower themr andβ the faster the termination of τ and increase in thesetended to make the duration of τ delivered to rock throughbit to increase proportionally to β is suggests that thehigher rock strength requires longer duration of τ fordrilling the rock

Figure 7 shows dimensionless Um state dened as ratioum a displacement um by β and uinfin caused by incident stresswave and loading condition e prediction of displacementof drill bit drilling displacement by characteristic of rockwas limited is study accordingly judged the di-mensionless drilling state Um only by using ratio uinfina displacement during loading and um considering inertiacaused during percussion (22)

Um umuinfin (22)

where the uinfin only by loading is dened as follows

mr = 1

mr = 2

mr = 05

0 5 10

15 20

4

3

2

Dam

ped

mod

es

1

0

mr = 1mr = 2

mr = 05

mr = 1

mr = 05

Figure 5 Damping mode capable of reaching maximum rockfracture displacement um given the piston-to-bit mass ratiomr and β1 overdamped mode 2 critically damped mode 3 underdampedmode 4 special underdamped mode

0 5 10

15 2000

02

06

10

14

18

20

04

08

12

16

Dim

ensio

nles

s pen

etra

tion

(Um

)

Section B

Section A

mr = 05mr = 1mr = 2

uinfin

Figure 6 State of reaching drill bit dimensionless penetration Umwith respect to mass ratio mr and β

Advances in Materials Science and Engineering 9

uinfin ρcAkv

τβv (23)

e e ect of initial condition on uinfin decreased with theincrease in β e um is determined by considering loadingcondition and inertia e ect is a ected by overresponse anddecreases as β increases

Section A refers to umlt uinfin with only τ and loadinge ect present (ie the condition for the normal state in theinitial mode) uinfin means only the τ e ect resulting fromloading exists Section B refers to umgt uinfin a state in whichum can be reached through τ and loading is exceeded issection is where the drill bit inertia and underdampedmotion appear e gradient of the curve for maximumdisplacement of the bit varies with β indicating that as therock strength increases the displacement allowing fracturedecreases Near value 2 β changes its gradient for maximumdisplacement is means that in Section A the rock is lesssti than the piston owing to the rock-sti ness e ect and inSection B the rock is sti er than the piston β is dened asthe sti ness ratio between the rock and the piston as in (11)Hence when βlt 2 a condition is established for rock fractureto occur in that the piston sti ness exceeds the rock sti ness

In examining um for piston-bit mass ratios of mrlt 2 therock strength that can be defeated by τ occurring in theinitial loading tends to increase is tendency is suspectedto result from the inertial e ect of the mass of the bit Forβgt 4 the um gradient tended to decrease with increasingmass of the bit Hence the eciency is considered to de-crease which is attributed to the e ect of the rockrsquos com-pressive strength e uniaxial compressive strength of veryhard rock is typically at least 200MPa while the tensilestrength of the H13 tool steels that are used mostly forpistons (ASTM A681 and DIN EN ISO 4957) is known to beat least 359ndash1170MPa [15]

Given the steels generally used to make pistons thee ective range of β is considered to be 034lt βlt 112 which

was calculated considering the strength ratio between thehard rock and an H13 steel tool Hence the e ective range ofβ could be no greater than 2 Rocks with a sti ness ratio ofβgt 2 might cause failure through the pistonrsquos plastic de-formation and fracture is study examined the e ectiveinterval of β and investigated the parameter combinationsgiving high drilling eciency in this interval

Figure 8 illustrates the motion of the bit generatedduring the percussion process and the energy transmissionrate (ie eciency) for the response characteristics τ andpercussion impact Analysis of mr found the maximumdrilling eciency in interval 1lt βlt 2 with a tendency ef-ciency decreasing after mr increased above a certain valueis result was attributed to the tendency whereby increasingmr converts the systemrsquos dynamic response to τ due to thepiston from underdamped to overdamped Furthermoreincreasing the mass of the bit can increase its internal energy(ie friction energy) to above the piston transmitted energy(ie kinetic and potential energy) Hence the bitrsquos dynamicresponse would be overdamped and could compromisedrilling eciency In other words the heavier the bit themore delayed the response to percussion

In Figure 8 the left side of red line (β18) represents thee ect of only incident stress wave meaning that the higherthe value in graph the higher the drilling eciency in in-cident stress wave e right side of red line representingunloading behavior after reaching maximum eciency inincident stress wave is a ected by both brittleness char-acteristics of rock and inertia e ect of drill bit

e piston should have twice the mass of the drill bit(iemr 05) for the most ecient drilling for βle 2 which isestablished to be the e ective interval of β and correspondsto both soft and hard rocks Drilling sti er rocks (2lt βlt 4)might be less ecient but increasing mr to 1 would helpimprove drilling eciency

00000

00002

00004

00006

00008

00010

00012

Dim

ensio

nles

s tim

e (t m

)

Section A

Section B

0 5 10

15 20

mr = 05mr = 1mr = 2

Figure 7 Dimensionless time capable of reaching Um for an in-cident stress-wave τ acting on the bit and β

0 2 4 6 8 10 12 14 16 18 20

= 1800

01

02

03

04

05

06

07

08

09

10

Effic

ienc

y

mr = 01mr = 025mr = 05mr = 1

mr = 2mr = 4mr = 10

Figure 8 Drilling eciency of percussive drilling systems withrespect to β and mr

10 Advances in Materials Science and Engineering

4 Results and Discussion

0is study defined equations for the motion of a drill bitwhen struck by a piston and expanded the dynamic prop-erties of the drill bit based on the conditions ζ and α It alsoanalyzed the relationship between the bit dynamics and theduration of the incident stress wave (τ) Underdamped drillbits displayed the greatest responsiveness and the fastesttransmission of τ Increasing the transmission of τ decreasedthe loading effect due to the drill bitrsquos inertia but led to a fastresponse 0e inertial effect of the drill bit was confirmed tobe inversely proportional to the transmission of τ

A percussive drilling system accomplishes its work viathe percussion impact of a piston In the systems studiedhere τ ended more quickly with lower mr and β values asthese values increased the duration of τ transmitted to therock via the drill bit increased in proportion to β 0e resultsalso showed that the maximum bit displacement (um) de-creased as β increased Analysis of the drilling dynamicsconfirmed the effective intervals of β With reference to therock strengths suggested by the International Society forRock Mechanics (ISRM) the effective interval of β is con-sidered to be no greater than 2

0e main purpose of this study was to examine theimpact energy transmission rate and drilling efficiency ina percussive drilling system (Figure 8) 0e results establishedamr value of 05 as the most efficient for rocks whose strengthcorresponds to the interval βlt 2 0e improved drilling ef-ficiency would lead to advantages such as reduced bit pro-duction costs Rocks stiffer than 2ltβlt 4 can be mostefficiently drilled by selecting mr 1

0emovement and response characteristics of bit duringpercussion process τ and transfer rate of energy caused bypercussion an efficiency depend on mr and β where in-crease in mr leads to overdamping of bit movement char-acteristics due to τ by piston and reduction in drillingefficiency due to internal energy effect by mass of drill bit Asthe increase in mass of drill bit for the section of 4lt βlt 6the drilling efficiency of percussive drilling system decreaseand for the very hard rock (UCSgt 200MPa) section of4lt βlt 6 it is considered that the reasonable applied mr ofdrilling tool is 1 or 2

0is study neglected the effects of the secondary incidentstress wave and also the effect of buttons embedded in thedrill bit Additionally for analytical simplification the pistonand drill bit were assumed to be of the same diameter andmaterial and a rectangular pulse with an incident stresswave of duration τ and amplitude 05ρcv was assumed 0eeffect of the flexural stress wave depending on the config-uration was not considered 0ese limitations of this papershould be addressed in further investigations that take intoaccount the configuration of the piston and drill bit theeffect of buttons and the drilling efficiency given differingshapes of the incident stress wave

5 Conclusion

0is paper aimed to identify the optimal design parametersfor percussive drilling systems by introducing a drill bit-rock

interaction model that could verify the bitrsquos motion duringpercussion and the resulting damping characteristics 0estudy analyzed drilling efficiency and drew the followingconclusions

Percussive drilling systems have six dynamic drill bitproperties that can be expandable 0is paper discussed thephysical meaning of the dimensionless parameters α and β0eir values determined the damping characteristics thatcan lead to the rockrsquos maximum fracture displacement 0efastest response of τ was observed for underdamped drill-bitmotion

Drilling was most efficient in the interval 1lt βlt 4 whereincreasing mr at a given β decreased drilling efficiency 0evalue of mr for efficient drilling was determined by the rockstrength (ie β)

0e results indicate that application of a piston-to-drill bitmass ratio of 05 (ie a piston mass twice that of the drill bitmass) to the rocks whose stiffness corresponds to βge 2 wouldlikely be most efficient and also reduce drill bit productioncosts Furthermore a selection ofmr 1 would be valid whendeveloping drill tools for boring complex rocks (1ltβlt 4) Atβgt 4 the best efficiency could be achieved when the bit massequals or exceeds the piston mass

Appendix

Theoretical Case of Bit Motion Condition(Drilling Dynamic Model)

0is paper proposed six conditions for the dynamics ofa drill bit struck by a piston impact as summarizedbelow 0e damping conditions for the damping ratiosand α from the drill-bit equations of motion (7ndash9) are asfollows

First for the overdamped condition of ζ1 gt 1 and αlt 1 τis transmitted to the drill bit indicating loading Equation(7) can be expressed as in (A1) 0e initial conditions are0lt tle τ σi 05ρcv F ku u0 0 and _u0 0

eurou +ρcA

mb

1113888 1113889 _u +k

mb

1113888 1113889u ρcA

mb

1113888 1113889v (A1)

where assuming the normal state u can be calculated asfollows

u ρcA

k1113874 1113875v (A2)

Next we examine the case where τ is complete and theloading effect is sustained owing to external forces and drill-bit inertia Applying the conditions of τ lt tle tm σi 0F ku u0 u(τ) and _u0 _u(τ) to (A1) results in thefollowing expression

eurou +ρcA

mb

1113888 1113889 _u +k

mb

1113888 1113889u 0 (A3)

where u 0 Last we examine the unloading conditions oftm lt t σi 0 F ckuminus ckuf u0 um and _u0 0 Apply-ing these to (9) when unloading gives

Advances in Materials Science and Engineering 11

eurou +ρcA

mb

1113888 1113889 _u + ck

mb

1113888 1113889u ck

mb

1113888 1113889uf (A4)

where bit displacement u uf is used to indicate the rockrsquosfracture displacement Furthermore considering the bitrsquosdamping ratio when unloading its behavior can be ex-pressed as

Damped mode

ζ2 gt 1 αlt1c

over( )

ζ2 1 α 1c

critical( )

ζ2 lt 1 αgt1c

under( )

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A5)

Second under critical damping where ζ1 1 and α 1 τis transmitted to the drill bit indicating loading Equation(7) can be expressed as in (A1) 0e initial conditions are0lt tle τ σi 05ρcv F ku u0 0 and _u0 0

In this case assuming the normal state u can be cal-culated using (A2)

0e following is the case where τ is complete and theloading effect is sustained owing to external forces and drillbit inertia Applying the conditions of τ lt tle tm σi 0F ku u0 u(τ) and _u0 _u(τ) to (A1) give (A3) whereu 0 Last we examine the unloading condition wheretm lt t σi 0 F ckuminus ckuf u0 um and _u0 0 Equation(9) when unloading can be expressed as (A4)

In this case bit displacement u uf is used to indicate therockrsquos fracture displacement Furthermore considering thebitrsquos damping ratio when unloading its behavior follows

Damped mode

ζ2 gt 1 αlt1c

(no case)

ζ2 1 α 1c

(no case)

ζ2 lt 1 αgt1c

under( )

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A6)

0e second condition of critical damping is defined asα 1When unloading drill bits c refers to the rockrsquos propertyeffect and is always less than 1 Hence when unloading noother cases but the underdamped mode are allowed In otherwords when unloading in the critically damped mode owingto the rock properties the bit is always underdamped

0e third condition is underdamping with ζ1 lt 1 andαgt 1 Here τ is transmitted to the drill bit meaning loadingEquation (7) can be expressed by (A1)0e initial conditionsare 0lt tle τ σi 05ρcv F ku u0 0 and _u0 0 Whereassuming the normal state u can be calculated as (A2)

0e following is the case where τ is complete and theloading effect is sustained owing to external forces and drillbit inertia Applying the conditions of τ lt tle tm σi 0F ku u0 u(τ) and _u0 _u(τ) to (A1) results in (A3)

where u 0 Last we examine the unloading conditionwhere tm lt t σi 0 F ckuminus ckuf u0 um and _u0 0Equation (9) when unloading can be expressed as (A4)

In this case the drill bit displacement u uf is used toindicate the rockrsquos fracture displacement Furthermoreconsidering the damping ratio in the drill bitrsquos unloadingthe bitrsquos behavior can be expressed as

Damped mode

ζ2 gt 1 αlt1c

(no case)

ζ2 1 α 1c

(no case)

ζ2 lt 1 αgt1c

under( )

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A7)

0e third condition is the critically damped modedefined as αgt 1 During drill bit unloading c refers to therock property effect and is always less than 1 Hencewhen unloading no other cases but underdamping areallowed

Next the underdamped condition is where ζ1 lt 1 andαgt 1 and τ is slower than tm (the time at which the rockrsquosmaximum fracture displacement is reached) that is tm lt τIn this case τ is transmitted to the drill bit meaningloading Equation (7) can be expressed by (A1) 0e initialconditions are 0lt tle tm σi 05ρcv F ku u0 0 and_u0 0 Here assuming the normal state u can be calculatedas (A2)

0e following case is where conversion is made tounloading after the drill bit reaches its maximum dis-placement but the incident stress wave is sustained Ap-plying the conditions of tm lt tle τ σi 05ρcvF ckuminus ckuf u0 um and _u0 0 to (9) results in

eurou +ρcA

mb

1113888 1113889 _u + ck

mb

1113888 1113889u ρcA

mb

1113888 1113889v + ck

mb

1113888 1113889uf (A8)

where the definition of u 1c(ρcAk)v + uf is possibleLast this unloading condition is where the incident

stress wave is resolved When unloading the drill bit usingthe conditions tlt τ σi 0 F ckuminus ckuf u0 u(τ) and_u0 _u(τ) allows (9) to be expressed as (A4)

In this case drill bit displacement u uf is used to in-dicate the rockrsquos fracture displacement 0e analysis of drillbit dynamics showed that when loading transitions tounloading in the underdamped condition the dynamicproperties of the bit exist only in the underdamping mode

Conflicts of Interest

The authors declare that there are no conflicts of interest

References

[1] C H Song K B Kwon J Y Park et al ldquoOptimum design ofthe internal flushing channel of a drill bit using RSM and CFDsimulationrdquo International Journal of Precision Engineeringand Manufacturing vol 15 no 6 pp 1041ndash1050 2014

12 Advances in Materials Science and Engineering

[2] X Li G Rupert D A Summers P Santi and D Liu ldquoAnalysisof impact hammer rebound to estimate rock drillabilityrdquo RockMechanics and Rock Engineering vol 33 no 1 pp 1ndash13 2000

[3] W A Hustrulid and C Fairhurst ldquoA theoretical and ex-perimental study of the percussive drilling of rock part Itheory of percussive drillingrdquo International Journal of RockMechanics and Mining Sciences amp Geomechanics Abstractsvol 8 no 4 pp 311ndash333 1971

[4] W A Hustrulid and C Fairhurst ldquoA theoretical and ex-perimental study of the percussive drilling of rock part IIforce-penetration and specific energy determinationrdquo In-ternational Journal of Rock Mechanics and Mining Sciences ampGeomechanics Abstracts vol 8 no 4 pp 335ndash356 1971

[5] W A Hustrulid and C Fairhurst ldquoA theoretical and ex-perimental study of the percussive drilling of rock part IIIexperimental verification of the mathematical theoryrdquo In-ternational Journal of Rock Mechanics and Mining Sciences ampGeomechanics Abstracts vol 9 no 3 pp 417-418 1972

[6] W A Hustrulid and C Fairhurst ldquoA theoretical and ex-perimental study of the percussive drilling of rock part IVapplication of the model to actual percussive drillingrdquo In-ternational Journal of Rock Mechanics and Mining Sciencesvol 9 no 3 pp 431ndash442 1972

[7] L E Chiang and D A Elias ldquoA 3D FEM methodology forsimulating the impact in rock-drilling hammersrdquo In-ternational Journal of Rock Mechanics and Mining Sciencesvol 45 no 5 pp 701ndash711 2008

[8] C H Song K B Kwon M G Cho J Y Oh D Y Shin andJ W Cho ldquoDevelopment of lab-scale rock drill apparatus fortesting performance of a drill bitrdquo International Journal ofPrecision Engineering and Manufacturing vol 16 no 7pp 1402ndash1414 2015

[9] K B Kwon C H Song J Y Park J Y Oh J W Lee andJ W Cho ldquoEvaluation of drilling efficiency by percussiontesting of a drill bit with new button arrangementrdquo In-ternational Journal of Precision Engineering andManufacturing vol 15 no 6 pp 1063ndash1068 2014

[10] B Lundberg andM Okrouhlik ldquoInfluence of 3D effects on theefficiency of percussive rock drillingrdquo International Journal ofimpact Engineering vol 25 no 4 pp 345ndash360 2001

[11] B Lundberg and P Collet ldquoOptimal wave shape with respectto efficiency in percussive drilling with detachable drill bitrdquoInternational Journal of impact Engineering vol 86 pp 179ndash187 2015

[12] X B Li G Rupert and D A Summers ldquoEnergy transmissionof down-hole hammer tool and its conditionalityrdquo Trans-actions of Nonferrous Metals Society of China vol 10 no 1pp 109ndash111 2000

[13] J W Cho S Jeon S H Yu and S H Chang ldquoOptimumspacing of TBM disc cutters a numerical simulation using thethree-dimensional dynamic fracturing methodrdquo Tunnellingand Underground Space Technology vol 25 no 3 pp 230ndash244 2010

[14] W Changming ldquoAn analytical study of percussive energytransfer in hydraulic rock drillsrdquo Mining Science and Tech-nology vol 13 no 1 pp 57ndash68 1991

[15] Online Materials Information Resource httpwwwmatwebcomsearchDataSheetaspxMatGUID8bc5d558f4174e6082ddf4966e382bd6ampckck1

Advances in Materials Science and Engineering 13

um(t)

1 +1minus

αminus 1

radic

2αminus 1

radic middot eminus(2α)(βτ)(1+

1minusα

radic)t minus

1 +1minus α

radic

21minus α

radic middot eminus(2α)(βτ)(1minus

1minusα

radic)t

1113890 1113891(overdamped)

1minus eminus2(βτ)t middot 2βτ

t + 11113890 1113891(critical damped)

1minus eminus2(βτ)t middot1

αminus 1

radic sin2

αminus 1

radic

αβτ

t1113888 1113889 + cos2

αminus 1

radic

αβτ

t1113888 11138891113890 1113891(underdamped)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(16)

Moreover for tmlt τ um is expressed as follows

um(t) 1minus eminus(2α)(βτ)

middot ⎡⎣1

αminus 1

radic sin2

αminus 1

radic

αβτ

t1113888 1113889

+ cos2

αminus 1

radic

αβτ

t1113888 1113889⎤⎦

(17)

0e initial conditions are 0lt tle τ F kui ui 0and _ui 0

Next the displacement value uf during unload-ing can be obtained via the equation of motion for a drill-bit (9) as shown in (18) 0e initial conditionsduring unloading are tgt τ F 0 u0 u(τ) and_u0 _u(τ)

uf(t)

α4

1minus α

radicτβ

middot eminus(2α)(1+

1minusα

radic)t

middot u02αβτ

(1 +1minus α

radic) minus _u0 +

4αβτu01113888 11138891113890 1113891

minusα

41minus α

radicτβ

middot eminus(2α)(1minus

1minusα

radic)t

middot u02αβτ

(1minus1minus α

radic) minus _u0 +

4αβτu01113888 11138891113890 1113891(overdamped)

eminus2(βτ)t middot _u0 + 2βτu01113888 1113889tminus u01113890 1113891(critical damped)

eminus(2α)(βτ)t middot ⎡⎣α

2αminus 1

radicτβ

1113888 1113889 _u0 middot sin2

αminus 1

radic

α1113888 1113889

βτ

t +1

αminus 1

radic u0 middot sin2

αminus 1

radic

αβτ

t1113888 1113889

+ u0 middot cos2

αminus 1

radic

αβτ

t1113888 1113889⎤⎦(underdamped)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(18)

Table 2 Dynamic drill-bit properties for loading unloading and τ and the damping modes capable of reaching um

Case no ConditionsDamping modes capable of reaching um

0lt tle τ τ lt tle tm tgt tm

1 αlt 1c

Over Overlowast Over2 α 1

cOver Overlowast Critical

3 1clt αlt 1 Over Overlowast Under

4 α 1 Critical Criticallowast Under5 αgt 1 tm gt τ Under Underlowast Under

6 αgt 1 tm lt τ0lt tle tm tm lt tle τ tgt τUnderlowast Under Under

lowastDamping modes capable of reaching um

8 Advances in Materials Science and Engineering

eeciency of the percussive drilling systems for given βcan be calculated using (14ndash17) e factor um which isimportant to the eciency calculations is discussed in depth

32 Results of Energy Transmission Eciency e results ofsimulations on the drill bit-rock interaction model shown inFigure 5 indicate the damped modes that can arrive at um (forvarying values of β) as the response by the drill bit at the time ofpiston impact e damped mode is determined by ζ and αwhich indictate the system responses Overdamped mode hasαlt 1 and can be expressed as follows

2mb

mh( )βlt 1

βlt1

2mr

(19)

e critical damped mode has α 1 and is expressed asfollows

2mb

mh( )β 1

β 1

2mr

(20)

e underdamped mode has αgt 1 and is expressed as

2mb

mh( )βgt 1

βgt1

2mr

(21)

where mbmh mr

e special underdamped mode refers to the case whereτ is faster than t indicating that the drill bit can arrive at umonly via an incident stress wave In this case the bit reachesits maximum displacement and transitions to the unloadingstate while the transmission of τ is underway

e damped mode of each mr shown in Figure 5 re-veals that the underdamped mode is dominant of thee ects Also overdamped and critical damped were ob-served where β was low is is possibly because of thee ects of the internal energy (ie compression strength)of rock and the percussive energy e overdamped andcritical damped were also observed is is possibly be-cause of internal energy (ie compression strength) ofrock and energy caused by percussion

Figure 6 shows dimensionless time tm tτ capable ofreaching Um for β and τ In Section A of tgt τ the bit reachesUm after termination of τ and in Section B of tlt τ the bitreachesUm before the termination of τe lower themr andβ the faster the termination of τ and increase in thesetended to make the duration of τ delivered to rock throughbit to increase proportionally to β is suggests that thehigher rock strength requires longer duration of τ fordrilling the rock

Figure 7 shows dimensionless Um state dened as ratioum a displacement um by β and uinfin caused by incident stresswave and loading condition e prediction of displacementof drill bit drilling displacement by characteristic of rockwas limited is study accordingly judged the di-mensionless drilling state Um only by using ratio uinfina displacement during loading and um considering inertiacaused during percussion (22)

Um umuinfin (22)

where the uinfin only by loading is dened as follows

mr = 1

mr = 2

mr = 05

0 5 10

15 20

4

3

2

Dam

ped

mod

es

1

0

mr = 1mr = 2

mr = 05

mr = 1

mr = 05

Figure 5 Damping mode capable of reaching maximum rockfracture displacement um given the piston-to-bit mass ratiomr and β1 overdamped mode 2 critically damped mode 3 underdampedmode 4 special underdamped mode

0 5 10

15 2000

02

06

10

14

18

20

04

08

12

16

Dim

ensio

nles

s pen

etra

tion

(Um

)

Section B

Section A

mr = 05mr = 1mr = 2

uinfin

Figure 6 State of reaching drill bit dimensionless penetration Umwith respect to mass ratio mr and β

Advances in Materials Science and Engineering 9

uinfin ρcAkv

τβv (23)

e e ect of initial condition on uinfin decreased with theincrease in β e um is determined by considering loadingcondition and inertia e ect is a ected by overresponse anddecreases as β increases

Section A refers to umlt uinfin with only τ and loadinge ect present (ie the condition for the normal state in theinitial mode) uinfin means only the τ e ect resulting fromloading exists Section B refers to umgt uinfin a state in whichum can be reached through τ and loading is exceeded issection is where the drill bit inertia and underdampedmotion appear e gradient of the curve for maximumdisplacement of the bit varies with β indicating that as therock strength increases the displacement allowing fracturedecreases Near value 2 β changes its gradient for maximumdisplacement is means that in Section A the rock is lesssti than the piston owing to the rock-sti ness e ect and inSection B the rock is sti er than the piston β is dened asthe sti ness ratio between the rock and the piston as in (11)Hence when βlt 2 a condition is established for rock fractureto occur in that the piston sti ness exceeds the rock sti ness

In examining um for piston-bit mass ratios of mrlt 2 therock strength that can be defeated by τ occurring in theinitial loading tends to increase is tendency is suspectedto result from the inertial e ect of the mass of the bit Forβgt 4 the um gradient tended to decrease with increasingmass of the bit Hence the eciency is considered to de-crease which is attributed to the e ect of the rockrsquos com-pressive strength e uniaxial compressive strength of veryhard rock is typically at least 200MPa while the tensilestrength of the H13 tool steels that are used mostly forpistons (ASTM A681 and DIN EN ISO 4957) is known to beat least 359ndash1170MPa [15]

Given the steels generally used to make pistons thee ective range of β is considered to be 034lt βlt 112 which

was calculated considering the strength ratio between thehard rock and an H13 steel tool Hence the e ective range ofβ could be no greater than 2 Rocks with a sti ness ratio ofβgt 2 might cause failure through the pistonrsquos plastic de-formation and fracture is study examined the e ectiveinterval of β and investigated the parameter combinationsgiving high drilling eciency in this interval

Figure 8 illustrates the motion of the bit generatedduring the percussion process and the energy transmissionrate (ie eciency) for the response characteristics τ andpercussion impact Analysis of mr found the maximumdrilling eciency in interval 1lt βlt 2 with a tendency ef-ciency decreasing after mr increased above a certain valueis result was attributed to the tendency whereby increasingmr converts the systemrsquos dynamic response to τ due to thepiston from underdamped to overdamped Furthermoreincreasing the mass of the bit can increase its internal energy(ie friction energy) to above the piston transmitted energy(ie kinetic and potential energy) Hence the bitrsquos dynamicresponse would be overdamped and could compromisedrilling eciency In other words the heavier the bit themore delayed the response to percussion

In Figure 8 the left side of red line (β18) represents thee ect of only incident stress wave meaning that the higherthe value in graph the higher the drilling eciency in in-cident stress wave e right side of red line representingunloading behavior after reaching maximum eciency inincident stress wave is a ected by both brittleness char-acteristics of rock and inertia e ect of drill bit

e piston should have twice the mass of the drill bit(iemr 05) for the most ecient drilling for βle 2 which isestablished to be the e ective interval of β and correspondsto both soft and hard rocks Drilling sti er rocks (2lt βlt 4)might be less ecient but increasing mr to 1 would helpimprove drilling eciency

00000

00002

00004

00006

00008

00010

00012

Dim

ensio

nles

s tim

e (t m

)

Section A

Section B

0 5 10

15 20

mr = 05mr = 1mr = 2

Figure 7 Dimensionless time capable of reaching Um for an in-cident stress-wave τ acting on the bit and β

0 2 4 6 8 10 12 14 16 18 20

= 1800

01

02

03

04

05

06

07

08

09

10

Effic

ienc

y

mr = 01mr = 025mr = 05mr = 1

mr = 2mr = 4mr = 10

Figure 8 Drilling eciency of percussive drilling systems withrespect to β and mr

10 Advances in Materials Science and Engineering

4 Results and Discussion

0is study defined equations for the motion of a drill bitwhen struck by a piston and expanded the dynamic prop-erties of the drill bit based on the conditions ζ and α It alsoanalyzed the relationship between the bit dynamics and theduration of the incident stress wave (τ) Underdamped drillbits displayed the greatest responsiveness and the fastesttransmission of τ Increasing the transmission of τ decreasedthe loading effect due to the drill bitrsquos inertia but led to a fastresponse 0e inertial effect of the drill bit was confirmed tobe inversely proportional to the transmission of τ

A percussive drilling system accomplishes its work viathe percussion impact of a piston In the systems studiedhere τ ended more quickly with lower mr and β values asthese values increased the duration of τ transmitted to therock via the drill bit increased in proportion to β 0e resultsalso showed that the maximum bit displacement (um) de-creased as β increased Analysis of the drilling dynamicsconfirmed the effective intervals of β With reference to therock strengths suggested by the International Society forRock Mechanics (ISRM) the effective interval of β is con-sidered to be no greater than 2

0e main purpose of this study was to examine theimpact energy transmission rate and drilling efficiency ina percussive drilling system (Figure 8) 0e results establishedamr value of 05 as the most efficient for rocks whose strengthcorresponds to the interval βlt 2 0e improved drilling ef-ficiency would lead to advantages such as reduced bit pro-duction costs Rocks stiffer than 2ltβlt 4 can be mostefficiently drilled by selecting mr 1

0emovement and response characteristics of bit duringpercussion process τ and transfer rate of energy caused bypercussion an efficiency depend on mr and β where in-crease in mr leads to overdamping of bit movement char-acteristics due to τ by piston and reduction in drillingefficiency due to internal energy effect by mass of drill bit Asthe increase in mass of drill bit for the section of 4lt βlt 6the drilling efficiency of percussive drilling system decreaseand for the very hard rock (UCSgt 200MPa) section of4lt βlt 6 it is considered that the reasonable applied mr ofdrilling tool is 1 or 2

0is study neglected the effects of the secondary incidentstress wave and also the effect of buttons embedded in thedrill bit Additionally for analytical simplification the pistonand drill bit were assumed to be of the same diameter andmaterial and a rectangular pulse with an incident stresswave of duration τ and amplitude 05ρcv was assumed 0eeffect of the flexural stress wave depending on the config-uration was not considered 0ese limitations of this papershould be addressed in further investigations that take intoaccount the configuration of the piston and drill bit theeffect of buttons and the drilling efficiency given differingshapes of the incident stress wave

5 Conclusion

0is paper aimed to identify the optimal design parametersfor percussive drilling systems by introducing a drill bit-rock

interaction model that could verify the bitrsquos motion duringpercussion and the resulting damping characteristics 0estudy analyzed drilling efficiency and drew the followingconclusions

Percussive drilling systems have six dynamic drill bitproperties that can be expandable 0is paper discussed thephysical meaning of the dimensionless parameters α and β0eir values determined the damping characteristics thatcan lead to the rockrsquos maximum fracture displacement 0efastest response of τ was observed for underdamped drill-bitmotion

Drilling was most efficient in the interval 1lt βlt 4 whereincreasing mr at a given β decreased drilling efficiency 0evalue of mr for efficient drilling was determined by the rockstrength (ie β)

0e results indicate that application of a piston-to-drill bitmass ratio of 05 (ie a piston mass twice that of the drill bitmass) to the rocks whose stiffness corresponds to βge 2 wouldlikely be most efficient and also reduce drill bit productioncosts Furthermore a selection ofmr 1 would be valid whendeveloping drill tools for boring complex rocks (1ltβlt 4) Atβgt 4 the best efficiency could be achieved when the bit massequals or exceeds the piston mass

Appendix

Theoretical Case of Bit Motion Condition(Drilling Dynamic Model)

0is paper proposed six conditions for the dynamics ofa drill bit struck by a piston impact as summarizedbelow 0e damping conditions for the damping ratiosand α from the drill-bit equations of motion (7ndash9) are asfollows

First for the overdamped condition of ζ1 gt 1 and αlt 1 τis transmitted to the drill bit indicating loading Equation(7) can be expressed as in (A1) 0e initial conditions are0lt tle τ σi 05ρcv F ku u0 0 and _u0 0

eurou +ρcA

mb

1113888 1113889 _u +k

mb

1113888 1113889u ρcA

mb

1113888 1113889v (A1)

where assuming the normal state u can be calculated asfollows

u ρcA

k1113874 1113875v (A2)

Next we examine the case where τ is complete and theloading effect is sustained owing to external forces and drill-bit inertia Applying the conditions of τ lt tle tm σi 0F ku u0 u(τ) and _u0 _u(τ) to (A1) results in thefollowing expression

eurou +ρcA

mb

1113888 1113889 _u +k

mb

1113888 1113889u 0 (A3)

where u 0 Last we examine the unloading conditions oftm lt t σi 0 F ckuminus ckuf u0 um and _u0 0 Apply-ing these to (9) when unloading gives

Advances in Materials Science and Engineering 11

eurou +ρcA

mb

1113888 1113889 _u + ck

mb

1113888 1113889u ck

mb

1113888 1113889uf (A4)

where bit displacement u uf is used to indicate the rockrsquosfracture displacement Furthermore considering the bitrsquosdamping ratio when unloading its behavior can be ex-pressed as

Damped mode

ζ2 gt 1 αlt1c

over( )

ζ2 1 α 1c

critical( )

ζ2 lt 1 αgt1c

under( )

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A5)

Second under critical damping where ζ1 1 and α 1 τis transmitted to the drill bit indicating loading Equation(7) can be expressed as in (A1) 0e initial conditions are0lt tle τ σi 05ρcv F ku u0 0 and _u0 0

In this case assuming the normal state u can be cal-culated using (A2)

0e following is the case where τ is complete and theloading effect is sustained owing to external forces and drillbit inertia Applying the conditions of τ lt tle tm σi 0F ku u0 u(τ) and _u0 _u(τ) to (A1) give (A3) whereu 0 Last we examine the unloading condition wheretm lt t σi 0 F ckuminus ckuf u0 um and _u0 0 Equation(9) when unloading can be expressed as (A4)

In this case bit displacement u uf is used to indicate therockrsquos fracture displacement Furthermore considering thebitrsquos damping ratio when unloading its behavior follows

Damped mode

ζ2 gt 1 αlt1c

(no case)

ζ2 1 α 1c

(no case)

ζ2 lt 1 αgt1c

under( )

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A6)

0e second condition of critical damping is defined asα 1When unloading drill bits c refers to the rockrsquos propertyeffect and is always less than 1 Hence when unloading noother cases but the underdamped mode are allowed In otherwords when unloading in the critically damped mode owingto the rock properties the bit is always underdamped

0e third condition is underdamping with ζ1 lt 1 andαgt 1 Here τ is transmitted to the drill bit meaning loadingEquation (7) can be expressed by (A1)0e initial conditionsare 0lt tle τ σi 05ρcv F ku u0 0 and _u0 0 Whereassuming the normal state u can be calculated as (A2)

0e following is the case where τ is complete and theloading effect is sustained owing to external forces and drillbit inertia Applying the conditions of τ lt tle tm σi 0F ku u0 u(τ) and _u0 _u(τ) to (A1) results in (A3)

where u 0 Last we examine the unloading conditionwhere tm lt t σi 0 F ckuminus ckuf u0 um and _u0 0Equation (9) when unloading can be expressed as (A4)

In this case the drill bit displacement u uf is used toindicate the rockrsquos fracture displacement Furthermoreconsidering the damping ratio in the drill bitrsquos unloadingthe bitrsquos behavior can be expressed as

Damped mode

ζ2 gt 1 αlt1c

(no case)

ζ2 1 α 1c

(no case)

ζ2 lt 1 αgt1c

under( )

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A7)

0e third condition is the critically damped modedefined as αgt 1 During drill bit unloading c refers to therock property effect and is always less than 1 Hencewhen unloading no other cases but underdamping areallowed

Next the underdamped condition is where ζ1 lt 1 andαgt 1 and τ is slower than tm (the time at which the rockrsquosmaximum fracture displacement is reached) that is tm lt τIn this case τ is transmitted to the drill bit meaningloading Equation (7) can be expressed by (A1) 0e initialconditions are 0lt tle tm σi 05ρcv F ku u0 0 and_u0 0 Here assuming the normal state u can be calculatedas (A2)

0e following case is where conversion is made tounloading after the drill bit reaches its maximum dis-placement but the incident stress wave is sustained Ap-plying the conditions of tm lt tle τ σi 05ρcvF ckuminus ckuf u0 um and _u0 0 to (9) results in

eurou +ρcA

mb

1113888 1113889 _u + ck

mb

1113888 1113889u ρcA

mb

1113888 1113889v + ck

mb

1113888 1113889uf (A8)

where the definition of u 1c(ρcAk)v + uf is possibleLast this unloading condition is where the incident

stress wave is resolved When unloading the drill bit usingthe conditions tlt τ σi 0 F ckuminus ckuf u0 u(τ) and_u0 _u(τ) allows (9) to be expressed as (A4)

In this case drill bit displacement u uf is used to in-dicate the rockrsquos fracture displacement 0e analysis of drillbit dynamics showed that when loading transitions tounloading in the underdamped condition the dynamicproperties of the bit exist only in the underdamping mode

Conflicts of Interest

The authors declare that there are no conflicts of interest

References

[1] C H Song K B Kwon J Y Park et al ldquoOptimum design ofthe internal flushing channel of a drill bit using RSM and CFDsimulationrdquo International Journal of Precision Engineeringand Manufacturing vol 15 no 6 pp 1041ndash1050 2014

12 Advances in Materials Science and Engineering

[2] X Li G Rupert D A Summers P Santi and D Liu ldquoAnalysisof impact hammer rebound to estimate rock drillabilityrdquo RockMechanics and Rock Engineering vol 33 no 1 pp 1ndash13 2000

[3] W A Hustrulid and C Fairhurst ldquoA theoretical and ex-perimental study of the percussive drilling of rock part Itheory of percussive drillingrdquo International Journal of RockMechanics and Mining Sciences amp Geomechanics Abstractsvol 8 no 4 pp 311ndash333 1971

[4] W A Hustrulid and C Fairhurst ldquoA theoretical and ex-perimental study of the percussive drilling of rock part IIforce-penetration and specific energy determinationrdquo In-ternational Journal of Rock Mechanics and Mining Sciences ampGeomechanics Abstracts vol 8 no 4 pp 335ndash356 1971

[5] W A Hustrulid and C Fairhurst ldquoA theoretical and ex-perimental study of the percussive drilling of rock part IIIexperimental verification of the mathematical theoryrdquo In-ternational Journal of Rock Mechanics and Mining Sciences ampGeomechanics Abstracts vol 9 no 3 pp 417-418 1972

[6] W A Hustrulid and C Fairhurst ldquoA theoretical and ex-perimental study of the percussive drilling of rock part IVapplication of the model to actual percussive drillingrdquo In-ternational Journal of Rock Mechanics and Mining Sciencesvol 9 no 3 pp 431ndash442 1972

[7] L E Chiang and D A Elias ldquoA 3D FEM methodology forsimulating the impact in rock-drilling hammersrdquo In-ternational Journal of Rock Mechanics and Mining Sciencesvol 45 no 5 pp 701ndash711 2008

[8] C H Song K B Kwon M G Cho J Y Oh D Y Shin andJ W Cho ldquoDevelopment of lab-scale rock drill apparatus fortesting performance of a drill bitrdquo International Journal ofPrecision Engineering and Manufacturing vol 16 no 7pp 1402ndash1414 2015

[9] K B Kwon C H Song J Y Park J Y Oh J W Lee andJ W Cho ldquoEvaluation of drilling efficiency by percussiontesting of a drill bit with new button arrangementrdquo In-ternational Journal of Precision Engineering andManufacturing vol 15 no 6 pp 1063ndash1068 2014

[10] B Lundberg andM Okrouhlik ldquoInfluence of 3D effects on theefficiency of percussive rock drillingrdquo International Journal ofimpact Engineering vol 25 no 4 pp 345ndash360 2001

[11] B Lundberg and P Collet ldquoOptimal wave shape with respectto efficiency in percussive drilling with detachable drill bitrdquoInternational Journal of impact Engineering vol 86 pp 179ndash187 2015

[12] X B Li G Rupert and D A Summers ldquoEnergy transmissionof down-hole hammer tool and its conditionalityrdquo Trans-actions of Nonferrous Metals Society of China vol 10 no 1pp 109ndash111 2000

[13] J W Cho S Jeon S H Yu and S H Chang ldquoOptimumspacing of TBM disc cutters a numerical simulation using thethree-dimensional dynamic fracturing methodrdquo Tunnellingand Underground Space Technology vol 25 no 3 pp 230ndash244 2010

[14] W Changming ldquoAn analytical study of percussive energytransfer in hydraulic rock drillsrdquo Mining Science and Tech-nology vol 13 no 1 pp 57ndash68 1991

[15] Online Materials Information Resource httpwwwmatwebcomsearchDataSheetaspxMatGUID8bc5d558f4174e6082ddf4966e382bd6ampckck1

Advances in Materials Science and Engineering 13

eeciency of the percussive drilling systems for given βcan be calculated using (14ndash17) e factor um which isimportant to the eciency calculations is discussed in depth

32 Results of Energy Transmission Eciency e results ofsimulations on the drill bit-rock interaction model shown inFigure 5 indicate the damped modes that can arrive at um (forvarying values of β) as the response by the drill bit at the time ofpiston impact e damped mode is determined by ζ and αwhich indictate the system responses Overdamped mode hasαlt 1 and can be expressed as follows

2mb

mh( )βlt 1

βlt1

2mr

(19)

e critical damped mode has α 1 and is expressed asfollows

2mb

mh( )β 1

β 1

2mr

(20)

e underdamped mode has αgt 1 and is expressed as

2mb

mh( )βgt 1

βgt1

2mr

(21)

where mbmh mr

e special underdamped mode refers to the case whereτ is faster than t indicating that the drill bit can arrive at umonly via an incident stress wave In this case the bit reachesits maximum displacement and transitions to the unloadingstate while the transmission of τ is underway

e damped mode of each mr shown in Figure 5 re-veals that the underdamped mode is dominant of thee ects Also overdamped and critical damped were ob-served where β was low is is possibly because of thee ects of the internal energy (ie compression strength)of rock and the percussive energy e overdamped andcritical damped were also observed is is possibly be-cause of internal energy (ie compression strength) ofrock and energy caused by percussion

Figure 6 shows dimensionless time tm tτ capable ofreaching Um for β and τ In Section A of tgt τ the bit reachesUm after termination of τ and in Section B of tlt τ the bitreachesUm before the termination of τe lower themr andβ the faster the termination of τ and increase in thesetended to make the duration of τ delivered to rock throughbit to increase proportionally to β is suggests that thehigher rock strength requires longer duration of τ fordrilling the rock

Figure 7 shows dimensionless Um state dened as ratioum a displacement um by β and uinfin caused by incident stresswave and loading condition e prediction of displacementof drill bit drilling displacement by characteristic of rockwas limited is study accordingly judged the di-mensionless drilling state Um only by using ratio uinfina displacement during loading and um considering inertiacaused during percussion (22)

Um umuinfin (22)

where the uinfin only by loading is dened as follows

mr = 1

mr = 2

mr = 05

0 5 10

15 20

4

3

2

Dam

ped

mod

es

1

0

mr = 1mr = 2

mr = 05

mr = 1

mr = 05

Figure 5 Damping mode capable of reaching maximum rockfracture displacement um given the piston-to-bit mass ratiomr and β1 overdamped mode 2 critically damped mode 3 underdampedmode 4 special underdamped mode

0 5 10

15 2000

02

06

10

14

18

20

04

08

12

16

Dim

ensio

nles

s pen

etra

tion

(Um

)

Section B

Section A

mr = 05mr = 1mr = 2

uinfin

Figure 6 State of reaching drill bit dimensionless penetration Umwith respect to mass ratio mr and β

Advances in Materials Science and Engineering 9

uinfin ρcAkv

τβv (23)

e e ect of initial condition on uinfin decreased with theincrease in β e um is determined by considering loadingcondition and inertia e ect is a ected by overresponse anddecreases as β increases

Section A refers to umlt uinfin with only τ and loadinge ect present (ie the condition for the normal state in theinitial mode) uinfin means only the τ e ect resulting fromloading exists Section B refers to umgt uinfin a state in whichum can be reached through τ and loading is exceeded issection is where the drill bit inertia and underdampedmotion appear e gradient of the curve for maximumdisplacement of the bit varies with β indicating that as therock strength increases the displacement allowing fracturedecreases Near value 2 β changes its gradient for maximumdisplacement is means that in Section A the rock is lesssti than the piston owing to the rock-sti ness e ect and inSection B the rock is sti er than the piston β is dened asthe sti ness ratio between the rock and the piston as in (11)Hence when βlt 2 a condition is established for rock fractureto occur in that the piston sti ness exceeds the rock sti ness

In examining um for piston-bit mass ratios of mrlt 2 therock strength that can be defeated by τ occurring in theinitial loading tends to increase is tendency is suspectedto result from the inertial e ect of the mass of the bit Forβgt 4 the um gradient tended to decrease with increasingmass of the bit Hence the eciency is considered to de-crease which is attributed to the e ect of the rockrsquos com-pressive strength e uniaxial compressive strength of veryhard rock is typically at least 200MPa while the tensilestrength of the H13 tool steels that are used mostly forpistons (ASTM A681 and DIN EN ISO 4957) is known to beat least 359ndash1170MPa [15]

Given the steels generally used to make pistons thee ective range of β is considered to be 034lt βlt 112 which

was calculated considering the strength ratio between thehard rock and an H13 steel tool Hence the e ective range ofβ could be no greater than 2 Rocks with a sti ness ratio ofβgt 2 might cause failure through the pistonrsquos plastic de-formation and fracture is study examined the e ectiveinterval of β and investigated the parameter combinationsgiving high drilling eciency in this interval

Figure 8 illustrates the motion of the bit generatedduring the percussion process and the energy transmissionrate (ie eciency) for the response characteristics τ andpercussion impact Analysis of mr found the maximumdrilling eciency in interval 1lt βlt 2 with a tendency ef-ciency decreasing after mr increased above a certain valueis result was attributed to the tendency whereby increasingmr converts the systemrsquos dynamic response to τ due to thepiston from underdamped to overdamped Furthermoreincreasing the mass of the bit can increase its internal energy(ie friction energy) to above the piston transmitted energy(ie kinetic and potential energy) Hence the bitrsquos dynamicresponse would be overdamped and could compromisedrilling eciency In other words the heavier the bit themore delayed the response to percussion

In Figure 8 the left side of red line (β18) represents thee ect of only incident stress wave meaning that the higherthe value in graph the higher the drilling eciency in in-cident stress wave e right side of red line representingunloading behavior after reaching maximum eciency inincident stress wave is a ected by both brittleness char-acteristics of rock and inertia e ect of drill bit

e piston should have twice the mass of the drill bit(iemr 05) for the most ecient drilling for βle 2 which isestablished to be the e ective interval of β and correspondsto both soft and hard rocks Drilling sti er rocks (2lt βlt 4)might be less ecient but increasing mr to 1 would helpimprove drilling eciency

00000

00002

00004

00006

00008

00010

00012

Dim

ensio

nles

s tim

e (t m

)

Section A

Section B

0 5 10

15 20

mr = 05mr = 1mr = 2

Figure 7 Dimensionless time capable of reaching Um for an in-cident stress-wave τ acting on the bit and β

0 2 4 6 8 10 12 14 16 18 20

= 1800

01

02

03

04

05

06

07

08

09

10

Effic

ienc

y

mr = 01mr = 025mr = 05mr = 1

mr = 2mr = 4mr = 10

Figure 8 Drilling eciency of percussive drilling systems withrespect to β and mr

10 Advances in Materials Science and Engineering

4 Results and Discussion

0is study defined equations for the motion of a drill bitwhen struck by a piston and expanded the dynamic prop-erties of the drill bit based on the conditions ζ and α It alsoanalyzed the relationship between the bit dynamics and theduration of the incident stress wave (τ) Underdamped drillbits displayed the greatest responsiveness and the fastesttransmission of τ Increasing the transmission of τ decreasedthe loading effect due to the drill bitrsquos inertia but led to a fastresponse 0e inertial effect of the drill bit was confirmed tobe inversely proportional to the transmission of τ

A percussive drilling system accomplishes its work viathe percussion impact of a piston In the systems studiedhere τ ended more quickly with lower mr and β values asthese values increased the duration of τ transmitted to therock via the drill bit increased in proportion to β 0e resultsalso showed that the maximum bit displacement (um) de-creased as β increased Analysis of the drilling dynamicsconfirmed the effective intervals of β With reference to therock strengths suggested by the International Society forRock Mechanics (ISRM) the effective interval of β is con-sidered to be no greater than 2

0e main purpose of this study was to examine theimpact energy transmission rate and drilling efficiency ina percussive drilling system (Figure 8) 0e results establishedamr value of 05 as the most efficient for rocks whose strengthcorresponds to the interval βlt 2 0e improved drilling ef-ficiency would lead to advantages such as reduced bit pro-duction costs Rocks stiffer than 2ltβlt 4 can be mostefficiently drilled by selecting mr 1

0emovement and response characteristics of bit duringpercussion process τ and transfer rate of energy caused bypercussion an efficiency depend on mr and β where in-crease in mr leads to overdamping of bit movement char-acteristics due to τ by piston and reduction in drillingefficiency due to internal energy effect by mass of drill bit Asthe increase in mass of drill bit for the section of 4lt βlt 6the drilling efficiency of percussive drilling system decreaseand for the very hard rock (UCSgt 200MPa) section of4lt βlt 6 it is considered that the reasonable applied mr ofdrilling tool is 1 or 2

0is study neglected the effects of the secondary incidentstress wave and also the effect of buttons embedded in thedrill bit Additionally for analytical simplification the pistonand drill bit were assumed to be of the same diameter andmaterial and a rectangular pulse with an incident stresswave of duration τ and amplitude 05ρcv was assumed 0eeffect of the flexural stress wave depending on the config-uration was not considered 0ese limitations of this papershould be addressed in further investigations that take intoaccount the configuration of the piston and drill bit theeffect of buttons and the drilling efficiency given differingshapes of the incident stress wave

5 Conclusion

0is paper aimed to identify the optimal design parametersfor percussive drilling systems by introducing a drill bit-rock

interaction model that could verify the bitrsquos motion duringpercussion and the resulting damping characteristics 0estudy analyzed drilling efficiency and drew the followingconclusions

Percussive drilling systems have six dynamic drill bitproperties that can be expandable 0is paper discussed thephysical meaning of the dimensionless parameters α and β0eir values determined the damping characteristics thatcan lead to the rockrsquos maximum fracture displacement 0efastest response of τ was observed for underdamped drill-bitmotion

Drilling was most efficient in the interval 1lt βlt 4 whereincreasing mr at a given β decreased drilling efficiency 0evalue of mr for efficient drilling was determined by the rockstrength (ie β)

0e results indicate that application of a piston-to-drill bitmass ratio of 05 (ie a piston mass twice that of the drill bitmass) to the rocks whose stiffness corresponds to βge 2 wouldlikely be most efficient and also reduce drill bit productioncosts Furthermore a selection ofmr 1 would be valid whendeveloping drill tools for boring complex rocks (1ltβlt 4) Atβgt 4 the best efficiency could be achieved when the bit massequals or exceeds the piston mass

Appendix

Theoretical Case of Bit Motion Condition(Drilling Dynamic Model)

0is paper proposed six conditions for the dynamics ofa drill bit struck by a piston impact as summarizedbelow 0e damping conditions for the damping ratiosand α from the drill-bit equations of motion (7ndash9) are asfollows

First for the overdamped condition of ζ1 gt 1 and αlt 1 τis transmitted to the drill bit indicating loading Equation(7) can be expressed as in (A1) 0e initial conditions are0lt tle τ σi 05ρcv F ku u0 0 and _u0 0

eurou +ρcA

mb

1113888 1113889 _u +k

mb

1113888 1113889u ρcA

mb

1113888 1113889v (A1)

where assuming the normal state u can be calculated asfollows

u ρcA

k1113874 1113875v (A2)

Next we examine the case where τ is complete and theloading effect is sustained owing to external forces and drill-bit inertia Applying the conditions of τ lt tle tm σi 0F ku u0 u(τ) and _u0 _u(τ) to (A1) results in thefollowing expression

eurou +ρcA

mb

1113888 1113889 _u +k

mb

1113888 1113889u 0 (A3)

where u 0 Last we examine the unloading conditions oftm lt t σi 0 F ckuminus ckuf u0 um and _u0 0 Apply-ing these to (9) when unloading gives

Advances in Materials Science and Engineering 11

eurou +ρcA

mb

1113888 1113889 _u + ck

mb

1113888 1113889u ck

mb

1113888 1113889uf (A4)

where bit displacement u uf is used to indicate the rockrsquosfracture displacement Furthermore considering the bitrsquosdamping ratio when unloading its behavior can be ex-pressed as

Damped mode

ζ2 gt 1 αlt1c

over( )

ζ2 1 α 1c

critical( )

ζ2 lt 1 αgt1c

under( )

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A5)

Second under critical damping where ζ1 1 and α 1 τis transmitted to the drill bit indicating loading Equation(7) can be expressed as in (A1) 0e initial conditions are0lt tle τ σi 05ρcv F ku u0 0 and _u0 0

In this case assuming the normal state u can be cal-culated using (A2)

0e following is the case where τ is complete and theloading effect is sustained owing to external forces and drillbit inertia Applying the conditions of τ lt tle tm σi 0F ku u0 u(τ) and _u0 _u(τ) to (A1) give (A3) whereu 0 Last we examine the unloading condition wheretm lt t σi 0 F ckuminus ckuf u0 um and _u0 0 Equation(9) when unloading can be expressed as (A4)

In this case bit displacement u uf is used to indicate therockrsquos fracture displacement Furthermore considering thebitrsquos damping ratio when unloading its behavior follows

Damped mode

ζ2 gt 1 αlt1c

(no case)

ζ2 1 α 1c

(no case)

ζ2 lt 1 αgt1c

under( )

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A6)

0e second condition of critical damping is defined asα 1When unloading drill bits c refers to the rockrsquos propertyeffect and is always less than 1 Hence when unloading noother cases but the underdamped mode are allowed In otherwords when unloading in the critically damped mode owingto the rock properties the bit is always underdamped

0e third condition is underdamping with ζ1 lt 1 andαgt 1 Here τ is transmitted to the drill bit meaning loadingEquation (7) can be expressed by (A1)0e initial conditionsare 0lt tle τ σi 05ρcv F ku u0 0 and _u0 0 Whereassuming the normal state u can be calculated as (A2)

0e following is the case where τ is complete and theloading effect is sustained owing to external forces and drillbit inertia Applying the conditions of τ lt tle tm σi 0F ku u0 u(τ) and _u0 _u(τ) to (A1) results in (A3)

where u 0 Last we examine the unloading conditionwhere tm lt t σi 0 F ckuminus ckuf u0 um and _u0 0Equation (9) when unloading can be expressed as (A4)

In this case the drill bit displacement u uf is used toindicate the rockrsquos fracture displacement Furthermoreconsidering the damping ratio in the drill bitrsquos unloadingthe bitrsquos behavior can be expressed as

Damped mode

ζ2 gt 1 αlt1c

(no case)

ζ2 1 α 1c

(no case)

ζ2 lt 1 αgt1c

under( )

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A7)

0e third condition is the critically damped modedefined as αgt 1 During drill bit unloading c refers to therock property effect and is always less than 1 Hencewhen unloading no other cases but underdamping areallowed

Next the underdamped condition is where ζ1 lt 1 andαgt 1 and τ is slower than tm (the time at which the rockrsquosmaximum fracture displacement is reached) that is tm lt τIn this case τ is transmitted to the drill bit meaningloading Equation (7) can be expressed by (A1) 0e initialconditions are 0lt tle tm σi 05ρcv F ku u0 0 and_u0 0 Here assuming the normal state u can be calculatedas (A2)

0e following case is where conversion is made tounloading after the drill bit reaches its maximum dis-placement but the incident stress wave is sustained Ap-plying the conditions of tm lt tle τ σi 05ρcvF ckuminus ckuf u0 um and _u0 0 to (9) results in

eurou +ρcA

mb

1113888 1113889 _u + ck

mb

1113888 1113889u ρcA

mb

1113888 1113889v + ck

mb

1113888 1113889uf (A8)

where the definition of u 1c(ρcAk)v + uf is possibleLast this unloading condition is where the incident

stress wave is resolved When unloading the drill bit usingthe conditions tlt τ σi 0 F ckuminus ckuf u0 u(τ) and_u0 _u(τ) allows (9) to be expressed as (A4)

In this case drill bit displacement u uf is used to in-dicate the rockrsquos fracture displacement 0e analysis of drillbit dynamics showed that when loading transitions tounloading in the underdamped condition the dynamicproperties of the bit exist only in the underdamping mode

Conflicts of Interest

The authors declare that there are no conflicts of interest

References

[1] C H Song K B Kwon J Y Park et al ldquoOptimum design ofthe internal flushing channel of a drill bit using RSM and CFDsimulationrdquo International Journal of Precision Engineeringand Manufacturing vol 15 no 6 pp 1041ndash1050 2014

12 Advances in Materials Science and Engineering

[2] X Li G Rupert D A Summers P Santi and D Liu ldquoAnalysisof impact hammer rebound to estimate rock drillabilityrdquo RockMechanics and Rock Engineering vol 33 no 1 pp 1ndash13 2000

[3] W A Hustrulid and C Fairhurst ldquoA theoretical and ex-perimental study of the percussive drilling of rock part Itheory of percussive drillingrdquo International Journal of RockMechanics and Mining Sciences amp Geomechanics Abstractsvol 8 no 4 pp 311ndash333 1971

[4] W A Hustrulid and C Fairhurst ldquoA theoretical and ex-perimental study of the percussive drilling of rock part IIforce-penetration and specific energy determinationrdquo In-ternational Journal of Rock Mechanics and Mining Sciences ampGeomechanics Abstracts vol 8 no 4 pp 335ndash356 1971

[5] W A Hustrulid and C Fairhurst ldquoA theoretical and ex-perimental study of the percussive drilling of rock part IIIexperimental verification of the mathematical theoryrdquo In-ternational Journal of Rock Mechanics and Mining Sciences ampGeomechanics Abstracts vol 9 no 3 pp 417-418 1972

[6] W A Hustrulid and C Fairhurst ldquoA theoretical and ex-perimental study of the percussive drilling of rock part IVapplication of the model to actual percussive drillingrdquo In-ternational Journal of Rock Mechanics and Mining Sciencesvol 9 no 3 pp 431ndash442 1972

[7] L E Chiang and D A Elias ldquoA 3D FEM methodology forsimulating the impact in rock-drilling hammersrdquo In-ternational Journal of Rock Mechanics and Mining Sciencesvol 45 no 5 pp 701ndash711 2008

[8] C H Song K B Kwon M G Cho J Y Oh D Y Shin andJ W Cho ldquoDevelopment of lab-scale rock drill apparatus fortesting performance of a drill bitrdquo International Journal ofPrecision Engineering and Manufacturing vol 16 no 7pp 1402ndash1414 2015

[9] K B Kwon C H Song J Y Park J Y Oh J W Lee andJ W Cho ldquoEvaluation of drilling efficiency by percussiontesting of a drill bit with new button arrangementrdquo In-ternational Journal of Precision Engineering andManufacturing vol 15 no 6 pp 1063ndash1068 2014

[10] B Lundberg andM Okrouhlik ldquoInfluence of 3D effects on theefficiency of percussive rock drillingrdquo International Journal ofimpact Engineering vol 25 no 4 pp 345ndash360 2001

[11] B Lundberg and P Collet ldquoOptimal wave shape with respectto efficiency in percussive drilling with detachable drill bitrdquoInternational Journal of impact Engineering vol 86 pp 179ndash187 2015

[12] X B Li G Rupert and D A Summers ldquoEnergy transmissionof down-hole hammer tool and its conditionalityrdquo Trans-actions of Nonferrous Metals Society of China vol 10 no 1pp 109ndash111 2000

[13] J W Cho S Jeon S H Yu and S H Chang ldquoOptimumspacing of TBM disc cutters a numerical simulation using thethree-dimensional dynamic fracturing methodrdquo Tunnellingand Underground Space Technology vol 25 no 3 pp 230ndash244 2010

[14] W Changming ldquoAn analytical study of percussive energytransfer in hydraulic rock drillsrdquo Mining Science and Tech-nology vol 13 no 1 pp 57ndash68 1991

[15] Online Materials Information Resource httpwwwmatwebcomsearchDataSheetaspxMatGUID8bc5d558f4174e6082ddf4966e382bd6ampckck1

Advances in Materials Science and Engineering 13

uinfin ρcAkv

τβv (23)

e e ect of initial condition on uinfin decreased with theincrease in β e um is determined by considering loadingcondition and inertia e ect is a ected by overresponse anddecreases as β increases

Section A refers to umlt uinfin with only τ and loadinge ect present (ie the condition for the normal state in theinitial mode) uinfin means only the τ e ect resulting fromloading exists Section B refers to umgt uinfin a state in whichum can be reached through τ and loading is exceeded issection is where the drill bit inertia and underdampedmotion appear e gradient of the curve for maximumdisplacement of the bit varies with β indicating that as therock strength increases the displacement allowing fracturedecreases Near value 2 β changes its gradient for maximumdisplacement is means that in Section A the rock is lesssti than the piston owing to the rock-sti ness e ect and inSection B the rock is sti er than the piston β is dened asthe sti ness ratio between the rock and the piston as in (11)Hence when βlt 2 a condition is established for rock fractureto occur in that the piston sti ness exceeds the rock sti ness

In examining um for piston-bit mass ratios of mrlt 2 therock strength that can be defeated by τ occurring in theinitial loading tends to increase is tendency is suspectedto result from the inertial e ect of the mass of the bit Forβgt 4 the um gradient tended to decrease with increasingmass of the bit Hence the eciency is considered to de-crease which is attributed to the e ect of the rockrsquos com-pressive strength e uniaxial compressive strength of veryhard rock is typically at least 200MPa while the tensilestrength of the H13 tool steels that are used mostly forpistons (ASTM A681 and DIN EN ISO 4957) is known to beat least 359ndash1170MPa [15]

Given the steels generally used to make pistons thee ective range of β is considered to be 034lt βlt 112 which

was calculated considering the strength ratio between thehard rock and an H13 steel tool Hence the e ective range ofβ could be no greater than 2 Rocks with a sti ness ratio ofβgt 2 might cause failure through the pistonrsquos plastic de-formation and fracture is study examined the e ectiveinterval of β and investigated the parameter combinationsgiving high drilling eciency in this interval

Figure 8 illustrates the motion of the bit generatedduring the percussion process and the energy transmissionrate (ie eciency) for the response characteristics τ andpercussion impact Analysis of mr found the maximumdrilling eciency in interval 1lt βlt 2 with a tendency ef-ciency decreasing after mr increased above a certain valueis result was attributed to the tendency whereby increasingmr converts the systemrsquos dynamic response to τ due to thepiston from underdamped to overdamped Furthermoreincreasing the mass of the bit can increase its internal energy(ie friction energy) to above the piston transmitted energy(ie kinetic and potential energy) Hence the bitrsquos dynamicresponse would be overdamped and could compromisedrilling eciency In other words the heavier the bit themore delayed the response to percussion

In Figure 8 the left side of red line (β18) represents thee ect of only incident stress wave meaning that the higherthe value in graph the higher the drilling eciency in in-cident stress wave e right side of red line representingunloading behavior after reaching maximum eciency inincident stress wave is a ected by both brittleness char-acteristics of rock and inertia e ect of drill bit

e piston should have twice the mass of the drill bit(iemr 05) for the most ecient drilling for βle 2 which isestablished to be the e ective interval of β and correspondsto both soft and hard rocks Drilling sti er rocks (2lt βlt 4)might be less ecient but increasing mr to 1 would helpimprove drilling eciency

00000

00002

00004

00006

00008

00010

00012

Dim

ensio

nles

s tim

e (t m

)

Section A

Section B

0 5 10

15 20

mr = 05mr = 1mr = 2

Figure 7 Dimensionless time capable of reaching Um for an in-cident stress-wave τ acting on the bit and β

0 2 4 6 8 10 12 14 16 18 20

= 1800

01

02

03

04

05

06

07

08

09

10

Effic

ienc

y

mr = 01mr = 025mr = 05mr = 1

mr = 2mr = 4mr = 10

Figure 8 Drilling eciency of percussive drilling systems withrespect to β and mr

10 Advances in Materials Science and Engineering

4 Results and Discussion

0is study defined equations for the motion of a drill bitwhen struck by a piston and expanded the dynamic prop-erties of the drill bit based on the conditions ζ and α It alsoanalyzed the relationship between the bit dynamics and theduration of the incident stress wave (τ) Underdamped drillbits displayed the greatest responsiveness and the fastesttransmission of τ Increasing the transmission of τ decreasedthe loading effect due to the drill bitrsquos inertia but led to a fastresponse 0e inertial effect of the drill bit was confirmed tobe inversely proportional to the transmission of τ

A percussive drilling system accomplishes its work viathe percussion impact of a piston In the systems studiedhere τ ended more quickly with lower mr and β values asthese values increased the duration of τ transmitted to therock via the drill bit increased in proportion to β 0e resultsalso showed that the maximum bit displacement (um) de-creased as β increased Analysis of the drilling dynamicsconfirmed the effective intervals of β With reference to therock strengths suggested by the International Society forRock Mechanics (ISRM) the effective interval of β is con-sidered to be no greater than 2

0e main purpose of this study was to examine theimpact energy transmission rate and drilling efficiency ina percussive drilling system (Figure 8) 0e results establishedamr value of 05 as the most efficient for rocks whose strengthcorresponds to the interval βlt 2 0e improved drilling ef-ficiency would lead to advantages such as reduced bit pro-duction costs Rocks stiffer than 2ltβlt 4 can be mostefficiently drilled by selecting mr 1

0emovement and response characteristics of bit duringpercussion process τ and transfer rate of energy caused bypercussion an efficiency depend on mr and β where in-crease in mr leads to overdamping of bit movement char-acteristics due to τ by piston and reduction in drillingefficiency due to internal energy effect by mass of drill bit Asthe increase in mass of drill bit for the section of 4lt βlt 6the drilling efficiency of percussive drilling system decreaseand for the very hard rock (UCSgt 200MPa) section of4lt βlt 6 it is considered that the reasonable applied mr ofdrilling tool is 1 or 2

0is study neglected the effects of the secondary incidentstress wave and also the effect of buttons embedded in thedrill bit Additionally for analytical simplification the pistonand drill bit were assumed to be of the same diameter andmaterial and a rectangular pulse with an incident stresswave of duration τ and amplitude 05ρcv was assumed 0eeffect of the flexural stress wave depending on the config-uration was not considered 0ese limitations of this papershould be addressed in further investigations that take intoaccount the configuration of the piston and drill bit theeffect of buttons and the drilling efficiency given differingshapes of the incident stress wave

5 Conclusion

0is paper aimed to identify the optimal design parametersfor percussive drilling systems by introducing a drill bit-rock

interaction model that could verify the bitrsquos motion duringpercussion and the resulting damping characteristics 0estudy analyzed drilling efficiency and drew the followingconclusions

Percussive drilling systems have six dynamic drill bitproperties that can be expandable 0is paper discussed thephysical meaning of the dimensionless parameters α and β0eir values determined the damping characteristics thatcan lead to the rockrsquos maximum fracture displacement 0efastest response of τ was observed for underdamped drill-bitmotion

Drilling was most efficient in the interval 1lt βlt 4 whereincreasing mr at a given β decreased drilling efficiency 0evalue of mr for efficient drilling was determined by the rockstrength (ie β)

0e results indicate that application of a piston-to-drill bitmass ratio of 05 (ie a piston mass twice that of the drill bitmass) to the rocks whose stiffness corresponds to βge 2 wouldlikely be most efficient and also reduce drill bit productioncosts Furthermore a selection ofmr 1 would be valid whendeveloping drill tools for boring complex rocks (1ltβlt 4) Atβgt 4 the best efficiency could be achieved when the bit massequals or exceeds the piston mass

Appendix

Theoretical Case of Bit Motion Condition(Drilling Dynamic Model)

0is paper proposed six conditions for the dynamics ofa drill bit struck by a piston impact as summarizedbelow 0e damping conditions for the damping ratiosand α from the drill-bit equations of motion (7ndash9) are asfollows

First for the overdamped condition of ζ1 gt 1 and αlt 1 τis transmitted to the drill bit indicating loading Equation(7) can be expressed as in (A1) 0e initial conditions are0lt tle τ σi 05ρcv F ku u0 0 and _u0 0

eurou +ρcA

mb

1113888 1113889 _u +k

mb

1113888 1113889u ρcA

mb

1113888 1113889v (A1)

where assuming the normal state u can be calculated asfollows

u ρcA

k1113874 1113875v (A2)

Next we examine the case where τ is complete and theloading effect is sustained owing to external forces and drill-bit inertia Applying the conditions of τ lt tle tm σi 0F ku u0 u(τ) and _u0 _u(τ) to (A1) results in thefollowing expression

eurou +ρcA

mb

1113888 1113889 _u +k

mb

1113888 1113889u 0 (A3)

where u 0 Last we examine the unloading conditions oftm lt t σi 0 F ckuminus ckuf u0 um and _u0 0 Apply-ing these to (9) when unloading gives

Advances in Materials Science and Engineering 11

eurou +ρcA

mb

1113888 1113889 _u + ck

mb

1113888 1113889u ck

mb

1113888 1113889uf (A4)

where bit displacement u uf is used to indicate the rockrsquosfracture displacement Furthermore considering the bitrsquosdamping ratio when unloading its behavior can be ex-pressed as

Damped mode

ζ2 gt 1 αlt1c

over( )

ζ2 1 α 1c

critical( )

ζ2 lt 1 αgt1c

under( )

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A5)

Second under critical damping where ζ1 1 and α 1 τis transmitted to the drill bit indicating loading Equation(7) can be expressed as in (A1) 0e initial conditions are0lt tle τ σi 05ρcv F ku u0 0 and _u0 0

In this case assuming the normal state u can be cal-culated using (A2)

0e following is the case where τ is complete and theloading effect is sustained owing to external forces and drillbit inertia Applying the conditions of τ lt tle tm σi 0F ku u0 u(τ) and _u0 _u(τ) to (A1) give (A3) whereu 0 Last we examine the unloading condition wheretm lt t σi 0 F ckuminus ckuf u0 um and _u0 0 Equation(9) when unloading can be expressed as (A4)

In this case bit displacement u uf is used to indicate therockrsquos fracture displacement Furthermore considering thebitrsquos damping ratio when unloading its behavior follows

Damped mode

ζ2 gt 1 αlt1c

(no case)

ζ2 1 α 1c

(no case)

ζ2 lt 1 αgt1c

under( )

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A6)

0e second condition of critical damping is defined asα 1When unloading drill bits c refers to the rockrsquos propertyeffect and is always less than 1 Hence when unloading noother cases but the underdamped mode are allowed In otherwords when unloading in the critically damped mode owingto the rock properties the bit is always underdamped

0e third condition is underdamping with ζ1 lt 1 andαgt 1 Here τ is transmitted to the drill bit meaning loadingEquation (7) can be expressed by (A1)0e initial conditionsare 0lt tle τ σi 05ρcv F ku u0 0 and _u0 0 Whereassuming the normal state u can be calculated as (A2)

0e following is the case where τ is complete and theloading effect is sustained owing to external forces and drillbit inertia Applying the conditions of τ lt tle tm σi 0F ku u0 u(τ) and _u0 _u(τ) to (A1) results in (A3)

where u 0 Last we examine the unloading conditionwhere tm lt t σi 0 F ckuminus ckuf u0 um and _u0 0Equation (9) when unloading can be expressed as (A4)

In this case the drill bit displacement u uf is used toindicate the rockrsquos fracture displacement Furthermoreconsidering the damping ratio in the drill bitrsquos unloadingthe bitrsquos behavior can be expressed as

Damped mode

ζ2 gt 1 αlt1c

(no case)

ζ2 1 α 1c

(no case)

ζ2 lt 1 αgt1c

under( )

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A7)

0e third condition is the critically damped modedefined as αgt 1 During drill bit unloading c refers to therock property effect and is always less than 1 Hencewhen unloading no other cases but underdamping areallowed

Next the underdamped condition is where ζ1 lt 1 andαgt 1 and τ is slower than tm (the time at which the rockrsquosmaximum fracture displacement is reached) that is tm lt τIn this case τ is transmitted to the drill bit meaningloading Equation (7) can be expressed by (A1) 0e initialconditions are 0lt tle tm σi 05ρcv F ku u0 0 and_u0 0 Here assuming the normal state u can be calculatedas (A2)

0e following case is where conversion is made tounloading after the drill bit reaches its maximum dis-placement but the incident stress wave is sustained Ap-plying the conditions of tm lt tle τ σi 05ρcvF ckuminus ckuf u0 um and _u0 0 to (9) results in

eurou +ρcA

mb

1113888 1113889 _u + ck

mb

1113888 1113889u ρcA

mb

1113888 1113889v + ck

mb

1113888 1113889uf (A8)

where the definition of u 1c(ρcAk)v + uf is possibleLast this unloading condition is where the incident

stress wave is resolved When unloading the drill bit usingthe conditions tlt τ σi 0 F ckuminus ckuf u0 u(τ) and_u0 _u(τ) allows (9) to be expressed as (A4)

In this case drill bit displacement u uf is used to in-dicate the rockrsquos fracture displacement 0e analysis of drillbit dynamics showed that when loading transitions tounloading in the underdamped condition the dynamicproperties of the bit exist only in the underdamping mode

Conflicts of Interest

The authors declare that there are no conflicts of interest

References

[1] C H Song K B Kwon J Y Park et al ldquoOptimum design ofthe internal flushing channel of a drill bit using RSM and CFDsimulationrdquo International Journal of Precision Engineeringand Manufacturing vol 15 no 6 pp 1041ndash1050 2014

12 Advances in Materials Science and Engineering

[2] X Li G Rupert D A Summers P Santi and D Liu ldquoAnalysisof impact hammer rebound to estimate rock drillabilityrdquo RockMechanics and Rock Engineering vol 33 no 1 pp 1ndash13 2000

[3] W A Hustrulid and C Fairhurst ldquoA theoretical and ex-perimental study of the percussive drilling of rock part Itheory of percussive drillingrdquo International Journal of RockMechanics and Mining Sciences amp Geomechanics Abstractsvol 8 no 4 pp 311ndash333 1971

[4] W A Hustrulid and C Fairhurst ldquoA theoretical and ex-perimental study of the percussive drilling of rock part IIforce-penetration and specific energy determinationrdquo In-ternational Journal of Rock Mechanics and Mining Sciences ampGeomechanics Abstracts vol 8 no 4 pp 335ndash356 1971

[5] W A Hustrulid and C Fairhurst ldquoA theoretical and ex-perimental study of the percussive drilling of rock part IIIexperimental verification of the mathematical theoryrdquo In-ternational Journal of Rock Mechanics and Mining Sciences ampGeomechanics Abstracts vol 9 no 3 pp 417-418 1972

[6] W A Hustrulid and C Fairhurst ldquoA theoretical and ex-perimental study of the percussive drilling of rock part IVapplication of the model to actual percussive drillingrdquo In-ternational Journal of Rock Mechanics and Mining Sciencesvol 9 no 3 pp 431ndash442 1972

[7] L E Chiang and D A Elias ldquoA 3D FEM methodology forsimulating the impact in rock-drilling hammersrdquo In-ternational Journal of Rock Mechanics and Mining Sciencesvol 45 no 5 pp 701ndash711 2008

[8] C H Song K B Kwon M G Cho J Y Oh D Y Shin andJ W Cho ldquoDevelopment of lab-scale rock drill apparatus fortesting performance of a drill bitrdquo International Journal ofPrecision Engineering and Manufacturing vol 16 no 7pp 1402ndash1414 2015

[9] K B Kwon C H Song J Y Park J Y Oh J W Lee andJ W Cho ldquoEvaluation of drilling efficiency by percussiontesting of a drill bit with new button arrangementrdquo In-ternational Journal of Precision Engineering andManufacturing vol 15 no 6 pp 1063ndash1068 2014

[10] B Lundberg andM Okrouhlik ldquoInfluence of 3D effects on theefficiency of percussive rock drillingrdquo International Journal ofimpact Engineering vol 25 no 4 pp 345ndash360 2001

[11] B Lundberg and P Collet ldquoOptimal wave shape with respectto efficiency in percussive drilling with detachable drill bitrdquoInternational Journal of impact Engineering vol 86 pp 179ndash187 2015

[12] X B Li G Rupert and D A Summers ldquoEnergy transmissionof down-hole hammer tool and its conditionalityrdquo Trans-actions of Nonferrous Metals Society of China vol 10 no 1pp 109ndash111 2000

[13] J W Cho S Jeon S H Yu and S H Chang ldquoOptimumspacing of TBM disc cutters a numerical simulation using thethree-dimensional dynamic fracturing methodrdquo Tunnellingand Underground Space Technology vol 25 no 3 pp 230ndash244 2010

[14] W Changming ldquoAn analytical study of percussive energytransfer in hydraulic rock drillsrdquo Mining Science and Tech-nology vol 13 no 1 pp 57ndash68 1991

[15] Online Materials Information Resource httpwwwmatwebcomsearchDataSheetaspxMatGUID8bc5d558f4174e6082ddf4966e382bd6ampckck1

Advances in Materials Science and Engineering 13

4 Results and Discussion

0is study defined equations for the motion of a drill bitwhen struck by a piston and expanded the dynamic prop-erties of the drill bit based on the conditions ζ and α It alsoanalyzed the relationship between the bit dynamics and theduration of the incident stress wave (τ) Underdamped drillbits displayed the greatest responsiveness and the fastesttransmission of τ Increasing the transmission of τ decreasedthe loading effect due to the drill bitrsquos inertia but led to a fastresponse 0e inertial effect of the drill bit was confirmed tobe inversely proportional to the transmission of τ

A percussive drilling system accomplishes its work viathe percussion impact of a piston In the systems studiedhere τ ended more quickly with lower mr and β values asthese values increased the duration of τ transmitted to therock via the drill bit increased in proportion to β 0e resultsalso showed that the maximum bit displacement (um) de-creased as β increased Analysis of the drilling dynamicsconfirmed the effective intervals of β With reference to therock strengths suggested by the International Society forRock Mechanics (ISRM) the effective interval of β is con-sidered to be no greater than 2

0e main purpose of this study was to examine theimpact energy transmission rate and drilling efficiency ina percussive drilling system (Figure 8) 0e results establishedamr value of 05 as the most efficient for rocks whose strengthcorresponds to the interval βlt 2 0e improved drilling ef-ficiency would lead to advantages such as reduced bit pro-duction costs Rocks stiffer than 2ltβlt 4 can be mostefficiently drilled by selecting mr 1

0emovement and response characteristics of bit duringpercussion process τ and transfer rate of energy caused bypercussion an efficiency depend on mr and β where in-crease in mr leads to overdamping of bit movement char-acteristics due to τ by piston and reduction in drillingefficiency due to internal energy effect by mass of drill bit Asthe increase in mass of drill bit for the section of 4lt βlt 6the drilling efficiency of percussive drilling system decreaseand for the very hard rock (UCSgt 200MPa) section of4lt βlt 6 it is considered that the reasonable applied mr ofdrilling tool is 1 or 2

0is study neglected the effects of the secondary incidentstress wave and also the effect of buttons embedded in thedrill bit Additionally for analytical simplification the pistonand drill bit were assumed to be of the same diameter andmaterial and a rectangular pulse with an incident stresswave of duration τ and amplitude 05ρcv was assumed 0eeffect of the flexural stress wave depending on the config-uration was not considered 0ese limitations of this papershould be addressed in further investigations that take intoaccount the configuration of the piston and drill bit theeffect of buttons and the drilling efficiency given differingshapes of the incident stress wave

5 Conclusion

0is paper aimed to identify the optimal design parametersfor percussive drilling systems by introducing a drill bit-rock

interaction model that could verify the bitrsquos motion duringpercussion and the resulting damping characteristics 0estudy analyzed drilling efficiency and drew the followingconclusions

Percussive drilling systems have six dynamic drill bitproperties that can be expandable 0is paper discussed thephysical meaning of the dimensionless parameters α and β0eir values determined the damping characteristics thatcan lead to the rockrsquos maximum fracture displacement 0efastest response of τ was observed for underdamped drill-bitmotion

Drilling was most efficient in the interval 1lt βlt 4 whereincreasing mr at a given β decreased drilling efficiency 0evalue of mr for efficient drilling was determined by the rockstrength (ie β)

0e results indicate that application of a piston-to-drill bitmass ratio of 05 (ie a piston mass twice that of the drill bitmass) to the rocks whose stiffness corresponds to βge 2 wouldlikely be most efficient and also reduce drill bit productioncosts Furthermore a selection ofmr 1 would be valid whendeveloping drill tools for boring complex rocks (1ltβlt 4) Atβgt 4 the best efficiency could be achieved when the bit massequals or exceeds the piston mass

Appendix

Theoretical Case of Bit Motion Condition(Drilling Dynamic Model)

0is paper proposed six conditions for the dynamics ofa drill bit struck by a piston impact as summarizedbelow 0e damping conditions for the damping ratiosand α from the drill-bit equations of motion (7ndash9) are asfollows

First for the overdamped condition of ζ1 gt 1 and αlt 1 τis transmitted to the drill bit indicating loading Equation(7) can be expressed as in (A1) 0e initial conditions are0lt tle τ σi 05ρcv F ku u0 0 and _u0 0

eurou +ρcA

mb

1113888 1113889 _u +k

mb

1113888 1113889u ρcA

mb

1113888 1113889v (A1)

where assuming the normal state u can be calculated asfollows

u ρcA

k1113874 1113875v (A2)

Next we examine the case where τ is complete and theloading effect is sustained owing to external forces and drill-bit inertia Applying the conditions of τ lt tle tm σi 0F ku u0 u(τ) and _u0 _u(τ) to (A1) results in thefollowing expression

eurou +ρcA

mb

1113888 1113889 _u +k

mb

1113888 1113889u 0 (A3)

where u 0 Last we examine the unloading conditions oftm lt t σi 0 F ckuminus ckuf u0 um and _u0 0 Apply-ing these to (9) when unloading gives

Advances in Materials Science and Engineering 11

eurou +ρcA

mb

1113888 1113889 _u + ck

mb

1113888 1113889u ck

mb

1113888 1113889uf (A4)

where bit displacement u uf is used to indicate the rockrsquosfracture displacement Furthermore considering the bitrsquosdamping ratio when unloading its behavior can be ex-pressed as

Damped mode

ζ2 gt 1 αlt1c

over( )

ζ2 1 α 1c

critical( )

ζ2 lt 1 αgt1c

under( )

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A5)

Second under critical damping where ζ1 1 and α 1 τis transmitted to the drill bit indicating loading Equation(7) can be expressed as in (A1) 0e initial conditions are0lt tle τ σi 05ρcv F ku u0 0 and _u0 0

In this case assuming the normal state u can be cal-culated using (A2)

0e following is the case where τ is complete and theloading effect is sustained owing to external forces and drillbit inertia Applying the conditions of τ lt tle tm σi 0F ku u0 u(τ) and _u0 _u(τ) to (A1) give (A3) whereu 0 Last we examine the unloading condition wheretm lt t σi 0 F ckuminus ckuf u0 um and _u0 0 Equation(9) when unloading can be expressed as (A4)

In this case bit displacement u uf is used to indicate therockrsquos fracture displacement Furthermore considering thebitrsquos damping ratio when unloading its behavior follows

Damped mode

ζ2 gt 1 αlt1c

(no case)

ζ2 1 α 1c

(no case)

ζ2 lt 1 αgt1c

under( )

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A6)

0e second condition of critical damping is defined asα 1When unloading drill bits c refers to the rockrsquos propertyeffect and is always less than 1 Hence when unloading noother cases but the underdamped mode are allowed In otherwords when unloading in the critically damped mode owingto the rock properties the bit is always underdamped

0e third condition is underdamping with ζ1 lt 1 andαgt 1 Here τ is transmitted to the drill bit meaning loadingEquation (7) can be expressed by (A1)0e initial conditionsare 0lt tle τ σi 05ρcv F ku u0 0 and _u0 0 Whereassuming the normal state u can be calculated as (A2)

0e following is the case where τ is complete and theloading effect is sustained owing to external forces and drillbit inertia Applying the conditions of τ lt tle tm σi 0F ku u0 u(τ) and _u0 _u(τ) to (A1) results in (A3)

where u 0 Last we examine the unloading conditionwhere tm lt t σi 0 F ckuminus ckuf u0 um and _u0 0Equation (9) when unloading can be expressed as (A4)

In this case the drill bit displacement u uf is used toindicate the rockrsquos fracture displacement Furthermoreconsidering the damping ratio in the drill bitrsquos unloadingthe bitrsquos behavior can be expressed as

Damped mode

ζ2 gt 1 αlt1c

(no case)

ζ2 1 α 1c

(no case)

ζ2 lt 1 αgt1c

under( )

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A7)

0e third condition is the critically damped modedefined as αgt 1 During drill bit unloading c refers to therock property effect and is always less than 1 Hencewhen unloading no other cases but underdamping areallowed

Next the underdamped condition is where ζ1 lt 1 andαgt 1 and τ is slower than tm (the time at which the rockrsquosmaximum fracture displacement is reached) that is tm lt τIn this case τ is transmitted to the drill bit meaningloading Equation (7) can be expressed by (A1) 0e initialconditions are 0lt tle tm σi 05ρcv F ku u0 0 and_u0 0 Here assuming the normal state u can be calculatedas (A2)

0e following case is where conversion is made tounloading after the drill bit reaches its maximum dis-placement but the incident stress wave is sustained Ap-plying the conditions of tm lt tle τ σi 05ρcvF ckuminus ckuf u0 um and _u0 0 to (9) results in

eurou +ρcA

mb

1113888 1113889 _u + ck

mb

1113888 1113889u ρcA

mb

1113888 1113889v + ck

mb

1113888 1113889uf (A8)

where the definition of u 1c(ρcAk)v + uf is possibleLast this unloading condition is where the incident

stress wave is resolved When unloading the drill bit usingthe conditions tlt τ σi 0 F ckuminus ckuf u0 u(τ) and_u0 _u(τ) allows (9) to be expressed as (A4)

In this case drill bit displacement u uf is used to in-dicate the rockrsquos fracture displacement 0e analysis of drillbit dynamics showed that when loading transitions tounloading in the underdamped condition the dynamicproperties of the bit exist only in the underdamping mode

Conflicts of Interest

The authors declare that there are no conflicts of interest

References

[1] C H Song K B Kwon J Y Park et al ldquoOptimum design ofthe internal flushing channel of a drill bit using RSM and CFDsimulationrdquo International Journal of Precision Engineeringand Manufacturing vol 15 no 6 pp 1041ndash1050 2014

12 Advances in Materials Science and Engineering

[2] X Li G Rupert D A Summers P Santi and D Liu ldquoAnalysisof impact hammer rebound to estimate rock drillabilityrdquo RockMechanics and Rock Engineering vol 33 no 1 pp 1ndash13 2000

[3] W A Hustrulid and C Fairhurst ldquoA theoretical and ex-perimental study of the percussive drilling of rock part Itheory of percussive drillingrdquo International Journal of RockMechanics and Mining Sciences amp Geomechanics Abstractsvol 8 no 4 pp 311ndash333 1971

[4] W A Hustrulid and C Fairhurst ldquoA theoretical and ex-perimental study of the percussive drilling of rock part IIforce-penetration and specific energy determinationrdquo In-ternational Journal of Rock Mechanics and Mining Sciences ampGeomechanics Abstracts vol 8 no 4 pp 335ndash356 1971

[5] W A Hustrulid and C Fairhurst ldquoA theoretical and ex-perimental study of the percussive drilling of rock part IIIexperimental verification of the mathematical theoryrdquo In-ternational Journal of Rock Mechanics and Mining Sciences ampGeomechanics Abstracts vol 9 no 3 pp 417-418 1972

[6] W A Hustrulid and C Fairhurst ldquoA theoretical and ex-perimental study of the percussive drilling of rock part IVapplication of the model to actual percussive drillingrdquo In-ternational Journal of Rock Mechanics and Mining Sciencesvol 9 no 3 pp 431ndash442 1972

[7] L E Chiang and D A Elias ldquoA 3D FEM methodology forsimulating the impact in rock-drilling hammersrdquo In-ternational Journal of Rock Mechanics and Mining Sciencesvol 45 no 5 pp 701ndash711 2008

[8] C H Song K B Kwon M G Cho J Y Oh D Y Shin andJ W Cho ldquoDevelopment of lab-scale rock drill apparatus fortesting performance of a drill bitrdquo International Journal ofPrecision Engineering and Manufacturing vol 16 no 7pp 1402ndash1414 2015

[9] K B Kwon C H Song J Y Park J Y Oh J W Lee andJ W Cho ldquoEvaluation of drilling efficiency by percussiontesting of a drill bit with new button arrangementrdquo In-ternational Journal of Precision Engineering andManufacturing vol 15 no 6 pp 1063ndash1068 2014

[10] B Lundberg andM Okrouhlik ldquoInfluence of 3D effects on theefficiency of percussive rock drillingrdquo International Journal ofimpact Engineering vol 25 no 4 pp 345ndash360 2001

[11] B Lundberg and P Collet ldquoOptimal wave shape with respectto efficiency in percussive drilling with detachable drill bitrdquoInternational Journal of impact Engineering vol 86 pp 179ndash187 2015

[12] X B Li G Rupert and D A Summers ldquoEnergy transmissionof down-hole hammer tool and its conditionalityrdquo Trans-actions of Nonferrous Metals Society of China vol 10 no 1pp 109ndash111 2000

[13] J W Cho S Jeon S H Yu and S H Chang ldquoOptimumspacing of TBM disc cutters a numerical simulation using thethree-dimensional dynamic fracturing methodrdquo Tunnellingand Underground Space Technology vol 25 no 3 pp 230ndash244 2010

[14] W Changming ldquoAn analytical study of percussive energytransfer in hydraulic rock drillsrdquo Mining Science and Tech-nology vol 13 no 1 pp 57ndash68 1991

[15] Online Materials Information Resource httpwwwmatwebcomsearchDataSheetaspxMatGUID8bc5d558f4174e6082ddf4966e382bd6ampckck1

Advances in Materials Science and Engineering 13

eurou +ρcA

mb

1113888 1113889 _u + ck

mb

1113888 1113889u ck

mb

1113888 1113889uf (A4)

where bit displacement u uf is used to indicate the rockrsquosfracture displacement Furthermore considering the bitrsquosdamping ratio when unloading its behavior can be ex-pressed as

Damped mode

ζ2 gt 1 αlt1c

over( )

ζ2 1 α 1c

critical( )

ζ2 lt 1 αgt1c

under( )

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A5)

Second under critical damping where ζ1 1 and α 1 τis transmitted to the drill bit indicating loading Equation(7) can be expressed as in (A1) 0e initial conditions are0lt tle τ σi 05ρcv F ku u0 0 and _u0 0

In this case assuming the normal state u can be cal-culated using (A2)

0e following is the case where τ is complete and theloading effect is sustained owing to external forces and drillbit inertia Applying the conditions of τ lt tle tm σi 0F ku u0 u(τ) and _u0 _u(τ) to (A1) give (A3) whereu 0 Last we examine the unloading condition wheretm lt t σi 0 F ckuminus ckuf u0 um and _u0 0 Equation(9) when unloading can be expressed as (A4)

In this case bit displacement u uf is used to indicate therockrsquos fracture displacement Furthermore considering thebitrsquos damping ratio when unloading its behavior follows

Damped mode

ζ2 gt 1 αlt1c

(no case)

ζ2 1 α 1c

(no case)

ζ2 lt 1 αgt1c

under( )

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A6)

0e second condition of critical damping is defined asα 1When unloading drill bits c refers to the rockrsquos propertyeffect and is always less than 1 Hence when unloading noother cases but the underdamped mode are allowed In otherwords when unloading in the critically damped mode owingto the rock properties the bit is always underdamped

0e third condition is underdamping with ζ1 lt 1 andαgt 1 Here τ is transmitted to the drill bit meaning loadingEquation (7) can be expressed by (A1)0e initial conditionsare 0lt tle τ σi 05ρcv F ku u0 0 and _u0 0 Whereassuming the normal state u can be calculated as (A2)

0e following is the case where τ is complete and theloading effect is sustained owing to external forces and drillbit inertia Applying the conditions of τ lt tle tm σi 0F ku u0 u(τ) and _u0 _u(τ) to (A1) results in (A3)

where u 0 Last we examine the unloading conditionwhere tm lt t σi 0 F ckuminus ckuf u0 um and _u0 0Equation (9) when unloading can be expressed as (A4)

In this case the drill bit displacement u uf is used toindicate the rockrsquos fracture displacement Furthermoreconsidering the damping ratio in the drill bitrsquos unloadingthe bitrsquos behavior can be expressed as

Damped mode

ζ2 gt 1 αlt1c

(no case)

ζ2 1 α 1c

(no case)

ζ2 lt 1 αgt1c

under( )

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A7)

0e third condition is the critically damped modedefined as αgt 1 During drill bit unloading c refers to therock property effect and is always less than 1 Hencewhen unloading no other cases but underdamping areallowed

Next the underdamped condition is where ζ1 lt 1 andαgt 1 and τ is slower than tm (the time at which the rockrsquosmaximum fracture displacement is reached) that is tm lt τIn this case τ is transmitted to the drill bit meaningloading Equation (7) can be expressed by (A1) 0e initialconditions are 0lt tle tm σi 05ρcv F ku u0 0 and_u0 0 Here assuming the normal state u can be calculatedas (A2)

0e following case is where conversion is made tounloading after the drill bit reaches its maximum dis-placement but the incident stress wave is sustained Ap-plying the conditions of tm lt tle τ σi 05ρcvF ckuminus ckuf u0 um and _u0 0 to (9) results in

eurou +ρcA

mb

1113888 1113889 _u + ck

mb

1113888 1113889u ρcA

mb

1113888 1113889v + ck

mb

1113888 1113889uf (A8)

where the definition of u 1c(ρcAk)v + uf is possibleLast this unloading condition is where the incident

stress wave is resolved When unloading the drill bit usingthe conditions tlt τ σi 0 F ckuminus ckuf u0 u(τ) and_u0 _u(τ) allows (9) to be expressed as (A4)

In this case drill bit displacement u uf is used to in-dicate the rockrsquos fracture displacement 0e analysis of drillbit dynamics showed that when loading transitions tounloading in the underdamped condition the dynamicproperties of the bit exist only in the underdamping mode

Conflicts of Interest

The authors declare that there are no conflicts of interest

References

[1] C H Song K B Kwon J Y Park et al ldquoOptimum design ofthe internal flushing channel of a drill bit using RSM and CFDsimulationrdquo International Journal of Precision Engineeringand Manufacturing vol 15 no 6 pp 1041ndash1050 2014

12 Advances in Materials Science and Engineering

[2] X Li G Rupert D A Summers P Santi and D Liu ldquoAnalysisof impact hammer rebound to estimate rock drillabilityrdquo RockMechanics and Rock Engineering vol 33 no 1 pp 1ndash13 2000

[3] W A Hustrulid and C Fairhurst ldquoA theoretical and ex-perimental study of the percussive drilling of rock part Itheory of percussive drillingrdquo International Journal of RockMechanics and Mining Sciences amp Geomechanics Abstractsvol 8 no 4 pp 311ndash333 1971

[4] W A Hustrulid and C Fairhurst ldquoA theoretical and ex-perimental study of the percussive drilling of rock part IIforce-penetration and specific energy determinationrdquo In-ternational Journal of Rock Mechanics and Mining Sciences ampGeomechanics Abstracts vol 8 no 4 pp 335ndash356 1971

[5] W A Hustrulid and C Fairhurst ldquoA theoretical and ex-perimental study of the percussive drilling of rock part IIIexperimental verification of the mathematical theoryrdquo In-ternational Journal of Rock Mechanics and Mining Sciences ampGeomechanics Abstracts vol 9 no 3 pp 417-418 1972

[6] W A Hustrulid and C Fairhurst ldquoA theoretical and ex-perimental study of the percussive drilling of rock part IVapplication of the model to actual percussive drillingrdquo In-ternational Journal of Rock Mechanics and Mining Sciencesvol 9 no 3 pp 431ndash442 1972

[7] L E Chiang and D A Elias ldquoA 3D FEM methodology forsimulating the impact in rock-drilling hammersrdquo In-ternational Journal of Rock Mechanics and Mining Sciencesvol 45 no 5 pp 701ndash711 2008

[8] C H Song K B Kwon M G Cho J Y Oh D Y Shin andJ W Cho ldquoDevelopment of lab-scale rock drill apparatus fortesting performance of a drill bitrdquo International Journal ofPrecision Engineering and Manufacturing vol 16 no 7pp 1402ndash1414 2015

[9] K B Kwon C H Song J Y Park J Y Oh J W Lee andJ W Cho ldquoEvaluation of drilling efficiency by percussiontesting of a drill bit with new button arrangementrdquo In-ternational Journal of Precision Engineering andManufacturing vol 15 no 6 pp 1063ndash1068 2014

[10] B Lundberg andM Okrouhlik ldquoInfluence of 3D effects on theefficiency of percussive rock drillingrdquo International Journal ofimpact Engineering vol 25 no 4 pp 345ndash360 2001

[11] B Lundberg and P Collet ldquoOptimal wave shape with respectto efficiency in percussive drilling with detachable drill bitrdquoInternational Journal of impact Engineering vol 86 pp 179ndash187 2015

[12] X B Li G Rupert and D A Summers ldquoEnergy transmissionof down-hole hammer tool and its conditionalityrdquo Trans-actions of Nonferrous Metals Society of China vol 10 no 1pp 109ndash111 2000

[13] J W Cho S Jeon S H Yu and S H Chang ldquoOptimumspacing of TBM disc cutters a numerical simulation using thethree-dimensional dynamic fracturing methodrdquo Tunnellingand Underground Space Technology vol 25 no 3 pp 230ndash244 2010

[14] W Changming ldquoAn analytical study of percussive energytransfer in hydraulic rock drillsrdquo Mining Science and Tech-nology vol 13 no 1 pp 57ndash68 1991

[15] Online Materials Information Resource httpwwwmatwebcomsearchDataSheetaspxMatGUID8bc5d558f4174e6082ddf4966e382bd6ampckck1

Advances in Materials Science and Engineering 13

[2] X Li G Rupert D A Summers P Santi and D Liu ldquoAnalysisof impact hammer rebound to estimate rock drillabilityrdquo RockMechanics and Rock Engineering vol 33 no 1 pp 1ndash13 2000

[3] W A Hustrulid and C Fairhurst ldquoA theoretical and ex-perimental study of the percussive drilling of rock part Itheory of percussive drillingrdquo International Journal of RockMechanics and Mining Sciences amp Geomechanics Abstractsvol 8 no 4 pp 311ndash333 1971

[4] W A Hustrulid and C Fairhurst ldquoA theoretical and ex-perimental study of the percussive drilling of rock part IIforce-penetration and specific energy determinationrdquo In-ternational Journal of Rock Mechanics and Mining Sciences ampGeomechanics Abstracts vol 8 no 4 pp 335ndash356 1971

[5] W A Hustrulid and C Fairhurst ldquoA theoretical and ex-perimental study of the percussive drilling of rock part IIIexperimental verification of the mathematical theoryrdquo In-ternational Journal of Rock Mechanics and Mining Sciences ampGeomechanics Abstracts vol 9 no 3 pp 417-418 1972

[6] W A Hustrulid and C Fairhurst ldquoA theoretical and ex-perimental study of the percussive drilling of rock part IVapplication of the model to actual percussive drillingrdquo In-ternational Journal of Rock Mechanics and Mining Sciencesvol 9 no 3 pp 431ndash442 1972

[7] L E Chiang and D A Elias ldquoA 3D FEM methodology forsimulating the impact in rock-drilling hammersrdquo In-ternational Journal of Rock Mechanics and Mining Sciencesvol 45 no 5 pp 701ndash711 2008

[8] C H Song K B Kwon M G Cho J Y Oh D Y Shin andJ W Cho ldquoDevelopment of lab-scale rock drill apparatus fortesting performance of a drill bitrdquo International Journal ofPrecision Engineering and Manufacturing vol 16 no 7pp 1402ndash1414 2015

[9] K B Kwon C H Song J Y Park J Y Oh J W Lee andJ W Cho ldquoEvaluation of drilling efficiency by percussiontesting of a drill bit with new button arrangementrdquo In-ternational Journal of Precision Engineering andManufacturing vol 15 no 6 pp 1063ndash1068 2014

[10] B Lundberg andM Okrouhlik ldquoInfluence of 3D effects on theefficiency of percussive rock drillingrdquo International Journal ofimpact Engineering vol 25 no 4 pp 345ndash360 2001

[11] B Lundberg and P Collet ldquoOptimal wave shape with respectto efficiency in percussive drilling with detachable drill bitrdquoInternational Journal of impact Engineering vol 86 pp 179ndash187 2015

[12] X B Li G Rupert and D A Summers ldquoEnergy transmissionof down-hole hammer tool and its conditionalityrdquo Trans-actions of Nonferrous Metals Society of China vol 10 no 1pp 109ndash111 2000

[13] J W Cho S Jeon S H Yu and S H Chang ldquoOptimumspacing of TBM disc cutters a numerical simulation using thethree-dimensional dynamic fracturing methodrdquo Tunnellingand Underground Space Technology vol 25 no 3 pp 230ndash244 2010

[14] W Changming ldquoAn analytical study of percussive energytransfer in hydraulic rock drillsrdquo Mining Science and Tech-nology vol 13 no 1 pp 57ndash68 1991

[15] Online Materials Information Resource httpwwwmatwebcomsearchDataSheetaspxMatGUID8bc5d558f4174e6082ddf4966e382bd6ampckck1

Advances in Materials Science and Engineering 13

CorrosionInternational Journal of

Hindawiwwwhindawicom Volume 2018

Advances in

Materials Science and EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Journal of

Chemistry

Analytical ChemistryInternational Journal of

Hindawiwwwhindawicom Volume 2018

ScienticaHindawiwwwhindawicom Volume 2018

Polymer ScienceInternational Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Advances in Condensed Matter Physics

Hindawiwwwhindawicom Volume 2018

International Journal of

BiomaterialsHindawiwwwhindawicom

Journal ofEngineeringVolume 2018

Applied ChemistryJournal of

Hindawiwwwhindawicom Volume 2018

NanotechnologyHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

High Energy PhysicsAdvances in

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

TribologyAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

ChemistryAdvances in

Hindawiwwwhindawicom Volume 2018

Advances inPhysical Chemistry

Hindawiwwwhindawicom Volume 2018

BioMed Research InternationalMaterials

Journal of

Hindawiwwwhindawicom Volume 2018

Na

nom

ate

ria

ls

Hindawiwwwhindawicom Volume 2018

Journal ofNanomaterials

Submit your manuscripts atwwwhindawicom