Arithmetic solution for the axial vibration of drill ... · dynamic design, modification, and use of a DTH drilling system. Keywords Axial vibration, drill string mode, DTH, periodic

Article

Arithmetic solution for the axialvibration of drill string coupling with adown-the-hole hammer in rock drilling

Changgen Bu, Xiaofeng Li, Long Sun and Boru Xia

Abstract

Down-the-hole hammer (DTH) drilling is an air hammer drilling technique designed for drilling through bedrock and

features a typical drill string length of 200 m or shorter due to its technical specifications. During DTH drilling of granite-

like hard rocks, the impacts of the piston-bit-rock system cause the drill string to generate severe vibration and noise

pollution. In addition, the rapid wear of the button bit and drill string significantly decreases the drilling efficiency. Based

on a distributed parameter drill string model of a DTH, this paper studies the phenomenon of the drill string’s axial

forced vibration with a periodic impacting force under DTH drilling in an innovative manner. With the focus of study on

the DTH button bit, the transient impact force on the button bit during the drilling of the piston-bit-rock system is

determined, and the impact force is converted to a periodic excitation force function using polynomial fitting. Then, the

periodic impulse is transformed into a harmonic series using Fourier transforms, and finally, the drill string vibration

response under the harmonic excitation force series is determined. The results reveal that a periodic impulse can mainly

be determined by the nature of the DTH drill string and rock and the impact frequency during drilling. Further evidence

demonstrates that at least one frequency component of the impulse harmonic series will be equal to the modal

frequencies of the drill string insofar as the condition 2nf0l ¼ iffiffiffiffiffiffiffiffiffiE=�p

is met; the coupling of the short drill string

with the DTH may cause resonance at a specific hole depth, whose resonance regions are adjacent to but not continuous

with the extension of the drill string. This work should serve as an important theoretical guide for designers in the

dynamic design, modification, and use of a DTH drilling system.

Keywords

Axial vibration, drill string mode, DTH, periodic impulse

1. Introduction

In the process of oil drilling, a rock bit, drag bit, andpolycrystalline diamond compacts (PDC) bit are oftenused to drill through rock. The interaction between therock bit and rock produces low-frequency excitationseven though it has been demonstrated to be relativelystable. When this frequency is similar to the naturalfrequency of the long drill string, the drill string willresonate. Then, the alternating load will cause fatiguefailure of the drill string when the drill string works fora long duration; meanwhile, the coupling vibrationincreases both the contact between the drill string andwall and the wear of the casing. The resulting drillstring failures are numerous, and thus, this studyfocuses on the hazards of this long drill string vibrationof a deep well in oilfield production. In early research,

Dareing (1984) established the motion equation of lon-gitudinal and torsional vibration of a bottom holeassembly (BHA) to study the dynamic characteristicsof the drill string, and the natural frequency wasobtained using a separation of variables method.Aarrestand et al. (1986) and (1988) studied the vibra-tion problem of drilling equipment based on earlieranalysis and then set the boundary conditions that

School of Engineering and Technology, China University of Geosciences

(Beijing), People’s Republic of China

Corresponding author:

Changgen Bu, School of Engineering and Technology, China University of

Geosciences (Beijing), People’s Republic of China.

Email: [email protected]

Received: 17 March 2014; accepted: 14 October 2014

Journal of Vibration and Control

2016, Vol. 22(13) 3090–3101

! The Author(s) 2014

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DOI: 10.1177/1077546314560041

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the top is free and the bottom is fixed. When the upperpart of the bit has a shock absorber with longitudinalrigidity, the first-order natural frequency of the drill setcan be solved using a simple mass-spring model.Apostal et al. (1990) developed a three-dimensionalfinite element model of vibrations of down hole equip-ment based on the finite element method for theforced-frequency-response induced by the periodicand harmonic excitation and conducted research onthe transient vibration analysis of a BHA by introdu-cing inertial force and friction factors. In the mid-to-late 1990s, Clayer et al. (1990) analyzed the effectof the top and bottom boundary conditions of thedrill string on drill string vibration when a rock bit isused for drilling by considering the surface equipmentand down hole condition as one system; these research-ers studied the formation mechanism of stick-slip andits incentive relationship with transverse vibration.Yigit and Christoforou (1996), (1998) and (2003) mod-eled the drill string based on the assumed modemethod. Their models consider the coupling betweenaxial and transverse vibrations and between torsionaland transverse vibrations, as well as that among axial,transverse, and torsional vibrations. Batako et al.(2003) and (2004) studied the problem of nonlinearself-excited vibration induced by the stick-slip frictionof a drill string and performed experiments on theself-excited impact vibration caused by stick-slip.

Liu et al. (1998) and (2000) constructed a mathem-atical model of longitudinal and torsional vibrationthrough theoretical analysis and simulation andsolved the model using the finite difference method;however, these researchers did not provide a methodto confirm the exciting force. Applying wave theory,Gao et al. (2000) developed longitudinal vibration dif-ferential equations that considered shock absorbers andthen developed the calculation equation for the naturalfrequency of a drill string. On-site measurements andcalculations indicate that not all natural frequencypoints can cause the drill string to vibrate strongly;on-site test data also verify the calculation equation.By calculating the natural frequency, Gao et al.(2000) analyzed the effect of shock absorbers on thenatural frequency and variation tendency of the naturalfrequency with well depth. However, the exciting forceof the bottom of the drill string was not found to revealthe interaction between the bit and rock using the stiff-ness coefficient. Qiu (2002) used the energy method tocalculate the kinetic and potential energy of the unitand established a finite element model of the drillstring system using the Lagrange equation. Then,these researchers developed a numerical calculationmethod to solve the differential equations of axialvibration coupling with lateral vibration and torsionalvibration and used their model to perform simulations.

However the interaction between the drill bit and rockwas not considered. Li et al. (2004) used the method offorce excitation and displacement excitation to solvethe longitudinal vibration of the drill string to reduceand eliminate on-site vibration and noted that thedisplacement excitation method was superior. Theresearchers then used the method of torque excitationand angle displacement excitation to study the torsionalvibration problems of the drill string and found that itis more realistic to use the angle displacement excitationmethod to study the torsional vibration of a drill string.

Caresta et al. (2013) predicted the local response fol-lowing an impact or a shock loading. The assumptionunderlying this approach is that the duration of theimpact is short enough that the reverberant field doesnot affect the contact force. The impulse as an inputfunction is expressed by a half-period sine. Theseassumptions are valid in many situations and lead toa useful analytical approximation. However, themethod does not reveal that if the impulse describedonly by the force amplitude and duration affects theresponse of an impact system when the impact durationmust not be much smaller than the dwell time.

Additional studies have considered the dynamics ofthe long drill string during drilling; however, it is diffi-cult to accurately describe the long drill string vibrationbased on existing theories because drill string stressesand movements are extremely complex in holes.Existing studies on down-the-hole hammer (DTH) per-cussive drilling systems have mainly focused on thedesign of the DTH in China and abroad; a few system-atic studies on the impact dynamic characteristics ofDTH percussive drilling systems coupled with shortdrill strings have been performed. As illustrated inFigure 1, the drilling hole is shallower (generally lessthan 200m) in DTH drilling than in oil drilling.Because an acutely periodic axial impacting existsduring drilling, longitudinal vibration of the short drillstring can occur easily due to the effect of the DTH.

Beginning with the flexible body impact of the DTHcoupled with the drill string, an intensive study on thevibration characteristics of a DTH drill string is per-formed to optimize and control the DTH drillingsystem, to reduce or prevent the resonance of theDTH drilling string system, to effectively avoid drillingaccidents and abnormal phenomena caused by vibra-tion, and to improve the drilling efficiency of the entiresystem.

The authors’ work provides two major contributionsthat distinguish it from the existing literature. First, aset of complete research methods are constructed basedon the axial shock and vibration of a DTH drill toperform the following: 1) convert the impact force ofthe piston-bit-rock system to a periodic excitation forcefunction using polynomial fitting; 2) transform the

Bu et al. 3091



periodic impulse into a harmonic series using Fouriertransforms; and 3) determine the drill string vibrationresponse under the harmonic excitation force series.Second, using these mathematical methods, it isproved that the coupling of the drill string with theDTH will be resonant at a specific hole depth duringthe drilling of hard rock.

2. Dynamic modeling of an elasticdrill string

2.1. Kinetic differential equationof a continuous column

As illustrated in Figure 2, the DTH drilling system hasa shorter drill string and lower load on the drill bit; thedrill string is typically stable in the hole. The top of thedrill string is regarded as a movable hinge, as it can

slide along the mast guide; the bottom of the drillstring is a free end or linear motion pair due to itsconstraints originating from the drill hole. Thebottom of the drill string is excited by F tð Þ, which isthe resultant force of the impulse between the drill bitand piston and the impacting force between the drill bitand rock.

Only the relative motion of the drill string is con-sidered (bulk motion is ignored); the cross-section ofthe drill string is taken as a plane, ignoring the effectof lateral deformation on the axial vibration. l is thelength of the drill string, S is the cross-sectional area, �is the material density, and E is the elastic modulus; thegravity effect is ignored. The transient vibration of therod will decay fast due to the structural damping, soonly the steady-state vibration is studied.

The partial differential equation of small elementscan be developed as follows according to theD’Alembert principle:

�S@2u

@t2þ c

@u

@t� ES

@2u

@x2¼ f ðx, tÞ ð1Þ

where f ðx, tÞ represents the longitudinal load on theunit length of the drill string and c denotes the viscousdamping coefficient on a unit length of drill string (Hu,2005). In equation (1), the axial free vibration equation

Figure 1. Structure diagram of a down-the-hole hammer

coupled with a drill string system.

Figure 2. Establishment and boundary constraints of the drill

string coordinates.

3092 Journal of Vibration and Control 22(13)



of the drill string can be obtained by assuming thatf ðx, tÞ ¼ 0:

�S@2u

@t2þ c

@u

@t� ES

@2u

@x2¼ 0 ð2Þ

By assuming that uðx, tÞ ¼ XðxÞqðtÞ satisfiesequation (2) and by substituting uðx, tÞ into equation(2), the following expression can be obtained:

€qðtÞ

qðtÞþ

c

�S

_qðtÞ

qðtÞ¼

E

�

€XðxÞ

XðxÞð3Þ

Then, two independent ordinary differential equa-tions are obtained by the separation of variables:

E

�X00ðxÞ þ !2

i XðxÞ ¼ 0 ð4Þ

€qðtÞ þc

�S_qðtÞ þ !2

i qðtÞ ¼ 0 ð5Þ

Solving equations (4) and (5) yields

uiðx, tÞ ¼ XiðxÞqiðtÞ

¼ Ai sin!i

axþ Bi cos

!i

ax

� �qiðtÞ ð6Þ

where a ¼ffiffiffiffiffiffiffiffiffiE=�p

is the axial wave velocity of the drillstring and Ai, Bi and !i are determined by the boundaryconditions and initial conditions.

2.2. Longitudinal forced vibration of the drillingstring under one harmonic force

The principal mode equation (equation (6)) corres-ponding to !i satisfies equation (2). Equation (6) is aparticular solution of equation (2). The general solutionof the longitudinal free vibration of the drill string isexpressed as a linear superposition of an infinitenumber of linearly independent particular solutions(principal vibration) as follows:

uðx, tÞ ¼X1i¼1

XiðxÞqiðtÞ ð7Þ

Substituting equation (7) into equation (1),equation (1) becomes

�SX1i¼1

Xi €qi þ cX1i¼1

Xi _qi � ESX1i¼1

X00i qi ¼ f ðx, tÞ ð8Þ

Multiplying both sides of equation (8) by Xj andintegrating this result with respect to x yields

X1i¼1

€qi

Z l

0

�SXiXj dxþX1i¼1

_qi

Z l

0

cXiXj dx

�X1i¼1

qi

Z l

0

ESX00i Xj dx ¼

Z l

0

f ðx, tÞXj dx ð9Þ

The vibration mode function of an elastic body’smulti degree-of-freedom system is orthogonal; there-fore, only i ¼ j is satisfied, and the following expres-sions can be obtained:

mj ¼

Z l

0

�SXiXj dx ¼

Z l

0

�SX2j dx

kj ¼ �

Z l

0

ESX00i Xj dx ¼

Z l

0

!2i �SX

2j dx

cj ¼ c

Z l

0

X2j ðxÞdx

8>>>>>>>>><>>>>>>>>>:

ð10Þ

By substituting equation (10) into equation (9), the j-order equation of a normal coordinate system is

mj €qj ðtÞ þ cj _qj ðtÞ þ kjqj ðtÞ ¼ Qj ðtÞ ð11Þ

where Qj ðtÞ ¼R l0 f ðx, tÞXj ðxÞdx is the generalized force

of the j-order normal coordinate.By introducing a Dirac delta function �, the concen-

trated load FðtÞ ¼ P0 sin!t can be converted to a dis-tributed force:

f ðx, tÞ ¼ P0 sin!t�ðxÞ ¼0 0 � x5 l

P0 sin!t x ¼ l

�ð12Þ

Then, the normal generalized force can be acquired:

Qj ðtÞ ¼

Z l

0

P0 sin!tXj ðxÞ�ðxÞdx ¼ P0 sin!tXj ðl Þ ð13Þ

In the DTH drilling system, the boundary conditionscan be defined by equation (14) for the following con-ditions: first, there is a movable hinge on both ends;second, there is a free one on both ends; and third,one end is a movable hinge, whereas the other end isfree or linear motion pair.

@uð0, tÞ

@x¼@uðl, tÞ

@x¼ 0 or

dX

dx

��x¼0

¼dX

dx

��x¼l

¼ 0 ð14Þ

By substituting equation (14) into equation (6), it isfound that A ¼ 0; thus, the modal frequency equation

Bu et al. 3093



sinð!l=aÞ ¼ 0 becomes

!i ¼i�

l

ffiffiffiffiE

�

si ¼ 0, 1, 2, . . . ð15Þ

Because ! 6¼ 0, the vibration mode functions of eachorder of modal frequency are as follows:

XiðxÞ ¼ Bi cosi�x

li ¼ 0, 1, 2, . . . ð16Þ

By substituting equation (16) into equation (10),there is obtained

mi ¼

Z l

0

�S Bi cosi�x

l

� �2

dx ¼1

2�SlB2

i ð17Þ

where Bi¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2mi=�Slp

ði ¼ 1, 2, 3 . . .Þ is the coefficientof the vibration mode functions; the normal vibrationmode function can be expressed as

XiðxÞ ¼ Bi cosi�x

l¼

ffiffiffiffiffiffiffiffi2mi

�Sl

scos

i�x

li ¼ 0, 1, 2, . . .

ð18Þ

Then, the normal generalized force is obtained asfollows:

QiðtÞ ¼ P0 sin!tXiðl Þ ¼

ffiffiffiffiffiffiffiffi2mi

�Sl

sP0 sin!t cos i� ð19Þ

By substituting equation (19) into equation (11), themodal damping ratio becomes �i ¼ ci=2mi!i; the experi-ment conducted by Wen et al. (2009) demonstrate thatthe modal damping ratios of each order vibration areidentical, and the value of a higher-order modal damp-ing ratio is slightly larger. For a small damping(0 � �i � 0:2) system, the modal damping ratios ofeach order vibration are typically identical (Wenet al., 2009). The modal damping ratio of the DTHdrilling system is taken to be 0.001 because air isthe circulating medium in DTH drilling. The steadyforced vibration response based on the normal coord-inates is

qiðtÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffi2

�Slmi

sP0 cos i� sinð!t� ’iÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið!2

i � !2Þ

2þ ð2�i!i!Þ

2q ð20Þ

Therefore, the axial steady forced vibration of thestring under the simple harmonic excitation force is

uðx, tÞ ¼X1i¼0

XiðxÞqiðtÞ

¼2

�Sl

X1i¼0

P0 sinð!t� ’iÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið!2

i � !2Þ

2þ ð2�i!i!Þ

2q

� cos i� cosi�x

lð21Þ

Equation (21) is the vibration response of the drillstring under the single harmonic force FðtÞ ¼ P0 sin!t.

3. Solution of the down-the-holehammer drill string impactvibration response

3.1. Periodic excitation of the down-the-holehammer drill string

The excitation force of a DTH on a drill string is thekey to solving the vibration response of a DTH drillstring. Bu et al. (2009) constructed a three-dimensionalmodel of the transient impacts of a DTH drillingsystem (piston-drill-rock) based on the finite elementmethod of impact dynamics in LS-DYNA program.He analyzed the exact process of DTH drilling rocksat a piston velocity of 7.2m/s. The impact curves of thepiston-bit and bit-rock (granite) systems were obtainedby simulation, as illustrated in Figure 3.

A drill bit is considered to be a point mass because theaxial size of a drill bit is considerably smaller than that ofthe drill string. The difference between the two forcesshown in Figure 3 can be described as FðtÞ ¼ Fb tð Þ�Fa tð Þ, which is the axial force of the drill bit during thedrilling process. Only when the axial force of the drill isgreater than zero (FðtÞ ¼ Fb tð Þ � Fa tð Þ4 0) can the drillbit impact the bottom of the drill string; thus, the impactforce above the time axis is considered as the effectiveimpact of the DTH drill string. F(t) is the effectiveimpact force of the drill bit on the drill string, for a cer-tain piston-bit-rock system and impact frequency, thelength of drill string has no influence on it.

In Figure 4, the dashed area is the effective impactforce in the drilling process. The cycle impact betweenthe drill bit and the bottom of the drill string is acquiredwhen the impact cycle of the piston is T¼ 0.05 s.Figure 5 presents two cycles of steady impact force.

3.2. Polynomial fitting of the impact force curve

As the time scale of the impact is quite short in compari-son to the impact cycle T, the impact force can be




approximated with a half-sine wave or even a delta func-tion, but using the polynomial fitting of the impact forcewould obtain a more common approach which can bemore applicable to arbitrary periodical exciting force,such as the sonic drill chiseling rock.

According to the polynomial fitting of the effectcurves using MATLAB software, the non-zero impactcurves are fitted to a quartic polynomial, as shown inFigure 6. Therefore, the excitation force of the drill bit

onto the bottom of the drill string with cycle T can bedescribed as follows:

FðtÞ ¼

�1:81e20t4þ2:13e17t3

�9:08e13t2þ1:35e10t

�22086:49 ðnT5t5nTþ �Þ

0 ðnTþ �5 t5ðnþ1ÞTÞ

8>>><>>>:

ð22Þ

Figure 3. Curves describing the force Fa(t) between the bit and piston and force Fb(t) between the bit and rock. Reproduced with

kind permission from Springer (Bu et al. 2009).

Figure 4. Effective impact force of the drill bit on the drill string.

Bu et al. 3095



where n¼ 0, 1, 2, . . . and �¼3:5� 10�4s.In total, 176 data points were used to fit the curve,

yielding a determination coefficient of 0.98261 for theimpact vibration force curve and a significance level Pof less than 0.0001; the fitting effect of the impact curveis remarkable, i.e., equation (22) can be used to fit theperiodic vibration force curve.

3.3. Fourier series of the fitting function

To solve the inhomogeneous equation (1) correspond-ing to the fitting function, the fitting function in

equation (22) is expanded by Fourier expansion inorder to obtain the sum of the harmonic series,namely equation (23a), whose curve is presented inFigure 6. The more series are used for the Fourierexpansion, the more accurate is the correspondingFourier function curve with the polynomial fittingcurve. After many trials, the superposition of 1000series can obtain good consistency.

FðtÞ ¼ a0 þX1n¼1

ðan cos n!tþbn sin n!tÞ n ¼ 0, 1, . . .

ð23aÞ

Figure 6. Exciting force curve, polynomial fitting curve, and corresponding Fourier function curve.

Figure 5. Impact pulse when the impact cycle is T¼ 0.05 s.




where ! ¼ 2�=T ¼ 40� and a0, an, and bn are given by

a0 ¼1

T

Z T

0

FðtÞdt ¼ 2762:9 ð23bÞ

an ¼2

T

Z T

0

FðtÞ cos n!tdt

¼�3:8427e3

�nþ

3:7e11

402n3�3�4:344e21

404n5�5

� �sin

7�n

500

þ�2:824e9

40�2n2þ2:424e17

403�4n4

� �cos

7�n

500

�1:35e10

40n2�2þ1:278e18

403n4�4ð23cÞ

bn ¼2

T

Z T

0

FðtÞ sin n!tdt

¼3:8427e3

�n�

3:7e11

402�3n3þ4:344e21

404�5n5

� �cos

7�n

500

þ�2:824e9

40�2n2þ2:424e17

403�4n4

� �sin

7�n

500

�22086:49

�nþ1:816e14

402�3n3�4:344e21

404�5n5ð23dÞ

The fundamental frequency of the Fourier seriesexpressed by equation (23a) is valid up to the periodT of the piston impact. The coefficients of the Fourierseries are mainly affected by the effective impact of thepiston-bit-rock system on the drill string.

3.4. Method used to determine the vibration ofthe drill string with cycle impact excitation

Any periodic function satisfying the Dirichlet conditioncan be expanded into a Fourier series. Therefore, theimpact periodic function can be decomposed into thealgebraic sum of a series of different frequency har-monic functions, as shown in equation (23a). To solvethe vibration response of the drill string coupling withthe DTH, as illustrated in Figure 2, the delta function �is introduced and the cycle impact force is transformedinto a distribution of functions f ðx, tÞ that satisfiesequation (1); f ðx, tÞ is given by

f ðx, tÞ ¼ FðtÞ�ðxÞ

¼ a0 þX1n¼1

ðan cos n!tþbn sin n!tÞ

!�ðxÞ ð24Þ

where n¼ 1, 2, . . . and a0, an, and bn are coefficients ofthe harmonic series.

In equation (24), ðan cos n!tþ bn sin n!tÞ is the nthcomponent of the steady-state harmonic vibration

force, i.e., equation (24) is the sum of n equation (12)of single harmonic vibration forces. a0 is the constantforce component of the cycle impact force, which hasno effect on the steady-state vibration of the drill string.In the elastic range, the superposition of the vibrationresponse induced by the multiple harmonic forces basedon one harmonic force versus the steady-state responseof equation (21) is adopted to obtain the steady-statevibration response of the drill string, as shown inFigure 2 under the DTH drilling cycle impact excita-tion. Then, the steady-state vibration response of thedrill string under the cycle impact function F(t) isobtained as

uðx, tÞ ¼2

�Sl

X1i¼0

a0

!2i

cos i� cosi�x

l

"

þX1n¼1

X1i¼0

an cosðn!t� ’iÞ

þbn sinðn!t� ’iÞ

� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið!2

i � n2!2Þ2þ ð2�in!!iÞ

2q

� cos i� cosi�x

l

ð25Þ

where ’i ¼ arctan ½2�i!i!=ð!2i � !

2Þ�.The drill string axial vibration displacement field

uðx, tÞ, the velocity field @uðx, tÞ=@t, and the stress field�ðx, tÞ ¼ E @uðx, tÞ=@x under periodic excitation func-tion F tð Þ are obtained by substituting equations (23b),(23c), (23d), !i, and n! into equation (25), where !i isthe modal frequency of the drill string for each orderand n! is the circular frequency of each harmonic func-tion of the cycle impact Fourier series. Equation (25)illustrates that the vibration response value tends tomaximum, i.e., the system will be resonant, whenthe DTH impact frequency or frequency multiplicationis close or equal to the modal frequency of thedrill string.

3.5. Modal frequency of the drill string andsteady-state response of the cycle impact

This paper considers a DTH alloy steel drill string with�89mm; the material properties and structural param-eters of this drill string are listed in Table 1.

Table 1. Structural parameters and materials properties of the

drill string.

Length

(m)

Cross-sectional

area (m2)

Density

(kg/m3)

Modulus of

elasticity (Pa)

l 0.0057 7,850 2.06 E11

Bu et al. 3097



fi is each modal frequency of the drill string and canbe obtained from equation (15) when both ends of theDTH satisfy the following constraint:

fi ¼i

2l

ffiffiffiffiE

�

s¼ 2561:3

i

lHz i ¼ 0, 1, 2, . . . ð26Þ

The modal frequency (fi) of the drill string in theequation above is inversely proportional to its length(l), and their relations are plotted in Figure 7.

The impulse of the bit onto the drill string will causethe drill string to resonate when each frequency of theFourier series of the impulse, e.g., f0, f1, . . . nf0, is closeor equal to each modal frequency. The length of a shortDTH drill string typically does not exceed 200m, andthe first-order modal frequency is not less than 12.8Hz.Equation (25) indicates that when the fundamentalfrequency of the impulse equals the first-order modalfrequency of the drill string, each frequency of theimpulse Fourier series is equal to each modal frequencyof the drill string. As illustrated in Figure 7, this equa-tion meets all orders of resonance conditions for thelength of the drill string l1 ¼ 128:07 m. The vibrationresponse of the drill string and the correspondingamplitude-frequency characteristics can be obtainedfrom equation (25), and taking 50 modes can obtainthe desired results, as demonstrated in Figure 8 inwhich the amplitude of the vibration response of thefundamental frequency is the largest.

When the length of the drill string is l3, the second-order harmonic frequency of the impulse equals the

first-order modal frequency of the drill string, andequation (25) is transformed into 2f0 ¼ 2561:3� 1=l3.Here, the length of the drill string is 64.03m, and cer-tain multiples of the frequency, i.e., 2f0, 4f0, 6f0 . . .,equal the first-, second-, third-, . . . order modal fre-quencies of the drill string, respectively, which isthe resonance frequency region. The amplitude of the64.03-m drill string is smaller than that of the 128.07-mdrill string; however, the drill string is shorter becausethe amplitude of the second-order harmonic frequencyis smaller than that of the fundamental frequency in theimpulse Fourier series.

When the length of the drill string is l2, the third-order harmonic frequency of the impulse equals thesecond-order modal frequency of the drill string, andequation (25) is transformed into 3f0 ¼ 2561:3� 2=l2.The length of the drill string is 85.38m. Certain mul-tiples of the frequency, i.e., 3f0, 6f0, 9f0 . . ., equals thesecond, fourth, sixth, . . . order modal frequency of thedrill string, respectively, which is the resonance fre-quency region. Figure 8 indicates that the amplitudeof the vibration response is the smallest for the lengthsl1, l2, l3 of the drill string when the excitation of thethird-order harmonic frequency component of theimpulse equals the second-order modal frequency ofthe drill string. This result occurs because the amplitudeof the third-order harmonic frequency is smaller thanthat of the fundamental frequency or the second-orderharmonic frequency in the impulse Fourier series.

When the length l4 of the drill string is equal to 60m,as demonstrated in Figure 7, there is no intersectionpoint among the modal frequencies of the drill string

Figure 7. Relations between the modal frequency of the drill string and the impulse frequency multiple of the Fourier series.




Figure 8. a) Time domain and b) frequency domain curves of the impact vibration of the down-the-hole hammer drill string.

Bu et al. 3099



and the low frequencies in the impulse Fourier series;the drill string mainly performs low-order forced vibra-tions, as illustrated in Figure 9, whose amplitude isconsiderably smaller than those of the resonant lengthsl1, l2, l3.

The aforementioned findings demonstrate that whenthe frequency multiple of the impulse componentequals the lower-order modal frequency of the drillstring, the condition under which the drill stringenters the state of resonance is

nf0 ¼i

2l

ffiffiffiffiE

�

sor l ¼

i

2nf0

ffiffiffiffiE

�

sð27Þ

where both the frequency multiple n and the modalorder i are positive integers. This result suggests thatwhen the length of the drill string satisfies equation(27), there should be at least one frequency of theimpulse component equal to the modal frequency ofthe short drill string within 200m; thus, the drillstring will reach the resonance point when the DTHis in operation. In particular, n and i take lowervalue, its resonance regions are adjacent but not con-tinuous with the extension of the drill string.

As observed in Figures 7 and 8, when the DTH drillsgranite, the impulse can cause the drill bit to impact thebottom of the drill string, shifting the drill string intothe higher-order mode of resonant vibration. The peri-odic impact force of the DTH can easily cause the drill

Figure 9. a) Time domain and b) frequency domain curves of the impact vibration of the 60-m drill string.




string to resonate during granite drilling. This result isconsistent with the excited vibration of the DTH in theon-site drilling of granite-like hard rock.

4. Conclusions

The following conclusions can be drawn by solving andanalyzing the axial vibration of drill string couplingwith a DTH in rock drilling:

1. When the DTH impacts granite with a frequency off0 ¼ 20Hz, the fundamental frequency and low-frequency multiple of the cycle impulse Fourierseries overlap the low modal-order frequency of thedrill string. In other words, when the length of thedrill string satisfies l ¼ i

ffiffiffiffiffiffiffiffiffiE=�p

=2nf0 or the frequencymultiple of the impulse component satisfies the con-dition nf0 ¼ i

ffiffiffiffiffiffiffiffiffiE=�p

=2l, at least one frequency of thecycle impulse component will be equal to the modalfrequency of the drill string; the short drill string ofthe DTH may resonate when it drills a hole of a spe-cific length. With the extension of the drill string, theresonance regions are adjacent to one another but notcontinuous. This result is consistent with the phenom-enon that the DTH excites drill string vibrationduring the on-site drilling of granite-like hard rock.

2. The arithmetic solution to the axial vibration of drillstring coupling with a DTH provides a new analysismethod for researching DTH drill string dynamics.The numerical analysis demonstrates that the funda-mental frequency of the cycle impulse Fourier seriesis determined by the impact frequency of the DTHdrilling; the amplitude of the Fourier series is deter-mined by the interaction properties of the DTH andthe rock.

Funding

This research was funded by the National Natural ScienceFoundation of China (grant numbers 51275493 and

50475056).

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Documents

Arithmetic solution for the axial vibration of drill ... · dynamic design, modification, and use of a DTH drilling system. Keywords Axial vibration, drill string mode, DTH, periodic