Research ArticleNonlinear Model and Qualitative Analysis for CoupledAxialTorsional Vibrations of Drill String

Fushen Ren12 Baojin Wang1 Suli Chen1 Zhigang Yao1 and Baojun Bai2

1Department of Mechanical Science and Engineering Northeast Petroleum University Daqing Heilongjiang 163318 China2Department of Petroleum Engineering Missouri University of Science and Technology Rolla MO 65401 USA

Correspondence should be addressed to Baojun Bai baibmstedu

Received 29 September 2015 Revised 29 November 2015 Accepted 30 November 2015

Academic Editor Emiliano Mucchi

Copyright copy 2016 Fushen Ren et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A nonlinear dynamics model and qualitative analysis are presented to study the key effective factors for coupled axialtorsionalvibrations of a drill string which is described as a simplified equivalent flexible shell under axial rotation Here after dimensionlessprocessing the mathematical models are obtained accounting for the coupling of axial and torsional vibrations using the nonlineardynamics qualitative method in which excitation loads and boundary conditions of the drill string are simplified to a rotatingflexible shell The analysis of dynamics responses is performed by means of the Runge-Kutta-Fehlberg method in which the rulesthat govern the changing of the torsional and axial excitation are revealed and suggestions for engineering applications are alsogiven The simulation analysis shows that when the drill string is in a lower-speed rotation zone the torsional excitation is the keyfactor in the coupling vibration and increasing the torsional stress of the drill string more easily leads to the coupling vibrationhowever when the drill string is in a higher-speed rotating zone the axial excitation is a key factor in the coupling vibration andthe axial stress in a particular interval more easily leads to the coupling vibration of the drill string

1 Introduction

Thedrill string consists of several drill pipes drill collars sta-bilizers and connections (crossover sub) subjected to someheavy and complex dynamic loadings caused by differentsources such as bit and drill string interactions with theformations torque exerted by the rotary table or top drivebuckling and misalignment By producing different statesof stress these loads might result in excess vibrations andlead to failure of the drilling tools Moreover rotation of therotary table or top drive on the surface might be transformedinto a turbulent movement in the downhole Three forms ofvibrations that have been identified for the drill string areaxial torsional and lateral vibrations as shown in Figure 1 [1]

The coupling vibration of the drill string can lead to severevibration and this energy boosts the amplitude of the stringvibration increases bending and impacts with the boreholeand leads to the early fatigue of tools and the reduction of bitlife Moreover impacts with the borehole wall tend to forman overgauge hole or produce problems with the directional

control of the well and also increase the surface torque [2]An analytical approach has been the basis for early analyses[3 4] Yigit and Christoforou [5 6] modeled the drill stringbased on the assumed mode method Their models accountfor the coupling between the axial and transverse vibrations[5] and between the torsional and transverse vibrations [6]Khan [7] employs the FDM (finite difference method) tosolve the axial and torsional vibrations of the drill stringneglecting the added mass and damping effects Shyu [8]studied the coupling between the axial and lateral vibrationsand the whirling of the drill string using FDM Christoforouand Yigit [9] extended their previous work to analyze thecoupled axialtorsionalflexural vibrations of drill strings bymeans of a simplified lumped parameter differential systemTrindade et al [10] introduced a nonlinear continuous beammodel to study the influence of geometrical nonlinearity incoupled axialtransversal vibrations of drill strings whichhas shown that the nonlinear model has strong quantita-tive and qualitative discrepancies with respect to a linearmodel

Hindawi Publishing CorporationShock and VibrationVolume 2016 Article ID 1646814 17 pageshttpdxdoiorg10115520161646814

2 Shock and Vibration

Lateral vibration Torsional vibration Axial vibration

Stabilizer Stabilizer

Figure 1 Three forms of drill string vibration

In accord with the geometric nonlinear characteristics ofdrill strings and by adopting the combination of nonlineardynamics and FEM (finite element method) Sampaio et al[11] established the mathematical coupling model of axialand torsional vibrations by comparing it with a linear modelfinding that the nonlinear model more accurately reflectedthe features of the vibration of the drill string Hakimi andMoradi [12] introduced a differential quadrature methodto analyze the vibration of the drill string Ren et al [13]and Ren and Yao [14] taking into account the flexible andgeometric large deformation of the drill string establishedthe mathematical model of bending nonlinear vibration forthe horizontal well In addition taking into account otherfactors of drill-string vibration scholars have carried outsome research such as on the (a) displacement and forceof the drill bit [15] (b) critical speed of the drill string[16] (c) nonlinear collision and friction contact betweenthe drill string and the borehole wall [17 18] (d) theoryand experiments about coupled solids-liquids [19] and (e)the weight on bit (WOB) and rotation speed of the drillstring [20ndash22] Goncalves and Del Prado [23] analyzed thenonlinear vibrations and dynamic instability of axially loadedcircular cylindrical shells under both static and harmonicforces based on Donnell shallow shell equations The mostdangerous region in parameter space is obtained and thetriggering mechanisms associated with the stability bound-aries are identified [23] Based on the Sanders-Koiter theoryStrozzi and Pellicano [24] analyzed the nonlinear vibrationsof functionally graded circular cylindrical shells and foundthat when these shells that were having an actual hardeningresponse were simulated with an insufficient expansion theirbehavior could appear spuriously softening

Based on the specific problem of the coupled vibration ofthe drill string scholars usually analyze the drill string to findthe relationship between the drilling process parameters andthe modal shapes and then change the drilling parametersto avoid the resonance of the drill string which is a verypractical control method however the key factor is to dis-cover the relationship between the key dynamics parametersand the coupled vibrations of the drill string Consideringthe complexity of the practical dynamics of the drill stringsystem and the comprehensiveness of the quantitative analy-sis directly this study employs the dimensionless method toinvestigate themechanismof the coupled vibration of the drillstring qualitatively This will reveal which key factors affectthe coupled vibration and how they function which can be abasis for the quantitative analysis of the coupling vibration ofthe drill string

u2(120579)

120590xx0

1205901205791205790

L

x

u1

z

h

Figure 2 The mechanical model of the rotation flexible shell

2 Nonlinear Dynamics Motion Equation

During the process of drilling the transverse vibration ofthe drill string which is intense at the bit in the bottom ofthe well and attenuates quickly along the drill string mainlycontributes to the bottom hole assembly (BHA) however theaxial and torsional vibration affects the entire drill string

In addition several stabilizers support the lower portionof the drill string and absorb most transverse vibrationenergy In other words the transversemotion of the stabilizedsection mainly contributes to the transverse dynamics at thebit The drill collars are assumed to be rigid for torsionalvibrations that is to say the torsional deformations areassumed to take place only in the drill pipe which can bejustified since the drill collars are much stiffer than the drillpipe in torsionTherefore the transverse vibration of the BHAis decoupled from the upper segments of the drill string andthe entire drill string is assumed to be fixed at the top and freeat the bit for the axial and torsional motion [9]

Based on the above analyses ignoring the transversevibration of the drill string this work only analyzes themechanism of the coupled axial and torsional vibration ofthe upper portion of the BHA Here assuming the centerlineof the shell before deformation as the axis the drill stringis simplified to a flexible rotation shell instead of simply asupported beam which will ignore most torsional vibrationThe dynamics model is shown in Figure 2 where 119906

1and

1199062are the displacement of axial 119909 and circumferential 119906

respectively ℎ is the thickness of the drill string 119871 is thelength of the drill string 120590

1199091199090is the initial axial stress and

1205901205791205790

is the initial torsional stressTo facilitate the calculation of the drill string strain select

the arbitrary point displacement of the drill string as follows[25]

1199061= 119906 (1a)

1199062= V +

119911

119877

V (1b)

where 119906 and V are arbitrary axial and circumferential dis-placements of the drill string on the middle surface 119911 =

0 respectively and 119877 is the middle radius of the drillstring

Since the drill string cannot be thin and the tangentialdisplacements 119906 and V of the drill string are not smallthe Flugge-Lurrsquoe-Byrne nonlinear shell theory is selected inwhich two hypotheses are removed fromDonnellrsquos nonlinearshell theory [26] in order to obtain more accurate nonlinear

Shock and Vibration 3

shell theories One is the thinness assumption and the otheris that the tangential displacements are infinitely small whichleads to neglecting the nonlinear terms that depend on 119906 andV

Based on the nonlinear shell theory [25] the relationshipof the stressdisplacement at any point on the drill string canbe expressed by

120576119909119909

=

120597119906

120597119909

+

1

2

(

120597119906

120597119909

)

2

+

1

2

(

120597V120597119909

)

2

+ 119911

1

119877

(

120597V120597119909

)

2

(2a)

120576120579120579

=

1

119877

120597V120597120579

+

1

21198772(

120597119906

120597120579

)

2

+

1

21198772(

120597V120597120579

)

2

+

1

21198772V2

minus

1

1198773119911 (

120597119906

120597120579

)

2

(2b)

120576119909120579

=

120597V120597119909

+

120597119906

119877120597119909

+

1

21198772(

120597119906

120597120579

)

2

+

1

119877

120597119906

120597119909

120597119906

120597120579

+

1

119877

120597V120597119909

120597V120597120579

+ 119911120581119909120579

(2c)

where

120581119909120579

=

1

119877

120597V120597119909

minus

1

1198772

120597119906

120597120579

minus

1

1198772

120597119906

120597119909

120597119906

120597120579

minus

1

1198772

120597V120597119909

120597V120597120579

(3)

where 120576119909119909

and 120576120579120579

are linear strain 120576119909120579

is shear strain andthey are the dimensionless physical quantities Here (2a)(2b) and (2c) make up most of the nonlinear term whichavoids the nonlinear coupled axial and torsional terms in thefollowing analyses due to oversimplification

Analogous to the case of plane stress the stress inthickness of the drill string is neglected [27] Taking intoaccount the single material of the drill string neglecting thegeometric imperfections and subject to the initial axial andtorsional stress the relationship between stress and strain canbe written as

120590119909119909

= 1198641(120576119909119909

+ 120581120576120579120579

) + 1205901199091199090

(4a)

120590120579120579

= 1198641(120576120579120579

+ 120581120576119909119909

) + 1205901205791205790

(4b)

120590119909120579

= 1198642120576119909120579

(4c)

where 1198641

= 119864(1 minus 1205812) 1198642

= 1198642(1 + 120581) 120581 is Poissonrsquos ratioand 119864 is the elasticity modulus

Based on (2a) (2b) and (2c) and (4a) (4b) and (4c)according to Hamiltonrsquos principle the dynamic equation ofthe partial differential can be obtained as follows

120588 = (

1

2

119877ℎ1205901199091199090

1198641+ 119877ℎ120581119864

1120576120579120579

+ 119877ℎ1205921198641120576119909119909

)

1205972119906

1205971199092

+ (

ℎ1205901205791205790

2119877

+

ℎ1198641120576120579120579

119877

+

ℎ1198641120581120576119909119909

119877

)(

1205972119906

1205971205792)

minus

1198621

2120587119871

+

1

2

119877ℎ

1205971205901199091199090

120597119909

+ 119877ℎ1198641(

120597120576119909119909

120597119909

+ 120592

120597120576120579120579

120597119909

) +

ℎ

2119877

1205971205901199091199090

1205971199092

120597119906

120597119909

+

ℎ1198641

119877

120597120576120579120579

120597120579

120597119906

120597120579

+ 119877ℎ1205811198641

120597120576120579120579

120597119909

120597119906

120597119909

+

ℎ1205811198641

119877

120597120576120579120579

120597120579

120597119906

120597120579

+ 119877ℎ1198641

120597120576120579120579

120597119909

120597119906

120597119909

(5a)

120588V = (

1

2

119877ℎ1205901199091199090

+ 119877ℎ1205921198641120576120579120579

+ 119877ℎ1198641120576119909119909

)

1205972V

1205971199092

+ (

ℎ1205901205791205790

2119877

+

ℎ1198641120576120579120579

119877

+

ℎ1198641120592120576119909119909

119877

)(

1205972V

1205971205792)

minus

1

2120587119871

1198622V +

ℎ

2

1205971205901205791205790

120597120579

+ ℎ1198641(120592

120597120576119909119909

120597120579

120597120576120579120579

120597120579

)

+

ℎ

2119877

1205971205901205791205790

120597120579

120597V120597120579

+

ℎ1205921198641

119877

120597120576120579120579

120597120579

120597V120597120579

+ 119877ℎ1205921198641

120597120576120579120579

120597119909

120597V120597119909

+

ℎ1198641

119877

120597120576120579120579

120597120579

120597V120597120579

+ 119877ℎ1198641

120597120576119909119909

120597119909

120597V120597119909

minus

ℎ1205921198641

119877

120576120579120579

+

119877ℎ

2

1205971205901199091199090

120597119909

120597V120597119909

minus

ℎ

2

1205921205901205791205790

119877

(5b)

where 120588 is the density of the drill string 119868 = int

12

minus12120588119889119911 is the

rotary inertia and 1198621and 119862

2are the damping coefficients of

the first modal and the second modal respectivelyThe modals of axial and torsional vibration are given as

[28]

119906 = 1199081(119905) sin(

120587119909

2119871

) cos 120579

V = 1199082(119905) cos(2120587119909

119871

) sin 120579

(6)

In the modals of the axial vibration and torsional vibrationthe number of longitudinal half-waves was equal to one andtwo respectively In (6) when 119909 = 0 then 119906 = 0 when119909 = 119871 then 119906 became the extreme value when 119909 = 0 or119909 = 119871 then V got the extreme value all of which meet theactual displacement boundary conditions of the drill stringSubstituting (6) into (2a) (2b) and (2c) and then substitutingthe result into (4a) (4b) and (4c) produce120590

119909119909119909=0= 1205901199091199090

and

4 Shock and Vibration

120590120579120579119909=0

= 1205901205791205790

which account for the boundary conditions ofthe force

Using the Galerkin method to disperse (5a) and (5b)substituting (6) into (5a) and (5b) both sides of (5a) and (5b)are multiplied by the mode of the right side of (6) integratedin the circumferential and thickness direction Making useof modal orthogonality the motion equation of the middlesurface displacement is obtained as follows

1= 119886111986211+

119877ℎ1205901199091199090

2119871120588

minus

ℎ1205901205791205790

2120588119877

1199081minus 11988621199081minus 11988631199082

+ 119886411990811199082

2minus 11988651199083

1

(7a)

2= 119886111986222minus 11988661199081+

1

120588

ℎ1205901205791205790

119877

1199082+ 119886711990821199082

1minus 11988681199083

2 (7b)

where 1198861 1198862 1198863 1198864 1198865 1198866 1198867 and 119886

8are the constant

coefficients as listed in the Appendix From (7a) and (7b)we know that the vibration is a nonlinear coupled equationof axial and torsional vibration and that unilaterally takinginto account the single direction of the vibration is obviouslymuch too simplified to describe the actual vibrationThe axialstress as external excitation directly affects the dynamicsresponse The torsional stress 120590

1205791205790 as parameter excitation

affects the vibration by changing the axial stiffness therebyaffecting the natural frequency of the drill string Assumingthe axial and torsional excitation frequencies are integer timesof the rotation frequency of the drill string the equationitself will have two linear natural frequencies Due to theexistence of the coupling and nonlinear terms there isa great difference between the actual vibration frequencyand the linear natural frequency In addition subharmonicresonance superharmonic resonance bifurcation and chaosphenomena are possible in the vibration system with theparameters changing and the energy translating between themodals

3 Nonlinear Dynamics Average Equation

To facilitate the analysis replace the coefficients in (7a) and(7b) by 119886

119894 119887119895and use the dimensionless transformation as

follows (in the following dimensionless equation the asteriskwas removed for the convenience of expression)

119909lowast

1=

1199091

119871

119909lowast

2=

1199092

119871

119905lowast

=

119877

1198712radic

119864

120588

119905

120590lowast

1199091199090=

1205901199091199090

119864

120590lowast

1205791205790=

1205901205791205790

119864

(8)

The multiscale transformation is employed as follows

119886119894997888rarr 120576119886

119894

119887119895997888rarr 120576119887

119895

119894 = 1 119895 = 2

(9)

where 120576 is a small parameter used as a perturbation parameterfor determining the approximate solution for the motion ofthe drill string [22]

Substituting (9) into (7a) and (7b) one can obtain amotion equation that includes the small parameter 120576 Hereassume the form of the equation as follows

119909119894(119905 120576) = 119910

1198940(1198790 1198791) + 120576119910

119894(1198790 1198791) + sdot sdot sdot (119894 = 1 2) (10)

where 1198790= 119905 and 119879

1= 120576119905

Here the differential operators are employed as in thefollowing form

119889

119889119905

=

120597

1205971198790

1205971198790

120597119905

+

120597

1205971198791

1205971198791

120597119905

+ sdot sdot sdot = 1198630+ 1205761198631+ sdot sdot sdot (11a)

1198892

1198891199052

= (1198630+ 1205761198631+ sdot sdot sdot)2

= 1198632

0+ 2120576119863

01198631+ sdot sdot sdot (11b)

where 1198630= 120597120597119879

0and 119863

1= 120597120597119879

1

Taking into account the principal parameter resonanceand 2 1 internal resonance the resonance relationship can beexpressed as follows

1205962

1= 41205962+ 1205761205901

1205962

2= 1205962+ 1205761205902

(12)

where 1205961and 120596

2are the axial and torsional linear natural

frequencies respectivelyTaking into account the vibration response caused by

the axial and torsional excitation in the case of the aboveresonance the dimensionless axial and torsional excitationin the frequency domain can be expanded into the followingform including the two resonance frequencies [29]

1205901199091199090

= 1199021119890119894119905

+ 1199021119890minus119894119905

+ 11990221198902119894119905

+ 1199022119890minus119894119905

(13a)

1205901205791205790

= 1199023119890119894119905

+ 1199023119890minus119894119905

+ 11990241198902119894119905

+ 1199024119890minus119894119905

(13b)

where 1199021and 119902

2are the amplitude of the low frequency and

high frequency of the initial axial stress respectively and 1199023

and 1199024are the amplitude of the low and high frequencies of

the initial torsional stress respectivelySubstituting (8)ndash(12) into (7a) and (7b) and comparing

the same step coefficient of the small parameter 120576 on bothsides of the equations one obtains a differential equation inthe following form

For 1205760

1198632

011991010

+ 41199101= 0 (14a)

Shock and Vibration 5

For 1205761

1198632

011991020

+ 1199102= 0 (14b)

The plural form solutions of (14a) and (14b) can be expressedas

11991010

= 1198601(1198791) 1198902119894119905

+ 1198601(1198791) 119890minus2119894119905

(15a)

11991020

= 1198602(1198791) 119890119894119905

+ 1198602(1198792) 119890minus119894119905

(15b)

where 1198601and 119860

2are the conjugation of 119860

1and 119860

2 respec-

tively The plural solutions of 1198601and 119860

2are formed as

follows

1198601=

1

2

1199091+

1

2

1198941199092

1198602=

1

2

1199093+

1

2

1198941199094

(16)

where 1199091 1199092 1199093 and 119909

4are the projections of the vibration

vector on the complex plane which is equal to the vibrationvector described as the amplitude and the phase

Substituting (15a) and (15b) into (13a) and (13b) oneobtains themotion equationwith the small parameter 120576 thenmaking the long term equal to zero one obtains the averagingequation in the rectangular coordinate as follows

1= minus

1

2

11988691199091minus

1

8

11988610

1199092(1199092

3+ 1199092

4)

+

3

16

11988611

1199092(1199092

1+ 1199092

2)

(17a)

2= minus

1

2

11988691199092minus

1

8

11988610

1199091(1199092

3+ 1199092

4)

minus

3

16

11988611

1199091(1199092

1+ 1199092

2) minus

1

2

1199022

(17b)

3= minus

1

2

11988612

1199093+

1

2

11988613

1199094+

1

4

11988614

1199094(1199092

1+ 1199092

2)

+

3

8

11988615

1199094(1199092

3+ 1199092

4)

(17c)

4= minus

1

2

11988612

1199094minus

1

2

11988613

1199093minus

1

4

11988614

1199093(1199092

1+ 1199092

2)

minus

3

8

11988615

1199093(1199092

3+ 1199092

4) minus 1199023

(17d)

where 1198869 11988610 11988611 11988612 11988613 11988614 and 119886

15are the constant

coefficients listed in the AppendixThe change rules for the amplitude and phase angles in

the motion equation are found in (17a) (17b) (17c) and (17d)using the form of the differential equation We find that 119902

1

and 1199024have disappeared in (17a) (17b) (17c) and (17d) and

the parameters that can affect the dynamics response are only1199022and 1199023 We also know that 119902

2is the high-frequency item of

the initial axial excitation and 1199023is the low-frequency item of

the initial torsional excitation corresponding to the dynamicresponse of the drill string in low- and high-rotation speedareas respectively So we try to reveal relationships betweenthe two parameters with the coupling vibration of the drillstring by numerical simulation

4 Numerical Simulation andQualitative Analysis

In the present section the numerical simulation resultsobtained from using the proposed model are discussed forthe upper portion of the BHA The Runge-Kutta-Fehlbergmethod with adaptive steps is employed to perform thesimulations aiming at obtaining the dynamic response of thecoupled axial and torsional vibration in the case of resonanceThe geometric properties of the upper segment are the length119871 = 1050m themiddle diameter119877 = 01057m the thicknessℎ = 00171m Poissonrsquos ratio 120581 = 026 the elastic modulus119864 = 210GPa the density 120588 = 7850 kgm3 and the dampingcoefficients 119862

1and 119862

2 calculated from the considerations of

Spanos et al [30] The representative simulation results areshown along with the changes in 119902

2and 1199023

41 Torsional Excitation The Runge-Kutta-Fehlberg methodis employed to analyze the average of (17a) (17b) (17c) and(17d) and bifurcations of the system under exciting forces areobtained as shown in Figure 3

From the bifurcation along with the change in tor-sional excitation 119902

3 the response of the coupled axial

and torsional vibrations progresses through a cycle fromperiodic motion to doubling periodic motion to period-multiplying motion to quasiperiodic motion while exhibit-ing this unique phenomenon of nonlinear dynamics bifur-cation When the coupled vibration response is quasiperi-odic motion the amplitude of the drill string is obviouslyhigher than the period motion With increasing amplitudeof the excitation the responses change from quasiperiodicmotion to periodic motion the amplitude of the vibrationdoes not increase Instead it decreases to a certain extentwhich is different from the results of the linear analysismethod

By only increasing the torsional excitation until 1199023

=

724 (see Figure 3) and leaving other initial conditions andparameters the same this creates the periodic responsesshown in Figure 4

When 1199023

= 726 the amplitude of the system increasescorrespondingly accordinglyThe period-doubling responsesof system are shown in Figure 5

When 1199023

= 75 the phase diagram of the coupled vibra-tion changes the jumping phenomenon is more obviouspresent period-multiplying responses are shown in Figure 6

By increasing the value of 1199023so that 119902

3= 768 the qual-

itative nature of the system response changes presentingcorresponding quasiperiodic motions as shown in Figure 7This shows that new similar trajectories are derived from theoriginal trajectories of the phase diagram and they representmultifrequency vibration characteristics in the oscillogramand vibration frequencies that are close to the state ofcontinuous change Then the system vibration is in a statebetween the period vibration and the chaotic vibration qual-itatively the frequency bandwidth of the coupled vibrationalso increases the chaotic motions of the drill string aremorelikely to occur which leads to violent vibration of the drillstring

6 Shock and Vibration

x1

7 8 9 1110q3

12

14

16

18

(a) Bifurcation diagram of the axial modal

x3

10 118 97

q3

minus05

00

05

10

(b) Bifurcation diagram of the torsional modal

Figure 3 Bifurcations of the system under torsional excitation

It is well-known that the torsional vibration of the drillstring is mainly generated from the rotational speed changeof the bit when breaking rock intermittently The excitationfrequency is related to the rotation speed stiffness of thedrill string and characteristics of the rock The higher therock hardness the bigger the torsional stress and the greaterthe torsional amplitude We know from (17a) (17b) (17c)and (17d) that torsional stress strongly affects the couplingvibration when in low-frequency excitation but when inhigh-frequency excitation the torsional stress does not effec-tively contribute to the coupling vibration So in the practicaldrilling process when the drill string is in a high-rotationspeed zone the torsional stress will not lead to couplingvibration however when the drill string is in a low-rotationspeed zone more attention should be paid to the torsionalstress and appropriate drilling process parameters should beadopted to reduce the torsional stress especially when stick-slip vibration occurs This is necessary because at this timethe drill string is in low torsional excitation and large torsionalstress where it is more prone to produce a coupling vibrationof the drill string as suggested by Christoforou and Yigit [9]

Based on the above simulation analysis we also foundthat torsional excitation will affect both torsional and axialvibration simultaneously Torsional and axial vibration aresimilar qualitatively and in the nature of the vibration andare synchronous in form proving that energy can transferbetween the torsional and the axial vibration modal

42 Axial Excitation The above analysis accounts for theimpact of torsional excitation on the coupling of the axial andtorsional vibrations Here the impact of axial excitation willbe investigated

Thebifurcation diagram (Figure 8) shows the preliminaryresonance tendency The system undergoes a loop fromchaos to periodic motion then to period-three motionthen to period-multiplying and back to chaos showing

the nonlinear dynamic phenomenon of the period-doublingbifurcation and chaos

Increasing axial excitation until 1199022

= 05025 and keep-ing the remaining other initial conditions and parametersunchanged the system prevents period-doubling responsesas shown in Figure 9

When 1199022

= 0635 the system presents period-threeresponses as shown in Figure 10

When 1199022= 07725 the phase diagram changes drastically

and presents a jumping phenomenon then the systemresponds chaotically as shown in Figure 11

When 1199022

= 079 the qualitative nature of the couplingvibration of the drill string changes the present period-doubled responses are shown in Figure 12 Here the originalphase trajectory of the chaotic motion (Figure 11) contractsfor two trajectories and presents vibration characteristicsof frequency-doubling in the phase diagram The systemvibration turns from chaotic vibration into doubling periodicmotion and the stability of the system increases

As can be seen from the above qualitative analysis of theaxial excitation of the coupling vibration we find the samephenomenon as in the torsional excitation analysis for thecoupling vibration namely that axial excitation affects bothtorsional and axial vibration simultaneously and is similarqualitatively and also synchronous in form

Along with the increase in the axial excitation the systemcoupling vibration turns from period-doubling to period-three to chaotic and finally to period-doubling So the axialexcitation parameter leads to a coupling vibration in somespecial zone In the process of practical drilling the axialjump mainly contributes to the axial excitation The higherthe rotation speed the higher the excitation frequency Thelarger the WOB becomes the larger both the axial amplitudeand the axial stress will be We know from (17a) (17b) (17c)and (17d) that the axial excitation contributes to the couplingvibration when the drill string is in a high-rotation speedzone Therefore in the process of automation drilling with

Shock and Vibration 7

x2

minus3

minus2

minus1

1

0

2

3

minus2minus4 20x1

(a) Phase diagram of the axial modal

1020 10301010 1040 10501000t

minus4

minus3

minus2

minus1

0

1

2

x1

(b) Oscillogram of the axial modal

00 05 10minus10 minus05minus15

x3

minus15

minus10

minus05

00

05

10

x4

(c) Phase diagram of the torsional modal

1010 1020 1030 1040 10501000t

minus15

minus10

minus05

00

05

10x3

(d) Oscillogram of the torsional modal

5

0 02

0

1

minus1

minus2

x3

minus5

x2

minus4

minus2 x1

(e) Three-dimensional phase diagram

minus15

minus10

minus05

00

05

10

15

x2

minus10 minus05 00 05 10 15minus15

x1

(f) Poincare section

Figure 4 Period responses of the system (1199023= 724)

8 Shock and Vibration

x2

minus3

minus2

minus1

1

0

2

3

0minus2 2minus4

x1

(a) Phase diagram of the axial modal

1010 1020 1030 1040 10501000t

minus4

minus3

minus2

minus1

0

1

2

x1

(b) Oscillogram of the axial modal

minus10 minus05 00 05 10minus15

x3

minus15

minus10

minus05

00

05

10

x4

(c) Phase diagram of the torsional modal

1010 1020 1030 1040 10501000t

minus15

minus10

minus05

00

05

10x3

(d) Oscillogram of the torsional modal

5

00

2

0

1

minus1

minus2

x3

minus5

x2

minus4

minus2 x 1

(e) Three-dimensional phase diagram

0500 10 15minus10 minus05minus15

x1

minus15

minus10

minus05

00

05

10

15

x2

(f) Poincare section

Figure 5 Period responses of the system (1199023= 726)

Shock and Vibration 9

x2

minus2 0 2minus4

x1

minus3

minus2

minus1

1

0

2

3

(a) Phase diagram of the axial modal

minus4

minus3

minus2

minus1

0

1

2

x1

1010 1020 1030 1040 10501000t

(b) Oscillogram of the axial modal

minus10 minus05 00 05 10minus15

x3

minus15

minus10

minus05

00

05

10

x4

(c) Phase diagram of the torsional modal

minus15

minus10

minus05

00

05

10x3

1010 1020 1030 1040 10501000t

(d) Oscillogram of the torsional modal

5

0 02

0

1

minus1

minus2

x3

minus5

x2

minus4

minus2 x 1

(e) Three-dimensional phase diagram

minus15

minus10

minus05

00

05

10

15

x2

0500 10 15minus10 minus05minus15

x1

(f) Poincare section

Figure 6 Period responses of the system (1199023= 75)

10 Shock and Vibration

x2

minus2 0 2minus4

x1

minus3

minus2

minus1

1

0

2

3

(a) Phase diagram of the axial modal

1020 1040 1060 10801000t

minus4

minus3

minus2

minus1

0

1

2

x1

(b) Oscillogram of the axial modal

minus10 minus05 00 05 10minus15

x3

minus15

minus10

minus05

00

05

10

x4

(c) Phase diagram of the torsional modal

1020 1040 1060 10801000t

minus15

minus10

minus05

00

05

10x3

(d) Oscillogram of the torsional modal

5

00

2

0

1

minus1

minus2

x3

minus5

x2

minus4

minus2 x 1

(e) Three-dimensional phase diagram

minus10 minus05 00 05 10 15minus15

x1

minus15

minus10

minus05

00

05

10

15

x2

(f) Poincare section

Figure 7 The almost-periodic responses of the system (1199023= 768)

Shock and Vibration 11

06 07 08 09 1005q2

x2

minus10

minus05

00

(a) Bifurcation diagram of the axial modal

x3

06 07 0805 1009q2

10

15

20

(b) Bifurcation diagram of the torsional modal

Figure 8 Bifurcation of the resonance frequency-multiplier

a constant WOB in the case of a lower-rotation speed andlower rock hardness high drill efficiency can be attained byproperly increasing the WOB In the case of high-rotationspeed the coupling vibration is sensitive to the axial stressso the proper WOB should be adopted to avoid the couplingvibration of the drill string based on geological parameters

5 Conclusions

In this paper wemodeled and analyzed drill-string vibrationsby focusing on the coupled axialtorsional vibrations bymeans of nonlinear dynamics and qualitative analysis aimingat revealing the key effective factors influencing the coupledvibrations Here the drill string was described as a simplifiedequivalent flexible shell under axial rotation in which theexcitation loads and boundary conditions of the drill stringwere simplified After dimensionless processing we built thedynamics motion equation and the average equation

We found that the low-frequency amplitude expressionof the torsional excitation and the high-frequency amplitudeexpression of the axial excitation in the average equation arethe factors that determine the coupling vibration Based onthis procedure numerical simulations were carried out withadaptive steps using the Runge-Kutta-Fehlberg method todiscover the response of the system vibrations Further wefound that a change in the torsional or the axial excitationaffects both the torsional and the axial vibration simultane-ously and that it is similar qualitatively and synchronous inform

The results of the simulation analysis show that when thedrill string is in a lower-speed rotation zone the torsionalexcitation mainly contributes to the coupling vibrationincreasing the torsional stress of the drill string that moreeasily leads to the coupling vibrationWhen in a higher-speedrotating zone the axial excitation mainly contributes to thecoupling vibration so in a particular interval it is more likelyto cause the coupling vibration of the drill string

Appendix

Consider

1198861= minus

1

2119871120588120587

1198862=

1

120588

(

1

12

ℎ31198642

1198773

minus

ℎ1198642

119877

minus

1

4

1205872119877ℎ1198641

1198712

)

1198863=

1

120588

(

8

45

ℎ31198642

1198711198772

minus

32

15

ℎ1198642

119871

minus

32

15

ℎ1205811198641

119871

)

1198864=

1

120588

(

1

4

1205872ℎ31205811198641

11987131198773

minus

13

16

1205872ℎ1205811198641

1198712119877

minus

1

4

ℎ1198642

1198773

minus

1

16

1205874119877ℎ1198641

1198714

)

1198865=

1

120588

(

1

384

1205872ℎ31198642

11987121198773

minus

9

512

1205874119877ℎ1198641

1198714

minus

3

32

ℎ31198642

1198775

minus

9

32

ℎ1198642

1198773

minus

1

32

1205872ℎ1198641

1198712119877

minus

1

64

1205872ℎ1205811198641

1198712119877

)

1198866=

13ℎ1205811198641

15120588119871

1198867=

1

120588

(

1

4

1205872ℎ31205811198641

11987121198773

+

13

16

1205872ℎ1205811198641

1198712119877

minus

1

4

ℎ1198641

1198773

+

1

16

1205874119877ℎ1198641

1198714

+

3

4

ℎ1198641

1198713

+

1

24

1205872ℎ31198642

11987121198773

+

9

2

1205874119877ℎ1198641

1198714

)

1198868=

1

120588

(

1

2

1205872ℎ1198642

1198712119877

+

3

2

1205874ℎ31198641

1198714119877

+

1205872ℎ1205811198641

1198712119877

+

3

4

ℎ1198641

1198713

12 Shock and Vibrationx2

minus10

minus05

00

05

10

minus05 00 05 10minus10

x1

(a) Phase diagram of the axial modal

minus10

minus05

00

05

10

x1

1020 1040 1060 10801000t

(b) Oscillogram of the axial modal

minus2

minus1

0

1

2

x4

15 20 25 3010x3

(c) Phase diagram of the torsional modal

10

15

20

25

30x3

1020 1040 1060 10801000t

(d) Oscillogram of the torsional modal

1

0 0

11

2

3

minus1minus1

x3

x2 x 1

(e) Three-dimensional phase diagram

minus15

minus10

minus05

00

05

10

15

x2

0500 10 15minus10 minus05minus15

x1

(f) Poincare section

Figure 9 Period responses of the system (1199022= 05025)

Shock and Vibration 13x2

minus05 00 05 10minus10

x1

minus10

minus05

00

05

10

(a) Phase diagram of the axial modal

1020 1040 1060 10801000t

minus10

minus05

00

05

10

x1

(b) Oscillogram of the axial modal

minus20

minus15

minus10

minus05

00

05

10

15

20

x4

15 20 25 3010x3

(c) Phase diagram of the torsional modal

10

15

20

25

30x3

1020 1040 1060 10801000t

(d) Oscillogram of the torsional modal

1

0 0

11

2

3

minus1 minus1

x3

x2 x 1

(e) Three-dimensional phase diagram

minus10 minus05minus15 05 10 1500x1

minus15

minus10

minus05

00

05

10

15

x2

(f) Poincare section

Figure 10 Period responses of the system (1199022= 0635)

14 Shock and Vibration

minus2

minus1

0

1

2

x2

10 2 3minus1minus2

x1

(a) Phase diagram of the axial modal

1050 1100 1150 12001000t

minus2

minus1

0

1

2

3

x1

(b) Oscillogram of the axial modal

2 4minus2 0minus4

x3

minus4

minus2

0

2

4

x4

(c) Phase diagram of the torsional modal

minus4

minus2

0

2

4x3

1050 1100 1150 12001000t

(d) Oscillogram of the torsional modal

2

00

2

4

0

5

minus2

x3

minus5

x2

minus2

x 1

(e) Three-dimensional phase diagram

minus15

minus10

minus05

00

05

10

15

x2

0500 10 15minus10 minus05minus15

x1

(f) Poincare section

Figure 11 Period responses of the system (1199022= 07725)

Shock and Vibration 15x2

minus05 00 05 10minus10

x1

minus10

minus05

00

05

10

(a) Phase diagram of the axial modal

1020 1040 1060 10801000t

minus10

minus05

00

05

10

x1

(b) Oscillogram of the axial modal

15 20 25 3010x3

minus2

minus1

0

1

2

x4

(c) Phase diagram of the torsional modal

10

15

20

25

30x3

1020 1040 1060 10801000t

(d) Oscillogram of the torsional modal

1

0 0

11

2

3

minus1minus1

x3

x2

x 1

(e) Three-dimensional phase diagram

minus10 minus05minus15 05 10 1500x1

minus15

minus10

minus05

00

05

10

15

x2

(f) Poincare section

Figure 12 Period-doubled responses of the system (1199022= 079)

16 Shock and Vibration

+

1

24

1205872ℎ31198642

11987121198773

+

9

2

1205874119877ℎ1198641

1198714

)

1198869=

1198621

2120587119877radic120588119864

11988610

=

1205872ℎ31198641120581

411987121205881198772

minus

ℎ1198641

41205881198772

minus

1205874ℎ1198771198641

161198714120588

minus

1205872ℎ1198641120581

161198712120588119877

11988611

= minus

91205874ℎ1198771198641

5121198714120588

minus

3ℎ31198641

321205881198775

minus

1205872ℎ1198642

321198712120588119877

minus

1205872ℎ31198642

38411987121205881198773

minus

9ℎ1198641

321205881198773

minus

1205872ℎ1198641120581

641198712120588119877

11988612

=

1198622

2120587119877radic120588119864

11988613

= minus

41205872119877ℎ1198642

1198712120588

minus

ℎ1198641

119877120588

minus

1205872ℎ31198642

31198712120588119877

11988614

=

1205872ℎ31198641120581

411987121205881198773

minus

ℎ1198641

41205881198773

minus

1205874ℎ1198771198641

161198714120588

minus

131205872ℎ1198641120581

161198712120588119877

11988615

= minus

1205872ℎ31198642

2411987121205881198773

minus

1205872ℎ1198642

21198712120588119877

minus

91205874ℎ1198641120581

21198714120588

minus

3ℎ1198641

41205881198773

minus

1205872ℎ1198641120581

1198712120588119877

minus

31205874ℎ31198641

21198714120588119877

(A1)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to acknowledge financial support byNatural Science Foundation of China Project 11372071 andPostdoctoral Fund of China Project 2013M541339

References

[1] P D Spanos A M Chevallier N Politis and B Payne ldquoOilwell drilling a vibration perspectiverdquo The Shock and VibrationDigest vol 35 pp 81ndash99 2013

[2] W D Aldred andM C Sheppard ldquoDrillstring vibrations a newgeneration mechanism and control strategiesrdquo in Proceedingsof the SPE Annual Technical Conference and Exhibition SPE-24582-MS pp 353ndash363 Washington DC USA October 1992

[3] T V Aarrestand H A Tonnesen and A Kyllingstad ldquoDrill-string vibrations theory and experiments on full-scale researchdrilling rigrdquo inProceedings of the IADCSPEDrilling ConferenceSPE Paper No 14760 pp 311ndash321 Dallas Tex USA 1986

[4] J D Jansen ldquoNon-linear rotor dynamics as applied to oilwelldrillstring vibrationsrdquo Journal of Sound and Vibration vol 147no 1 pp 115ndash135 1991

[5] A S Yigit and A P Christoforou ldquoCoupled axial and transversevibrations of oilwell drillstringsrdquo Journal of Sound and Vibra-tion vol 195 no 4 pp 617ndash627 1996

[6] A S Yigit and A P Christoforou ldquoCoupled torsional and bend-ing vibration of drill-strings subject to impact with frictionrdquoJournal of Sound and Vibration vol 215 no 1 pp 167ndash181 1998

[7] K Z Khan Longitudinal and torsional vibration of drill-strings [MS thesis] Massachusetts Institute of TechnologyCambridge Mass USA 1986

[8] R J Shyu Bending of rotating drill-strings [PhD thesis] Mas-sachusetts Institute of Technology 1989

[9] A P Christoforou and A S Yigit ldquoFully coupled vibrations ofactively controlled drillstringsrdquo Journal of Sound and Vibrationvol 267 no 5 pp 1029ndash1045 2003

[10] M A Trindade C Wolter and R Sampaio ldquoKarhunen-Loevedecomposition of coupled axialbending vibrations of beamssubject to impactsrdquo Journal of Sound and Vibration vol 279 no3ndash5 pp 1015ndash1036 2005

[11] R Sampaio M T Piovan and G Venero Lozano ldquoCoupledaxialtorsional vibrations of drill-strings bymeans of non-linearmodelrdquo Mechanics Research Communications vol 34 no 5-6pp 497ndash502 2007

[12] H Hakimi and S Moradi ldquoDrillstring vibration analysis usingdifferential quadrature methodrdquo Journal of Petroleum Scienceand Engineering vol 70 no 3-4 pp 235ndash242 2010

[13] F S Ren S Chen and Z G Yao ldquoDynamics analysis andvibration suppression of a flexible rotation beamrdquo AppliedMechanics and Materials vol 214 pp 165ndash172 2012

[14] F-S Ren and Z-G Yao ldquoStudy on nonlinear dynamics andbifurcations in rotating compressive-drill stringrdquo EngineeringMechanics vol 30 no 10 pp 251ndash256 2013

[15] Z-F Li Y-G Zhang X-T Hou W-D Liu and G-Q XuldquoAnalysis of longitudinal and torsion vibration of drill-stringsrdquoEngineering Mechanics vol 21 no 6 pp 203ndash210 2004 (Chi-nese)

[16] G J Sheu and S M Yang ldquoDynamic analysis of a spinningRayleigh beamrdquo International Journal of Mechanical Sciencesvol 47 no 2 pp 157ndash169 2005

[17] X Zhou X Qichong D Hu et al ldquoThe nonlinear dynam-ics analysis of bottom drillstring of air drillingrdquo Journal ofChongqing University of Science and Technology vol 4 pp 119ndash121 2013 (Chinese)

[18] Y-J Jia P Jiang X-H Zhu and Y-F Zhang ldquoNonlinearbuckling simulation of drillstring system during pilot holedrilling in horizontal directional drillingrdquo Journal of SystemSimulation vol 25 no 4 pp 821ndash825 2013 (Chinese)

[19] W-S Xiao Z-Y Liu H-Y Wang X-F Wang L Fu and Y-C Yin ldquoTensional vibration analysis of drill-string by FSIrdquo OilField Equipment vol 42 pp 23ndash26 2013 (Chinese)

[20] D Shao Z Guan X Wen and Y Shi ldquoExperiment on lateralvibration characteristics of horizontal rotary drilling stringrdquoJournal of China University of Petroleum vol 37 pp 100ndash1052013 (Chinese)

[21] W Zhang X Zhu Z Zhou and W Ma ldquoEffect of rotationalspeeds mutation on the dynamic characteristics of the drill-string system in a vertical wellrdquo Journal of PetrochemicalUniversities vol 26 no 2 pp 47ndash51 2013 (Chinese)

[22] Y Tang ldquoNonlinear vibrations of axially accelerating viscoelas-tic Timoshenko beamsrdquo Chinese Journal of Theoretical andApplied Mechanics vol 45 no 6 pp 965ndash973 2013 (Chinese)

Shock and Vibration 17

[23] P B Goncalves and Z J G N Del Prado ldquoNonlinear oscilla-tions and stability of parametrically excited cylindrical shellsrdquoMeccanica vol 37 no 6 pp 569ndash597 2002

[24] M Strozzi and F Pellicano ldquoNonlinear vibrations of function-ally graded cylindrical shellsrdquo Thin-Walled Structures vol 67pp 63ndash77 2013

[25] M Amabili Nonlinear Vibrations and Stability of Shells andPlates Cambridge University Press Cambridge CambridgeUK 2008

[26] A W Leissa Vibrations of Shells Government Printing OfficeWashington DC USA 1973

[27] N Yamaki Elastic Stability of Circular Cylindrical Shells North-Holland Amsterdam The Netherlands 1984

[28] J N ReddyMechanics of LaminatedComposite Plates and ShellsCRC Press Boca Raton Fla USA 2004

[29] A H Nayfeh and D T Mook Nonlinear Oscillations Wiley-VCH Press Weinheim Germany 2004

[30] P D Spanos A K Sengupta R A Cunningham and P RPaslay ldquoModeling of roller cone bit lift-off dynamics in rotarydrillingrdquo Journal of Energy Resources Technology vol 117 no 3pp 197ndash207 1995

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2 Shock and Vibration

Lateral vibration Torsional vibration Axial vibration

Stabilizer Stabilizer

Figure 1 Three forms of drill string vibration

In accord with the geometric nonlinear characteristics ofdrill strings and by adopting the combination of nonlineardynamics and FEM (finite element method) Sampaio et al[11] established the mathematical coupling model of axialand torsional vibrations by comparing it with a linear modelfinding that the nonlinear model more accurately reflectedthe features of the vibration of the drill string Hakimi andMoradi [12] introduced a differential quadrature methodto analyze the vibration of the drill string Ren et al [13]and Ren and Yao [14] taking into account the flexible andgeometric large deformation of the drill string establishedthe mathematical model of bending nonlinear vibration forthe horizontal well In addition taking into account otherfactors of drill-string vibration scholars have carried outsome research such as on the (a) displacement and forceof the drill bit [15] (b) critical speed of the drill string[16] (c) nonlinear collision and friction contact betweenthe drill string and the borehole wall [17 18] (d) theoryand experiments about coupled solids-liquids [19] and (e)the weight on bit (WOB) and rotation speed of the drillstring [20ndash22] Goncalves and Del Prado [23] analyzed thenonlinear vibrations and dynamic instability of axially loadedcircular cylindrical shells under both static and harmonicforces based on Donnell shallow shell equations The mostdangerous region in parameter space is obtained and thetriggering mechanisms associated with the stability bound-aries are identified [23] Based on the Sanders-Koiter theoryStrozzi and Pellicano [24] analyzed the nonlinear vibrationsof functionally graded circular cylindrical shells and foundthat when these shells that were having an actual hardeningresponse were simulated with an insufficient expansion theirbehavior could appear spuriously softening

Based on the specific problem of the coupled vibration ofthe drill string scholars usually analyze the drill string to findthe relationship between the drilling process parameters andthe modal shapes and then change the drilling parametersto avoid the resonance of the drill string which is a verypractical control method however the key factor is to dis-cover the relationship between the key dynamics parametersand the coupled vibrations of the drill string Consideringthe complexity of the practical dynamics of the drill stringsystem and the comprehensiveness of the quantitative analy-sis directly this study employs the dimensionless method toinvestigate themechanismof the coupled vibration of the drillstring qualitatively This will reveal which key factors affectthe coupled vibration and how they function which can be abasis for the quantitative analysis of the coupling vibration ofthe drill string

u2(120579)

120590xx0

1205901205791205790

L

x

u1

z

h

Figure 2 The mechanical model of the rotation flexible shell

2 Nonlinear Dynamics Motion Equation

During the process of drilling the transverse vibration ofthe drill string which is intense at the bit in the bottom ofthe well and attenuates quickly along the drill string mainlycontributes to the bottom hole assembly (BHA) however theaxial and torsional vibration affects the entire drill string

In addition several stabilizers support the lower portionof the drill string and absorb most transverse vibrationenergy In other words the transversemotion of the stabilizedsection mainly contributes to the transverse dynamics at thebit The drill collars are assumed to be rigid for torsionalvibrations that is to say the torsional deformations areassumed to take place only in the drill pipe which can bejustified since the drill collars are much stiffer than the drillpipe in torsionTherefore the transverse vibration of the BHAis decoupled from the upper segments of the drill string andthe entire drill string is assumed to be fixed at the top and freeat the bit for the axial and torsional motion [9]

Based on the above analyses ignoring the transversevibration of the drill string this work only analyzes themechanism of the coupled axial and torsional vibration ofthe upper portion of the BHA Here assuming the centerlineof the shell before deformation as the axis the drill stringis simplified to a flexible rotation shell instead of simply asupported beam which will ignore most torsional vibrationThe dynamics model is shown in Figure 2 where 119906

1and

1199062are the displacement of axial 119909 and circumferential 119906

respectively ℎ is the thickness of the drill string 119871 is thelength of the drill string 120590

1199091199090is the initial axial stress and

1205901205791205790

is the initial torsional stressTo facilitate the calculation of the drill string strain select

the arbitrary point displacement of the drill string as follows[25]

1199061= 119906 (1a)

1199062= V +

119911

119877

V (1b)

where 119906 and V are arbitrary axial and circumferential dis-placements of the drill string on the middle surface 119911 =

0 respectively and 119877 is the middle radius of the drillstring

Since the drill string cannot be thin and the tangentialdisplacements 119906 and V of the drill string are not smallthe Flugge-Lurrsquoe-Byrne nonlinear shell theory is selected inwhich two hypotheses are removed fromDonnellrsquos nonlinearshell theory [26] in order to obtain more accurate nonlinear

Shock and Vibration 3

shell theories One is the thinness assumption and the otheris that the tangential displacements are infinitely small whichleads to neglecting the nonlinear terms that depend on 119906 andV

Based on the nonlinear shell theory [25] the relationshipof the stressdisplacement at any point on the drill string canbe expressed by

120576119909119909

=

120597119906

120597119909

+

1

2

(

120597119906

120597119909

)

2

+

1

2

(

120597V120597119909

)

2

+ 119911

1

119877

(

120597V120597119909

)

2

(2a)

120576120579120579

=

1

119877

120597V120597120579

+

1

21198772(

120597119906

120597120579

)

2

+

1

21198772(

120597V120597120579

)

2

+

1

21198772V2

minus

1

1198773119911 (

120597119906

120597120579

)

2

(2b)

120576119909120579

=

120597V120597119909

+

120597119906

119877120597119909

+

1

21198772(

120597119906

120597120579

)

2

+

1

119877

120597119906

120597119909

120597119906

120597120579

+

1

119877

120597V120597119909

120597V120597120579

+ 119911120581119909120579

(2c)

where

120581119909120579

=

1

119877

120597V120597119909

minus

1

1198772

120597119906

120597120579

minus

1

1198772

120597119906

120597119909

120597119906

120597120579

minus

1

1198772

120597V120597119909

120597V120597120579

(3)

where 120576119909119909

and 120576120579120579

are linear strain 120576119909120579

is shear strain andthey are the dimensionless physical quantities Here (2a)(2b) and (2c) make up most of the nonlinear term whichavoids the nonlinear coupled axial and torsional terms in thefollowing analyses due to oversimplification

Analogous to the case of plane stress the stress inthickness of the drill string is neglected [27] Taking intoaccount the single material of the drill string neglecting thegeometric imperfections and subject to the initial axial andtorsional stress the relationship between stress and strain canbe written as

120590119909119909

= 1198641(120576119909119909

+ 120581120576120579120579

) + 1205901199091199090

(4a)

120590120579120579

= 1198641(120576120579120579

+ 120581120576119909119909

) + 1205901205791205790

(4b)

120590119909120579

= 1198642120576119909120579

(4c)

where 1198641

= 119864(1 minus 1205812) 1198642

= 1198642(1 + 120581) 120581 is Poissonrsquos ratioand 119864 is the elasticity modulus

Based on (2a) (2b) and (2c) and (4a) (4b) and (4c)according to Hamiltonrsquos principle the dynamic equation ofthe partial differential can be obtained as follows

120588 = (

1

2

119877ℎ1205901199091199090

1198641+ 119877ℎ120581119864

1120576120579120579

+ 119877ℎ1205921198641120576119909119909

)

1205972119906

1205971199092

+ (

ℎ1205901205791205790

2119877

+

ℎ1198641120576120579120579

119877

+

ℎ1198641120581120576119909119909

119877

)(

1205972119906

1205971205792)

minus

1198621

2120587119871

+

1

2

119877ℎ

1205971205901199091199090

120597119909

+ 119877ℎ1198641(

120597120576119909119909

120597119909

+ 120592

120597120576120579120579

120597119909

) +

ℎ

2119877

1205971205901199091199090

1205971199092

120597119906

120597119909

+

ℎ1198641

119877

120597120576120579120579

120597120579

120597119906

120597120579

+ 119877ℎ1205811198641

120597120576120579120579

120597119909

120597119906

120597119909

+

ℎ1205811198641

119877

120597120576120579120579

120597120579

120597119906

120597120579

+ 119877ℎ1198641

120597120576120579120579

120597119909

120597119906

120597119909

(5a)

120588V = (

1

2

119877ℎ1205901199091199090

+ 119877ℎ1205921198641120576120579120579

+ 119877ℎ1198641120576119909119909

)

1205972V

1205971199092

+ (

ℎ1205901205791205790

2119877

+

ℎ1198641120576120579120579

119877

+

ℎ1198641120592120576119909119909

119877

)(

1205972V

1205971205792)

minus

1

2120587119871

1198622V +

ℎ

2

1205971205901205791205790

120597120579

+ ℎ1198641(120592

120597120576119909119909

120597120579

120597120576120579120579

120597120579

)

+

ℎ

2119877

1205971205901205791205790

120597120579

120597V120597120579

+

ℎ1205921198641

119877

120597120576120579120579

120597120579

120597V120597120579

+ 119877ℎ1205921198641

120597120576120579120579

120597119909

120597V120597119909

+

ℎ1198641

119877

120597120576120579120579

120597120579

120597V120597120579

+ 119877ℎ1198641

120597120576119909119909

120597119909

120597V120597119909

minus

ℎ1205921198641

119877

120576120579120579

+

119877ℎ

2

1205971205901199091199090

120597119909

120597V120597119909

minus

ℎ

2

1205921205901205791205790

119877

(5b)

where 120588 is the density of the drill string 119868 = int

12

minus12120588119889119911 is the

rotary inertia and 1198621and 119862

2are the damping coefficients of

the first modal and the second modal respectivelyThe modals of axial and torsional vibration are given as

[28]

119906 = 1199081(119905) sin(

120587119909

2119871

) cos 120579

V = 1199082(119905) cos(2120587119909

119871

) sin 120579

(6)

In the modals of the axial vibration and torsional vibrationthe number of longitudinal half-waves was equal to one andtwo respectively In (6) when 119909 = 0 then 119906 = 0 when119909 = 119871 then 119906 became the extreme value when 119909 = 0 or119909 = 119871 then V got the extreme value all of which meet theactual displacement boundary conditions of the drill stringSubstituting (6) into (2a) (2b) and (2c) and then substitutingthe result into (4a) (4b) and (4c) produce120590

119909119909119909=0= 1205901199091199090

and

4 Shock and Vibration

120590120579120579119909=0

= 1205901205791205790

which account for the boundary conditions ofthe force

Using the Galerkin method to disperse (5a) and (5b)substituting (6) into (5a) and (5b) both sides of (5a) and (5b)are multiplied by the mode of the right side of (6) integratedin the circumferential and thickness direction Making useof modal orthogonality the motion equation of the middlesurface displacement is obtained as follows

1= 119886111986211+

119877ℎ1205901199091199090

2119871120588

minus

ℎ1205901205791205790

2120588119877

1199081minus 11988621199081minus 11988631199082

+ 119886411990811199082

2minus 11988651199083

1

(7a)

2= 119886111986222minus 11988661199081+

1

120588

ℎ1205901205791205790

119877

1199082+ 119886711990821199082

1minus 11988681199083

2 (7b)

where 1198861 1198862 1198863 1198864 1198865 1198866 1198867 and 119886

8are the constant

coefficients as listed in the Appendix From (7a) and (7b)we know that the vibration is a nonlinear coupled equationof axial and torsional vibration and that unilaterally takinginto account the single direction of the vibration is obviouslymuch too simplified to describe the actual vibrationThe axialstress as external excitation directly affects the dynamicsresponse The torsional stress 120590

1205791205790 as parameter excitation

affects the vibration by changing the axial stiffness therebyaffecting the natural frequency of the drill string Assumingthe axial and torsional excitation frequencies are integer timesof the rotation frequency of the drill string the equationitself will have two linear natural frequencies Due to theexistence of the coupling and nonlinear terms there isa great difference between the actual vibration frequencyand the linear natural frequency In addition subharmonicresonance superharmonic resonance bifurcation and chaosphenomena are possible in the vibration system with theparameters changing and the energy translating between themodals

3 Nonlinear Dynamics Average Equation

To facilitate the analysis replace the coefficients in (7a) and(7b) by 119886

119894 119887119895and use the dimensionless transformation as

follows (in the following dimensionless equation the asteriskwas removed for the convenience of expression)

119909lowast

1=

1199091

119871

119909lowast

2=

1199092

119871

119905lowast

=

119877

1198712radic

119864

120588

119905

120590lowast

1199091199090=

1205901199091199090

119864

120590lowast

1205791205790=

1205901205791205790

119864

(8)

The multiscale transformation is employed as follows

119886119894997888rarr 120576119886

119894

119887119895997888rarr 120576119887

119895

119894 = 1 119895 = 2

(9)

where 120576 is a small parameter used as a perturbation parameterfor determining the approximate solution for the motion ofthe drill string [22]

Substituting (9) into (7a) and (7b) one can obtain amotion equation that includes the small parameter 120576 Hereassume the form of the equation as follows

119909119894(119905 120576) = 119910

1198940(1198790 1198791) + 120576119910

119894(1198790 1198791) + sdot sdot sdot (119894 = 1 2) (10)

where 1198790= 119905 and 119879

1= 120576119905

Here the differential operators are employed as in thefollowing form

119889

119889119905

=

120597

1205971198790

1205971198790

120597119905

+

120597

1205971198791

1205971198791

120597119905

+ sdot sdot sdot = 1198630+ 1205761198631+ sdot sdot sdot (11a)

1198892

1198891199052

= (1198630+ 1205761198631+ sdot sdot sdot)2

= 1198632

0+ 2120576119863

01198631+ sdot sdot sdot (11b)

where 1198630= 120597120597119879

0and 119863

1= 120597120597119879

1

Taking into account the principal parameter resonanceand 2 1 internal resonance the resonance relationship can beexpressed as follows

1205962

1= 41205962+ 1205761205901

1205962

2= 1205962+ 1205761205902

(12)

where 1205961and 120596

2are the axial and torsional linear natural

frequencies respectivelyTaking into account the vibration response caused by

the axial and torsional excitation in the case of the aboveresonance the dimensionless axial and torsional excitationin the frequency domain can be expanded into the followingform including the two resonance frequencies [29]

1205901199091199090

= 1199021119890119894119905

+ 1199021119890minus119894119905

+ 11990221198902119894119905

+ 1199022119890minus119894119905

(13a)

1205901205791205790

= 1199023119890119894119905

+ 1199023119890minus119894119905

+ 11990241198902119894119905

+ 1199024119890minus119894119905

(13b)

where 1199021and 119902

2are the amplitude of the low frequency and

high frequency of the initial axial stress respectively and 1199023

and 1199024are the amplitude of the low and high frequencies of

the initial torsional stress respectivelySubstituting (8)ndash(12) into (7a) and (7b) and comparing

the same step coefficient of the small parameter 120576 on bothsides of the equations one obtains a differential equation inthe following form

For 1205760

1198632

011991010

+ 41199101= 0 (14a)

Shock and Vibration 5

For 1205761

1198632

011991020

+ 1199102= 0 (14b)

The plural form solutions of (14a) and (14b) can be expressedas

11991010

= 1198601(1198791) 1198902119894119905

+ 1198601(1198791) 119890minus2119894119905

(15a)

11991020

= 1198602(1198791) 119890119894119905

+ 1198602(1198792) 119890minus119894119905

(15b)

where 1198601and 119860

2are the conjugation of 119860

1and 119860

2 respec-

tively The plural solutions of 1198601and 119860

2are formed as

follows

1198601=

1

2

1199091+

1

2

1198941199092

1198602=

1

2

1199093+

1

2

1198941199094

(16)

where 1199091 1199092 1199093 and 119909

4are the projections of the vibration

vector on the complex plane which is equal to the vibrationvector described as the amplitude and the phase

Substituting (15a) and (15b) into (13a) and (13b) oneobtains themotion equationwith the small parameter 120576 thenmaking the long term equal to zero one obtains the averagingequation in the rectangular coordinate as follows

1= minus

1

2

11988691199091minus

1

8

11988610

1199092(1199092

3+ 1199092

4)

+

3

16

11988611

1199092(1199092

1+ 1199092

2)

(17a)

2= minus

1

2

11988691199092minus

1

8

11988610

1199091(1199092

3+ 1199092

4)

minus

3

16

11988611

1199091(1199092

1+ 1199092

2) minus

1

2

1199022

(17b)

3= minus

1

2

11988612

1199093+

1

2

11988613

1199094+

1

4

11988614

1199094(1199092

1+ 1199092

2)

+

3

8

11988615

1199094(1199092

3+ 1199092

4)

(17c)

4= minus

1

2

11988612

1199094minus

1

2

11988613

1199093minus

1

4

11988614

1199093(1199092

1+ 1199092

2)

minus

3

8

11988615

1199093(1199092

3+ 1199092

4) minus 1199023

(17d)

where 1198869 11988610 11988611 11988612 11988613 11988614 and 119886

15are the constant

coefficients listed in the AppendixThe change rules for the amplitude and phase angles in

the motion equation are found in (17a) (17b) (17c) and (17d)using the form of the differential equation We find that 119902

1

and 1199024have disappeared in (17a) (17b) (17c) and (17d) and

the parameters that can affect the dynamics response are only1199022and 1199023 We also know that 119902

2is the high-frequency item of

the initial axial excitation and 1199023is the low-frequency item of

the initial torsional excitation corresponding to the dynamicresponse of the drill string in low- and high-rotation speedareas respectively So we try to reveal relationships betweenthe two parameters with the coupling vibration of the drillstring by numerical simulation

4 Numerical Simulation andQualitative Analysis

In the present section the numerical simulation resultsobtained from using the proposed model are discussed forthe upper portion of the BHA The Runge-Kutta-Fehlbergmethod with adaptive steps is employed to perform thesimulations aiming at obtaining the dynamic response of thecoupled axial and torsional vibration in the case of resonanceThe geometric properties of the upper segment are the length119871 = 1050m themiddle diameter119877 = 01057m the thicknessℎ = 00171m Poissonrsquos ratio 120581 = 026 the elastic modulus119864 = 210GPa the density 120588 = 7850 kgm3 and the dampingcoefficients 119862

1and 119862

2 calculated from the considerations of

Spanos et al [30] The representative simulation results areshown along with the changes in 119902

2and 1199023

41 Torsional Excitation The Runge-Kutta-Fehlberg methodis employed to analyze the average of (17a) (17b) (17c) and(17d) and bifurcations of the system under exciting forces areobtained as shown in Figure 3

From the bifurcation along with the change in tor-sional excitation 119902

3 the response of the coupled axial

and torsional vibrations progresses through a cycle fromperiodic motion to doubling periodic motion to period-multiplying motion to quasiperiodic motion while exhibit-ing this unique phenomenon of nonlinear dynamics bifur-cation When the coupled vibration response is quasiperi-odic motion the amplitude of the drill string is obviouslyhigher than the period motion With increasing amplitudeof the excitation the responses change from quasiperiodicmotion to periodic motion the amplitude of the vibrationdoes not increase Instead it decreases to a certain extentwhich is different from the results of the linear analysismethod

By only increasing the torsional excitation until 1199023

=

724 (see Figure 3) and leaving other initial conditions andparameters the same this creates the periodic responsesshown in Figure 4

When 1199023

= 726 the amplitude of the system increasescorrespondingly accordinglyThe period-doubling responsesof system are shown in Figure 5

When 1199023

= 75 the phase diagram of the coupled vibra-tion changes the jumping phenomenon is more obviouspresent period-multiplying responses are shown in Figure 6

By increasing the value of 1199023so that 119902

3= 768 the qual-

itative nature of the system response changes presentingcorresponding quasiperiodic motions as shown in Figure 7This shows that new similar trajectories are derived from theoriginal trajectories of the phase diagram and they representmultifrequency vibration characteristics in the oscillogramand vibration frequencies that are close to the state ofcontinuous change Then the system vibration is in a statebetween the period vibration and the chaotic vibration qual-itatively the frequency bandwidth of the coupled vibrationalso increases the chaotic motions of the drill string aremorelikely to occur which leads to violent vibration of the drillstring

6 Shock and Vibration

x1

7 8 9 1110q3

12

14

16

18

(a) Bifurcation diagram of the axial modal

x3

10 118 97

q3

minus05

00

05

10

(b) Bifurcation diagram of the torsional modal

Figure 3 Bifurcations of the system under torsional excitation

It is well-known that the torsional vibration of the drillstring is mainly generated from the rotational speed changeof the bit when breaking rock intermittently The excitationfrequency is related to the rotation speed stiffness of thedrill string and characteristics of the rock The higher therock hardness the bigger the torsional stress and the greaterthe torsional amplitude We know from (17a) (17b) (17c)and (17d) that torsional stress strongly affects the couplingvibration when in low-frequency excitation but when inhigh-frequency excitation the torsional stress does not effec-tively contribute to the coupling vibration So in the practicaldrilling process when the drill string is in a high-rotationspeed zone the torsional stress will not lead to couplingvibration however when the drill string is in a low-rotationspeed zone more attention should be paid to the torsionalstress and appropriate drilling process parameters should beadopted to reduce the torsional stress especially when stick-slip vibration occurs This is necessary because at this timethe drill string is in low torsional excitation and large torsionalstress where it is more prone to produce a coupling vibrationof the drill string as suggested by Christoforou and Yigit [9]

Based on the above simulation analysis we also foundthat torsional excitation will affect both torsional and axialvibration simultaneously Torsional and axial vibration aresimilar qualitatively and in the nature of the vibration andare synchronous in form proving that energy can transferbetween the torsional and the axial vibration modal

42 Axial Excitation The above analysis accounts for theimpact of torsional excitation on the coupling of the axial andtorsional vibrations Here the impact of axial excitation willbe investigated

Thebifurcation diagram (Figure 8) shows the preliminaryresonance tendency The system undergoes a loop fromchaos to periodic motion then to period-three motionthen to period-multiplying and back to chaos showing

the nonlinear dynamic phenomenon of the period-doublingbifurcation and chaos

Increasing axial excitation until 1199022

= 05025 and keep-ing the remaining other initial conditions and parametersunchanged the system prevents period-doubling responsesas shown in Figure 9

When 1199022

= 0635 the system presents period-threeresponses as shown in Figure 10

When 1199022= 07725 the phase diagram changes drastically

and presents a jumping phenomenon then the systemresponds chaotically as shown in Figure 11

When 1199022

= 079 the qualitative nature of the couplingvibration of the drill string changes the present period-doubled responses are shown in Figure 12 Here the originalphase trajectory of the chaotic motion (Figure 11) contractsfor two trajectories and presents vibration characteristicsof frequency-doubling in the phase diagram The systemvibration turns from chaotic vibration into doubling periodicmotion and the stability of the system increases

As can be seen from the above qualitative analysis of theaxial excitation of the coupling vibration we find the samephenomenon as in the torsional excitation analysis for thecoupling vibration namely that axial excitation affects bothtorsional and axial vibration simultaneously and is similarqualitatively and also synchronous in form

Along with the increase in the axial excitation the systemcoupling vibration turns from period-doubling to period-three to chaotic and finally to period-doubling So the axialexcitation parameter leads to a coupling vibration in somespecial zone In the process of practical drilling the axialjump mainly contributes to the axial excitation The higherthe rotation speed the higher the excitation frequency Thelarger the WOB becomes the larger both the axial amplitudeand the axial stress will be We know from (17a) (17b) (17c)and (17d) that the axial excitation contributes to the couplingvibration when the drill string is in a high-rotation speedzone Therefore in the process of automation drilling with

Shock and Vibration 7

x2

minus3

minus2

minus1

1

0

2

3

minus2minus4 20x1

(a) Phase diagram of the axial modal

1020 10301010 1040 10501000t

minus4

minus3

minus2

minus1

0

1

2

x1

(b) Oscillogram of the axial modal

00 05 10minus10 minus05minus15

x3

minus15

minus10

minus05

00

05

10

x4

(c) Phase diagram of the torsional modal

1010 1020 1030 1040 10501000t

minus15

minus10

minus05

00

05

10x3

(d) Oscillogram of the torsional modal

5

0 02

0

1

minus1

minus2

x3

minus5

x2

minus4

minus2 x1

(e) Three-dimensional phase diagram

minus15

minus10

minus05

00

05

10

15

x2

minus10 minus05 00 05 10 15minus15

x1

(f) Poincare section

Figure 4 Period responses of the system (1199023= 724)

8 Shock and Vibration

x2

minus3

minus2

minus1

1

0

2

3

0minus2 2minus4

x1

(a) Phase diagram of the axial modal

1010 1020 1030 1040 10501000t

minus4

minus3

minus2

minus1

0

1

2

x1

(b) Oscillogram of the axial modal

minus10 minus05 00 05 10minus15

x3

minus15

minus10

minus05

00

05

10

x4

(c) Phase diagram of the torsional modal

1010 1020 1030 1040 10501000t

minus15

minus10

minus05

00

05

10x3

(d) Oscillogram of the torsional modal

5

00

2

0

1

minus1

minus2

x3

minus5

x2

minus4

minus2 x 1

(e) Three-dimensional phase diagram

0500 10 15minus10 minus05minus15

x1

minus15

minus10

minus05

00

05

10

15

x2

(f) Poincare section

Figure 5 Period responses of the system (1199023= 726)

Shock and Vibration 9

x2

minus2 0 2minus4

x1

minus3

minus2

minus1

1

0

2

3

(a) Phase diagram of the axial modal

minus4

minus3

minus2

minus1

0

1

2

x1

1010 1020 1030 1040 10501000t

(b) Oscillogram of the axial modal

minus10 minus05 00 05 10minus15

x3

minus15

minus10

minus05

00

05

10

x4

(c) Phase diagram of the torsional modal

minus15

minus10

minus05

00

05

10x3

1010 1020 1030 1040 10501000t

(d) Oscillogram of the torsional modal

5

0 02

0

1

minus1

minus2

x3

minus5

x2

minus4

minus2 x 1

(e) Three-dimensional phase diagram

minus15

minus10

minus05

00

05

10

15

x2

0500 10 15minus10 minus05minus15

x1

(f) Poincare section

Figure 6 Period responses of the system (1199023= 75)

10 Shock and Vibration

x2

minus2 0 2minus4

x1

minus3

minus2

minus1

1

0

2

3

(a) Phase diagram of the axial modal

1020 1040 1060 10801000t

minus4

minus3

minus2

minus1

0

1

2

x1

(b) Oscillogram of the axial modal

minus10 minus05 00 05 10minus15

x3

minus15

minus10

minus05

00

05

10

x4

(c) Phase diagram of the torsional modal

1020 1040 1060 10801000t

minus15

minus10

minus05

00

05

10x3

(d) Oscillogram of the torsional modal

5

00

2

0

1

minus1

minus2

x3

minus5

x2

minus4

minus2 x 1

(e) Three-dimensional phase diagram

minus10 minus05 00 05 10 15minus15

x1

minus15

minus10

minus05

00

05

10

15

x2

(f) Poincare section

Figure 7 The almost-periodic responses of the system (1199023= 768)

Shock and Vibration 11

06 07 08 09 1005q2

x2

minus10

minus05

00

(a) Bifurcation diagram of the axial modal

x3

06 07 0805 1009q2

10

15

20

(b) Bifurcation diagram of the torsional modal

Figure 8 Bifurcation of the resonance frequency-multiplier

a constant WOB in the case of a lower-rotation speed andlower rock hardness high drill efficiency can be attained byproperly increasing the WOB In the case of high-rotationspeed the coupling vibration is sensitive to the axial stressso the proper WOB should be adopted to avoid the couplingvibration of the drill string based on geological parameters

5 Conclusions

In this paper wemodeled and analyzed drill-string vibrationsby focusing on the coupled axialtorsional vibrations bymeans of nonlinear dynamics and qualitative analysis aimingat revealing the key effective factors influencing the coupledvibrations Here the drill string was described as a simplifiedequivalent flexible shell under axial rotation in which theexcitation loads and boundary conditions of the drill stringwere simplified After dimensionless processing we built thedynamics motion equation and the average equation

We found that the low-frequency amplitude expressionof the torsional excitation and the high-frequency amplitudeexpression of the axial excitation in the average equation arethe factors that determine the coupling vibration Based onthis procedure numerical simulations were carried out withadaptive steps using the Runge-Kutta-Fehlberg method todiscover the response of the system vibrations Further wefound that a change in the torsional or the axial excitationaffects both the torsional and the axial vibration simultane-ously and that it is similar qualitatively and synchronous inform

The results of the simulation analysis show that when thedrill string is in a lower-speed rotation zone the torsionalexcitation mainly contributes to the coupling vibrationincreasing the torsional stress of the drill string that moreeasily leads to the coupling vibrationWhen in a higher-speedrotating zone the axial excitation mainly contributes to thecoupling vibration so in a particular interval it is more likelyto cause the coupling vibration of the drill string

Appendix

Consider

1198861= minus

1

2119871120588120587

1198862=

1

120588

(

1

12

ℎ31198642

1198773

minus

ℎ1198642

119877

minus

1

4

1205872119877ℎ1198641

1198712

)

1198863=

1

120588

(

8

45

ℎ31198642

1198711198772

minus

32

15

ℎ1198642

119871

minus

32

15

ℎ1205811198641

119871

)

1198864=

1

120588

(

1

4

1205872ℎ31205811198641

11987131198773

minus

13

16

1205872ℎ1205811198641

1198712119877

minus

1

4

ℎ1198642

1198773

minus

1

16

1205874119877ℎ1198641

1198714

)

1198865=

1

120588

(

1

384

1205872ℎ31198642

11987121198773

minus

9

512

1205874119877ℎ1198641

1198714

minus

3

32

ℎ31198642

1198775

minus

9

32

ℎ1198642

1198773

minus

1

32

1205872ℎ1198641

1198712119877

minus

1

64

1205872ℎ1205811198641

1198712119877

)

1198866=

13ℎ1205811198641

15120588119871

1198867=

1

120588

(

1

4

1205872ℎ31205811198641

11987121198773

+

13

16

1205872ℎ1205811198641

1198712119877

minus

1

4

ℎ1198641

1198773

+

1

16

1205874119877ℎ1198641

1198714

+

3

4

ℎ1198641

1198713

+

1

24

1205872ℎ31198642

11987121198773

+

9

2

1205874119877ℎ1198641

1198714

)

1198868=

1

120588

(

1

2

1205872ℎ1198642

1198712119877

+

3

2

1205874ℎ31198641

1198714119877

+

1205872ℎ1205811198641

1198712119877

+

3

4

ℎ1198641

1198713

12 Shock and Vibrationx2

minus10

minus05

00

05

10

minus05 00 05 10minus10

x1

(a) Phase diagram of the axial modal

minus10

minus05

00

05

10

x1

1020 1040 1060 10801000t

(b) Oscillogram of the axial modal

minus2

minus1

0

1

2

x4

15 20 25 3010x3

(c) Phase diagram of the torsional modal

10

15

20

25

30x3

1020 1040 1060 10801000t

(d) Oscillogram of the torsional modal

1

0 0

11

2

3

minus1minus1

x3

x2 x 1

(e) Three-dimensional phase diagram

minus15

minus10

minus05

00

05

10

15

x2

0500 10 15minus10 minus05minus15

x1

(f) Poincare section

Figure 9 Period responses of the system (1199022= 05025)

Shock and Vibration 13x2

minus05 00 05 10minus10

x1

minus10

minus05

00

05

10

(a) Phase diagram of the axial modal

1020 1040 1060 10801000t

minus10

minus05

00

05

10

x1

(b) Oscillogram of the axial modal

minus20

minus15

minus10

minus05

00

05

10

15

20

x4

15 20 25 3010x3

(c) Phase diagram of the torsional modal

10

15

20

25

30x3

1020 1040 1060 10801000t

(d) Oscillogram of the torsional modal

1

0 0

11

2

3

minus1 minus1

x3

x2 x 1

(e) Three-dimensional phase diagram

minus10 minus05minus15 05 10 1500x1

minus15

minus10

minus05

00

05

10

15

x2

(f) Poincare section

Figure 10 Period responses of the system (1199022= 0635)

14 Shock and Vibration

minus2

minus1

0

1

2

x2

10 2 3minus1minus2

x1

(a) Phase diagram of the axial modal

1050 1100 1150 12001000t

minus2

minus1

0

1

2

3

x1

(b) Oscillogram of the axial modal

2 4minus2 0minus4

x3

minus4

minus2

0

2

4

x4

(c) Phase diagram of the torsional modal

minus4

minus2

0

2

4x3

1050 1100 1150 12001000t

(d) Oscillogram of the torsional modal

2

00

2

4

0

5

minus2

x3

minus5

x2

minus2

x 1

(e) Three-dimensional phase diagram

minus15

minus10

minus05

00

05

10

15

x2

0500 10 15minus10 minus05minus15

x1

(f) Poincare section

Figure 11 Period responses of the system (1199022= 07725)

Shock and Vibration 15x2

minus05 00 05 10minus10

x1

minus10

minus05

00

05

10

(a) Phase diagram of the axial modal

1020 1040 1060 10801000t

minus10

minus05

00

05

10

x1

(b) Oscillogram of the axial modal

15 20 25 3010x3

minus2

minus1

0

1

2

x4

(c) Phase diagram of the torsional modal

10

15

20

25

30x3

1020 1040 1060 10801000t

(d) Oscillogram of the torsional modal

1

0 0

11

2

3

minus1minus1

x3

x2

x 1

(e) Three-dimensional phase diagram

minus10 minus05minus15 05 10 1500x1

minus15

minus10

minus05

00

05

10

15

x2

(f) Poincare section

Figure 12 Period-doubled responses of the system (1199022= 079)

16 Shock and Vibration

+

1

24

1205872ℎ31198642

11987121198773

+

9

2

1205874119877ℎ1198641

1198714

)

1198869=

1198621

2120587119877radic120588119864

11988610

=

1205872ℎ31198641120581

411987121205881198772

minus

ℎ1198641

41205881198772

minus

1205874ℎ1198771198641

161198714120588

minus

1205872ℎ1198641120581

161198712120588119877

11988611

= minus

91205874ℎ1198771198641

5121198714120588

minus

3ℎ31198641

321205881198775

minus

1205872ℎ1198642

321198712120588119877

minus

1205872ℎ31198642

38411987121205881198773

minus

9ℎ1198641

321205881198773

minus

1205872ℎ1198641120581

641198712120588119877

11988612

=

1198622

2120587119877radic120588119864

11988613

= minus

41205872119877ℎ1198642

1198712120588

minus

ℎ1198641

119877120588

minus

1205872ℎ31198642

31198712120588119877

11988614

=

1205872ℎ31198641120581

411987121205881198773

minus

ℎ1198641

41205881198773

minus

1205874ℎ1198771198641

161198714120588

minus

131205872ℎ1198641120581

161198712120588119877

11988615

= minus

1205872ℎ31198642

2411987121205881198773

minus

1205872ℎ1198642

21198712120588119877

minus

91205874ℎ1198641120581

21198714120588

minus

3ℎ1198641

41205881198773

minus

1205872ℎ1198641120581

1198712120588119877

minus

31205874ℎ31198641

21198714120588119877

(A1)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to acknowledge financial support byNatural Science Foundation of China Project 11372071 andPostdoctoral Fund of China Project 2013M541339

References

[1] P D Spanos A M Chevallier N Politis and B Payne ldquoOilwell drilling a vibration perspectiverdquo The Shock and VibrationDigest vol 35 pp 81ndash99 2013

[2] W D Aldred andM C Sheppard ldquoDrillstring vibrations a newgeneration mechanism and control strategiesrdquo in Proceedingsof the SPE Annual Technical Conference and Exhibition SPE-24582-MS pp 353ndash363 Washington DC USA October 1992

[3] T V Aarrestand H A Tonnesen and A Kyllingstad ldquoDrill-string vibrations theory and experiments on full-scale researchdrilling rigrdquo inProceedings of the IADCSPEDrilling ConferenceSPE Paper No 14760 pp 311ndash321 Dallas Tex USA 1986

[4] J D Jansen ldquoNon-linear rotor dynamics as applied to oilwelldrillstring vibrationsrdquo Journal of Sound and Vibration vol 147no 1 pp 115ndash135 1991

[5] A S Yigit and A P Christoforou ldquoCoupled axial and transversevibrations of oilwell drillstringsrdquo Journal of Sound and Vibra-tion vol 195 no 4 pp 617ndash627 1996

[6] A S Yigit and A P Christoforou ldquoCoupled torsional and bend-ing vibration of drill-strings subject to impact with frictionrdquoJournal of Sound and Vibration vol 215 no 1 pp 167ndash181 1998

[7] K Z Khan Longitudinal and torsional vibration of drill-strings [MS thesis] Massachusetts Institute of TechnologyCambridge Mass USA 1986

[8] R J Shyu Bending of rotating drill-strings [PhD thesis] Mas-sachusetts Institute of Technology 1989

[9] A P Christoforou and A S Yigit ldquoFully coupled vibrations ofactively controlled drillstringsrdquo Journal of Sound and Vibrationvol 267 no 5 pp 1029ndash1045 2003

[10] M A Trindade C Wolter and R Sampaio ldquoKarhunen-Loevedecomposition of coupled axialbending vibrations of beamssubject to impactsrdquo Journal of Sound and Vibration vol 279 no3ndash5 pp 1015ndash1036 2005

[11] R Sampaio M T Piovan and G Venero Lozano ldquoCoupledaxialtorsional vibrations of drill-strings bymeans of non-linearmodelrdquo Mechanics Research Communications vol 34 no 5-6pp 497ndash502 2007

[12] H Hakimi and S Moradi ldquoDrillstring vibration analysis usingdifferential quadrature methodrdquo Journal of Petroleum Scienceand Engineering vol 70 no 3-4 pp 235ndash242 2010

[13] F S Ren S Chen and Z G Yao ldquoDynamics analysis andvibration suppression of a flexible rotation beamrdquo AppliedMechanics and Materials vol 214 pp 165ndash172 2012

[14] F-S Ren and Z-G Yao ldquoStudy on nonlinear dynamics andbifurcations in rotating compressive-drill stringrdquo EngineeringMechanics vol 30 no 10 pp 251ndash256 2013

[15] Z-F Li Y-G Zhang X-T Hou W-D Liu and G-Q XuldquoAnalysis of longitudinal and torsion vibration of drill-stringsrdquoEngineering Mechanics vol 21 no 6 pp 203ndash210 2004 (Chi-nese)

[16] G J Sheu and S M Yang ldquoDynamic analysis of a spinningRayleigh beamrdquo International Journal of Mechanical Sciencesvol 47 no 2 pp 157ndash169 2005

[17] X Zhou X Qichong D Hu et al ldquoThe nonlinear dynam-ics analysis of bottom drillstring of air drillingrdquo Journal ofChongqing University of Science and Technology vol 4 pp 119ndash121 2013 (Chinese)

[18] Y-J Jia P Jiang X-H Zhu and Y-F Zhang ldquoNonlinearbuckling simulation of drillstring system during pilot holedrilling in horizontal directional drillingrdquo Journal of SystemSimulation vol 25 no 4 pp 821ndash825 2013 (Chinese)

[19] W-S Xiao Z-Y Liu H-Y Wang X-F Wang L Fu and Y-C Yin ldquoTensional vibration analysis of drill-string by FSIrdquo OilField Equipment vol 42 pp 23ndash26 2013 (Chinese)

[20] D Shao Z Guan X Wen and Y Shi ldquoExperiment on lateralvibration characteristics of horizontal rotary drilling stringrdquoJournal of China University of Petroleum vol 37 pp 100ndash1052013 (Chinese)

[21] W Zhang X Zhu Z Zhou and W Ma ldquoEffect of rotationalspeeds mutation on the dynamic characteristics of the drill-string system in a vertical wellrdquo Journal of PetrochemicalUniversities vol 26 no 2 pp 47ndash51 2013 (Chinese)

[22] Y Tang ldquoNonlinear vibrations of axially accelerating viscoelas-tic Timoshenko beamsrdquo Chinese Journal of Theoretical andApplied Mechanics vol 45 no 6 pp 965ndash973 2013 (Chinese)

Shock and Vibration 17

[23] P B Goncalves and Z J G N Del Prado ldquoNonlinear oscilla-tions and stability of parametrically excited cylindrical shellsrdquoMeccanica vol 37 no 6 pp 569ndash597 2002

[24] M Strozzi and F Pellicano ldquoNonlinear vibrations of function-ally graded cylindrical shellsrdquo Thin-Walled Structures vol 67pp 63ndash77 2013

[25] M Amabili Nonlinear Vibrations and Stability of Shells andPlates Cambridge University Press Cambridge CambridgeUK 2008

[26] A W Leissa Vibrations of Shells Government Printing OfficeWashington DC USA 1973

[27] N Yamaki Elastic Stability of Circular Cylindrical Shells North-Holland Amsterdam The Netherlands 1984

[28] J N ReddyMechanics of LaminatedComposite Plates and ShellsCRC Press Boca Raton Fla USA 2004

[29] A H Nayfeh and D T Mook Nonlinear Oscillations Wiley-VCH Press Weinheim Germany 2004

[30] P D Spanos A K Sengupta R A Cunningham and P RPaslay ldquoModeling of roller cone bit lift-off dynamics in rotarydrillingrdquo Journal of Energy Resources Technology vol 117 no 3pp 197ndash207 1995

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Submit your manuscripts athttpwwwhindawicom

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Shock and Vibration

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DistributedSensor Networks

International Journal of

Shock and Vibration 3

shell theories One is the thinness assumption and the otheris that the tangential displacements are infinitely small whichleads to neglecting the nonlinear terms that depend on 119906 andV

Based on the nonlinear shell theory [25] the relationshipof the stressdisplacement at any point on the drill string canbe expressed by

120576119909119909

=

120597119906

120597119909

+

1

2

(

120597119906

120597119909

)

2

+

1

2

(

120597V120597119909

)

2

+ 119911

1

119877

(

120597V120597119909

)

2

(2a)

120576120579120579

=

1

119877

120597V120597120579

+

1

21198772(

120597119906

120597120579

)

2

+

1

21198772(

120597V120597120579

)

2

+

1

21198772V2

minus

1

1198773119911 (

120597119906

120597120579

)

2

(2b)

120576119909120579

=

120597V120597119909

+

120597119906

119877120597119909

+

1

21198772(

120597119906

120597120579

)

2

+

1

119877

120597119906

120597119909

120597119906

120597120579

+

1

119877

120597V120597119909

120597V120597120579

+ 119911120581119909120579

(2c)

where

120581119909120579

=

1

119877

120597V120597119909

minus

1

1198772

120597119906

120597120579

minus

1

1198772

120597119906

120597119909

120597119906

120597120579

minus

1

1198772

120597V120597119909

120597V120597120579

(3)

where 120576119909119909

and 120576120579120579

are linear strain 120576119909120579

is shear strain andthey are the dimensionless physical quantities Here (2a)(2b) and (2c) make up most of the nonlinear term whichavoids the nonlinear coupled axial and torsional terms in thefollowing analyses due to oversimplification

Analogous to the case of plane stress the stress inthickness of the drill string is neglected [27] Taking intoaccount the single material of the drill string neglecting thegeometric imperfections and subject to the initial axial andtorsional stress the relationship between stress and strain canbe written as

120590119909119909

= 1198641(120576119909119909

+ 120581120576120579120579

) + 1205901199091199090

(4a)

120590120579120579

= 1198641(120576120579120579

+ 120581120576119909119909

) + 1205901205791205790

(4b)

120590119909120579

= 1198642120576119909120579

(4c)

where 1198641

= 119864(1 minus 1205812) 1198642

= 1198642(1 + 120581) 120581 is Poissonrsquos ratioand 119864 is the elasticity modulus

Based on (2a) (2b) and (2c) and (4a) (4b) and (4c)according to Hamiltonrsquos principle the dynamic equation ofthe partial differential can be obtained as follows

120588 = (

1

2

119877ℎ1205901199091199090

1198641+ 119877ℎ120581119864

1120576120579120579

+ 119877ℎ1205921198641120576119909119909

)

1205972119906

1205971199092

+ (

ℎ1205901205791205790

2119877

+

ℎ1198641120576120579120579

119877

+

ℎ1198641120581120576119909119909

119877

)(

1205972119906

1205971205792)

minus

1198621

2120587119871

+

1

2

119877ℎ

1205971205901199091199090

120597119909

+ 119877ℎ1198641(

120597120576119909119909

120597119909

+ 120592

120597120576120579120579

120597119909

) +

ℎ

2119877

1205971205901199091199090

1205971199092

120597119906

120597119909

+

ℎ1198641

119877

120597120576120579120579

120597120579

120597119906

120597120579

+ 119877ℎ1205811198641

120597120576120579120579

120597119909

120597119906

120597119909

+

ℎ1205811198641

119877

120597120576120579120579

120597120579

120597119906

120597120579

+ 119877ℎ1198641

120597120576120579120579

120597119909

120597119906

120597119909

(5a)

120588V = (

1

2

119877ℎ1205901199091199090

+ 119877ℎ1205921198641120576120579120579

+ 119877ℎ1198641120576119909119909

)

1205972V

1205971199092

+ (

ℎ1205901205791205790

2119877

+

ℎ1198641120576120579120579

119877

+

ℎ1198641120592120576119909119909

119877

)(

1205972V

1205971205792)

minus

1

2120587119871

1198622V +

ℎ

2

1205971205901205791205790

120597120579

+ ℎ1198641(120592

120597120576119909119909

120597120579

120597120576120579120579

120597120579

)

+

ℎ

2119877

1205971205901205791205790

120597120579

120597V120597120579

+

ℎ1205921198641

119877

120597120576120579120579

120597120579

120597V120597120579

+ 119877ℎ1205921198641

120597120576120579120579

120597119909

120597V120597119909

+

ℎ1198641

119877

120597120576120579120579

120597120579

120597V120597120579

+ 119877ℎ1198641

120597120576119909119909

120597119909

120597V120597119909

minus

ℎ1205921198641

119877

120576120579120579

+

119877ℎ

2

1205971205901199091199090

120597119909

120597V120597119909

minus

ℎ

2

1205921205901205791205790

119877

(5b)

where 120588 is the density of the drill string 119868 = int

12

minus12120588119889119911 is the

rotary inertia and 1198621and 119862

2are the damping coefficients of

the first modal and the second modal respectivelyThe modals of axial and torsional vibration are given as

[28]

119906 = 1199081(119905) sin(

120587119909

2119871

) cos 120579

V = 1199082(119905) cos(2120587119909

119871

) sin 120579

(6)

In the modals of the axial vibration and torsional vibrationthe number of longitudinal half-waves was equal to one andtwo respectively In (6) when 119909 = 0 then 119906 = 0 when119909 = 119871 then 119906 became the extreme value when 119909 = 0 or119909 = 119871 then V got the extreme value all of which meet theactual displacement boundary conditions of the drill stringSubstituting (6) into (2a) (2b) and (2c) and then substitutingthe result into (4a) (4b) and (4c) produce120590

119909119909119909=0= 1205901199091199090

and

4 Shock and Vibration

120590120579120579119909=0

= 1205901205791205790

which account for the boundary conditions ofthe force

Using the Galerkin method to disperse (5a) and (5b)substituting (6) into (5a) and (5b) both sides of (5a) and (5b)are multiplied by the mode of the right side of (6) integratedin the circumferential and thickness direction Making useof modal orthogonality the motion equation of the middlesurface displacement is obtained as follows

1= 119886111986211+

119877ℎ1205901199091199090

2119871120588

minus

ℎ1205901205791205790

2120588119877

1199081minus 11988621199081minus 11988631199082

+ 119886411990811199082

2minus 11988651199083

1

(7a)

2= 119886111986222minus 11988661199081+

1

120588

ℎ1205901205791205790

119877

1199082+ 119886711990821199082

1minus 11988681199083

2 (7b)

where 1198861 1198862 1198863 1198864 1198865 1198866 1198867 and 119886

8are the constant

coefficients as listed in the Appendix From (7a) and (7b)we know that the vibration is a nonlinear coupled equationof axial and torsional vibration and that unilaterally takinginto account the single direction of the vibration is obviouslymuch too simplified to describe the actual vibrationThe axialstress as external excitation directly affects the dynamicsresponse The torsional stress 120590

1205791205790 as parameter excitation

affects the vibration by changing the axial stiffness therebyaffecting the natural frequency of the drill string Assumingthe axial and torsional excitation frequencies are integer timesof the rotation frequency of the drill string the equationitself will have two linear natural frequencies Due to theexistence of the coupling and nonlinear terms there isa great difference between the actual vibration frequencyand the linear natural frequency In addition subharmonicresonance superharmonic resonance bifurcation and chaosphenomena are possible in the vibration system with theparameters changing and the energy translating between themodals

3 Nonlinear Dynamics Average Equation

To facilitate the analysis replace the coefficients in (7a) and(7b) by 119886

119894 119887119895and use the dimensionless transformation as

follows (in the following dimensionless equation the asteriskwas removed for the convenience of expression)

119909lowast

1=

1199091

119871

119909lowast

2=

1199092

119871

119905lowast

=

119877

1198712radic

119864

120588

119905

120590lowast

1199091199090=

1205901199091199090

119864

120590lowast

1205791205790=

1205901205791205790

119864

(8)

The multiscale transformation is employed as follows

119886119894997888rarr 120576119886

119894

119887119895997888rarr 120576119887

119895

119894 = 1 119895 = 2

(9)

where 120576 is a small parameter used as a perturbation parameterfor determining the approximate solution for the motion ofthe drill string [22]

Substituting (9) into (7a) and (7b) one can obtain amotion equation that includes the small parameter 120576 Hereassume the form of the equation as follows

119909119894(119905 120576) = 119910

1198940(1198790 1198791) + 120576119910

119894(1198790 1198791) + sdot sdot sdot (119894 = 1 2) (10)

where 1198790= 119905 and 119879

1= 120576119905

Here the differential operators are employed as in thefollowing form

119889

119889119905

=

120597

1205971198790

1205971198790

120597119905

+

120597

1205971198791

1205971198791

120597119905

+ sdot sdot sdot = 1198630+ 1205761198631+ sdot sdot sdot (11a)

1198892

1198891199052

= (1198630+ 1205761198631+ sdot sdot sdot)2

= 1198632

0+ 2120576119863

01198631+ sdot sdot sdot (11b)

where 1198630= 120597120597119879

0and 119863

1= 120597120597119879

1

Taking into account the principal parameter resonanceand 2 1 internal resonance the resonance relationship can beexpressed as follows

1205962

1= 41205962+ 1205761205901

1205962

2= 1205962+ 1205761205902

(12)

where 1205961and 120596

2are the axial and torsional linear natural

frequencies respectivelyTaking into account the vibration response caused by

the axial and torsional excitation in the case of the aboveresonance the dimensionless axial and torsional excitationin the frequency domain can be expanded into the followingform including the two resonance frequencies [29]

1205901199091199090

= 1199021119890119894119905

+ 1199021119890minus119894119905

+ 11990221198902119894119905

+ 1199022119890minus119894119905

(13a)

1205901205791205790

= 1199023119890119894119905

+ 1199023119890minus119894119905

+ 11990241198902119894119905

+ 1199024119890minus119894119905

(13b)

where 1199021and 119902

2are the amplitude of the low frequency and

high frequency of the initial axial stress respectively and 1199023

and 1199024are the amplitude of the low and high frequencies of

the initial torsional stress respectivelySubstituting (8)ndash(12) into (7a) and (7b) and comparing

the same step coefficient of the small parameter 120576 on bothsides of the equations one obtains a differential equation inthe following form

For 1205760

1198632

011991010

+ 41199101= 0 (14a)

Shock and Vibration 5

For 1205761

1198632

011991020

+ 1199102= 0 (14b)

The plural form solutions of (14a) and (14b) can be expressedas

11991010

= 1198601(1198791) 1198902119894119905

+ 1198601(1198791) 119890minus2119894119905

(15a)

11991020

= 1198602(1198791) 119890119894119905

+ 1198602(1198792) 119890minus119894119905

(15b)

where 1198601and 119860

2are the conjugation of 119860

1and 119860

2 respec-

tively The plural solutions of 1198601and 119860

2are formed as

follows

1198601=

1

2

1199091+

1

2

1198941199092

1198602=

1

2

1199093+

1

2

1198941199094

(16)

where 1199091 1199092 1199093 and 119909

4are the projections of the vibration

vector on the complex plane which is equal to the vibrationvector described as the amplitude and the phase

Substituting (15a) and (15b) into (13a) and (13b) oneobtains themotion equationwith the small parameter 120576 thenmaking the long term equal to zero one obtains the averagingequation in the rectangular coordinate as follows

1= minus

1

2

11988691199091minus

1

8

11988610

1199092(1199092

3+ 1199092

4)

+

3

16

11988611

1199092(1199092

1+ 1199092

2)

(17a)

2= minus

1

2

11988691199092minus

1

8

11988610

1199091(1199092

3+ 1199092

4)

minus

3

16

11988611

1199091(1199092

1+ 1199092

2) minus

1

2

1199022

(17b)

3= minus

1

2

11988612

1199093+

1

2

11988613

1199094+

1

4

11988614

1199094(1199092

1+ 1199092

2)

+

3

8

11988615

1199094(1199092

3+ 1199092

4)

(17c)

4= minus

1

2

11988612

1199094minus

1

2

11988613

1199093minus

1

4

11988614

1199093(1199092

1+ 1199092

2)

minus

3

8

11988615

1199093(1199092

3+ 1199092

4) minus 1199023

(17d)

where 1198869 11988610 11988611 11988612 11988613 11988614 and 119886

15are the constant

coefficients listed in the AppendixThe change rules for the amplitude and phase angles in

the motion equation are found in (17a) (17b) (17c) and (17d)using the form of the differential equation We find that 119902

1

and 1199024have disappeared in (17a) (17b) (17c) and (17d) and

the parameters that can affect the dynamics response are only1199022and 1199023 We also know that 119902

2is the high-frequency item of

the initial axial excitation and 1199023is the low-frequency item of

the initial torsional excitation corresponding to the dynamicresponse of the drill string in low- and high-rotation speedareas respectively So we try to reveal relationships betweenthe two parameters with the coupling vibration of the drillstring by numerical simulation

4 Numerical Simulation andQualitative Analysis

In the present section the numerical simulation resultsobtained from using the proposed model are discussed forthe upper portion of the BHA The Runge-Kutta-Fehlbergmethod with adaptive steps is employed to perform thesimulations aiming at obtaining the dynamic response of thecoupled axial and torsional vibration in the case of resonanceThe geometric properties of the upper segment are the length119871 = 1050m themiddle diameter119877 = 01057m the thicknessℎ = 00171m Poissonrsquos ratio 120581 = 026 the elastic modulus119864 = 210GPa the density 120588 = 7850 kgm3 and the dampingcoefficients 119862

1and 119862

2 calculated from the considerations of

Spanos et al [30] The representative simulation results areshown along with the changes in 119902

2and 1199023

41 Torsional Excitation The Runge-Kutta-Fehlberg methodis employed to analyze the average of (17a) (17b) (17c) and(17d) and bifurcations of the system under exciting forces areobtained as shown in Figure 3

From the bifurcation along with the change in tor-sional excitation 119902

3 the response of the coupled axial

and torsional vibrations progresses through a cycle fromperiodic motion to doubling periodic motion to period-multiplying motion to quasiperiodic motion while exhibit-ing this unique phenomenon of nonlinear dynamics bifur-cation When the coupled vibration response is quasiperi-odic motion the amplitude of the drill string is obviouslyhigher than the period motion With increasing amplitudeof the excitation the responses change from quasiperiodicmotion to periodic motion the amplitude of the vibrationdoes not increase Instead it decreases to a certain extentwhich is different from the results of the linear analysismethod

By only increasing the torsional excitation until 1199023

=

724 (see Figure 3) and leaving other initial conditions andparameters the same this creates the periodic responsesshown in Figure 4

When 1199023

= 726 the amplitude of the system increasescorrespondingly accordinglyThe period-doubling responsesof system are shown in Figure 5

When 1199023

= 75 the phase diagram of the coupled vibra-tion changes the jumping phenomenon is more obviouspresent period-multiplying responses are shown in Figure 6

By increasing the value of 1199023so that 119902

3= 768 the qual-

itative nature of the system response changes presentingcorresponding quasiperiodic motions as shown in Figure 7This shows that new similar trajectories are derived from theoriginal trajectories of the phase diagram and they representmultifrequency vibration characteristics in the oscillogramand vibration frequencies that are close to the state ofcontinuous change Then the system vibration is in a statebetween the period vibration and the chaotic vibration qual-itatively the frequency bandwidth of the coupled vibrationalso increases the chaotic motions of the drill string aremorelikely to occur which leads to violent vibration of the drillstring

6 Shock and Vibration

x1

7 8 9 1110q3

12

14

16

18

(a) Bifurcation diagram of the axial modal

x3

10 118 97

q3

minus05

00

05

10

(b) Bifurcation diagram of the torsional modal

Figure 3 Bifurcations of the system under torsional excitation

It is well-known that the torsional vibration of the drillstring is mainly generated from the rotational speed changeof the bit when breaking rock intermittently The excitationfrequency is related to the rotation speed stiffness of thedrill string and characteristics of the rock The higher therock hardness the bigger the torsional stress and the greaterthe torsional amplitude We know from (17a) (17b) (17c)and (17d) that torsional stress strongly affects the couplingvibration when in low-frequency excitation but when inhigh-frequency excitation the torsional stress does not effec-tively contribute to the coupling vibration So in the practicaldrilling process when the drill string is in a high-rotationspeed zone the torsional stress will not lead to couplingvibration however when the drill string is in a low-rotationspeed zone more attention should be paid to the torsionalstress and appropriate drilling process parameters should beadopted to reduce the torsional stress especially when stick-slip vibration occurs This is necessary because at this timethe drill string is in low torsional excitation and large torsionalstress where it is more prone to produce a coupling vibrationof the drill string as suggested by Christoforou and Yigit [9]

Based on the above simulation analysis we also foundthat torsional excitation will affect both torsional and axialvibration simultaneously Torsional and axial vibration aresimilar qualitatively and in the nature of the vibration andare synchronous in form proving that energy can transferbetween the torsional and the axial vibration modal

42 Axial Excitation The above analysis accounts for theimpact of torsional excitation on the coupling of the axial andtorsional vibrations Here the impact of axial excitation willbe investigated

Thebifurcation diagram (Figure 8) shows the preliminaryresonance tendency The system undergoes a loop fromchaos to periodic motion then to period-three motionthen to period-multiplying and back to chaos showing

the nonlinear dynamic phenomenon of the period-doublingbifurcation and chaos

Increasing axial excitation until 1199022

= 05025 and keep-ing the remaining other initial conditions and parametersunchanged the system prevents period-doubling responsesas shown in Figure 9

When 1199022

= 0635 the system presents period-threeresponses as shown in Figure 10

When 1199022= 07725 the phase diagram changes drastically

and presents a jumping phenomenon then the systemresponds chaotically as shown in Figure 11

When 1199022

= 079 the qualitative nature of the couplingvibration of the drill string changes the present period-doubled responses are shown in Figure 12 Here the originalphase trajectory of the chaotic motion (Figure 11) contractsfor two trajectories and presents vibration characteristicsof frequency-doubling in the phase diagram The systemvibration turns from chaotic vibration into doubling periodicmotion and the stability of the system increases

As can be seen from the above qualitative analysis of theaxial excitation of the coupling vibration we find the samephenomenon as in the torsional excitation analysis for thecoupling vibration namely that axial excitation affects bothtorsional and axial vibration simultaneously and is similarqualitatively and also synchronous in form

Along with the increase in the axial excitation the systemcoupling vibration turns from period-doubling to period-three to chaotic and finally to period-doubling So the axialexcitation parameter leads to a coupling vibration in somespecial zone In the process of practical drilling the axialjump mainly contributes to the axial excitation The higherthe rotation speed the higher the excitation frequency Thelarger the WOB becomes the larger both the axial amplitudeand the axial stress will be We know from (17a) (17b) (17c)and (17d) that the axial excitation contributes to the couplingvibration when the drill string is in a high-rotation speedzone Therefore in the process of automation drilling with

Shock and Vibration 7

x2

minus3

minus2

minus1

1

0

2

3

minus2minus4 20x1

(a) Phase diagram of the axial modal

1020 10301010 1040 10501000t

minus4

minus3

minus2

minus1

0

1

2

x1

(b) Oscillogram of the axial modal

00 05 10minus10 minus05minus15

x3

minus15

minus10

minus05

00

05

10

x4

(c) Phase diagram of the torsional modal

1010 1020 1030 1040 10501000t

minus15

minus10

minus05

00

05

10x3

(d) Oscillogram of the torsional modal

5

0 02

0

1

minus1

minus2

x3

minus5

x2

minus4

minus2 x1

(e) Three-dimensional phase diagram

minus15

minus10

minus05

00

05

10

15

x2

minus10 minus05 00 05 10 15minus15

x1

(f) Poincare section

Figure 4 Period responses of the system (1199023= 724)

8 Shock and Vibration

x2

minus3

minus2

minus1

1

0

2

3

0minus2 2minus4

x1

(a) Phase diagram of the axial modal

1010 1020 1030 1040 10501000t

minus4

minus3

minus2

minus1

0

1

2

x1

(b) Oscillogram of the axial modal

minus10 minus05 00 05 10minus15

x3

minus15

minus10

minus05

00

05

10

x4

(c) Phase diagram of the torsional modal

1010 1020 1030 1040 10501000t

minus15

minus10

minus05

00

05

10x3

(d) Oscillogram of the torsional modal

5

00

2

0

1

minus1

minus2

x3

minus5

x2

minus4

minus2 x 1

(e) Three-dimensional phase diagram

0500 10 15minus10 minus05minus15

x1

minus15

minus10

minus05

00

05

10

15

x2

(f) Poincare section

Figure 5 Period responses of the system (1199023= 726)

Shock and Vibration 9

x2

minus2 0 2minus4

x1

minus3

minus2

minus1

1

0

2

3

(a) Phase diagram of the axial modal

minus4

minus3

minus2

minus1

0

1

2

x1

1010 1020 1030 1040 10501000t

(b) Oscillogram of the axial modal

minus10 minus05 00 05 10minus15

x3

minus15

minus10

minus05

00

05

10

x4

(c) Phase diagram of the torsional modal

minus15

minus10

minus05

00

05

10x3

1010 1020 1030 1040 10501000t

(d) Oscillogram of the torsional modal

5

0 02

0

1

minus1

minus2

x3

minus5

x2

minus4

minus2 x 1

(e) Three-dimensional phase diagram

minus15

minus10

minus05

00

05

10

15

x2

0500 10 15minus10 minus05minus15

x1

(f) Poincare section

Figure 6 Period responses of the system (1199023= 75)

10 Shock and Vibration

x2

minus2 0 2minus4

x1

minus3

minus2

minus1

1

0

2

3

(a) Phase diagram of the axial modal

1020 1040 1060 10801000t

minus4

minus3

minus2

minus1

0

1

2

x1

(b) Oscillogram of the axial modal

minus10 minus05 00 05 10minus15

x3

minus15

minus10

minus05

00

05

10

x4

(c) Phase diagram of the torsional modal

1020 1040 1060 10801000t

minus15

minus10

minus05

00

05

10x3

(d) Oscillogram of the torsional modal

5

00

2

0

1

minus1

minus2

x3

minus5

x2

minus4

minus2 x 1

(e) Three-dimensional phase diagram

minus10 minus05 00 05 10 15minus15

x1

minus15

minus10

minus05

00

05

10

15

x2

(f) Poincare section

Figure 7 The almost-periodic responses of the system (1199023= 768)

Shock and Vibration 11

06 07 08 09 1005q2

x2

minus10

minus05

00

(a) Bifurcation diagram of the axial modal

x3

06 07 0805 1009q2

10

15

20

(b) Bifurcation diagram of the torsional modal

Figure 8 Bifurcation of the resonance frequency-multiplier

a constant WOB in the case of a lower-rotation speed andlower rock hardness high drill efficiency can be attained byproperly increasing the WOB In the case of high-rotationspeed the coupling vibration is sensitive to the axial stressso the proper WOB should be adopted to avoid the couplingvibration of the drill string based on geological parameters

5 Conclusions

In this paper wemodeled and analyzed drill-string vibrationsby focusing on the coupled axialtorsional vibrations bymeans of nonlinear dynamics and qualitative analysis aimingat revealing the key effective factors influencing the coupledvibrations Here the drill string was described as a simplifiedequivalent flexible shell under axial rotation in which theexcitation loads and boundary conditions of the drill stringwere simplified After dimensionless processing we built thedynamics motion equation and the average equation

We found that the low-frequency amplitude expressionof the torsional excitation and the high-frequency amplitudeexpression of the axial excitation in the average equation arethe factors that determine the coupling vibration Based onthis procedure numerical simulations were carried out withadaptive steps using the Runge-Kutta-Fehlberg method todiscover the response of the system vibrations Further wefound that a change in the torsional or the axial excitationaffects both the torsional and the axial vibration simultane-ously and that it is similar qualitatively and synchronous inform

The results of the simulation analysis show that when thedrill string is in a lower-speed rotation zone the torsionalexcitation mainly contributes to the coupling vibrationincreasing the torsional stress of the drill string that moreeasily leads to the coupling vibrationWhen in a higher-speedrotating zone the axial excitation mainly contributes to thecoupling vibration so in a particular interval it is more likelyto cause the coupling vibration of the drill string

Appendix

Consider

1198861= minus

1

2119871120588120587

1198862=

1

120588

(

1

12

ℎ31198642

1198773

minus

ℎ1198642

119877

minus

1

4

1205872119877ℎ1198641

1198712

)

1198863=

1

120588

(

8

45

ℎ31198642

1198711198772

minus

32

15

ℎ1198642

119871

minus

32

15

ℎ1205811198641

119871

)

1198864=

1

120588

(

1

4

1205872ℎ31205811198641

11987131198773

minus

13

16

1205872ℎ1205811198641

1198712119877

minus

1

4

ℎ1198642

1198773

minus

1

16

1205874119877ℎ1198641

1198714

)

1198865=

1

120588

(

1

384

1205872ℎ31198642

11987121198773

minus

9

512

1205874119877ℎ1198641

1198714

minus

3

32

ℎ31198642

1198775

minus

9

32

ℎ1198642

1198773

minus

1

32

1205872ℎ1198641

1198712119877

minus

1

64

1205872ℎ1205811198641

1198712119877

)

1198866=

13ℎ1205811198641

15120588119871

1198867=

1

120588

(

1

4

1205872ℎ31205811198641

11987121198773

+

13

16

1205872ℎ1205811198641

1198712119877

minus

1

4

ℎ1198641

1198773

+

1

16

1205874119877ℎ1198641

1198714

+

3

4

ℎ1198641

1198713

+

1

24

1205872ℎ31198642

11987121198773

+

9

2

1205874119877ℎ1198641

1198714

)

1198868=

1

120588

(

1

2

1205872ℎ1198642

1198712119877

+

3

2

1205874ℎ31198641

1198714119877

+

1205872ℎ1205811198641

1198712119877

+

3

4

ℎ1198641

1198713

12 Shock and Vibrationx2

minus10

minus05

00

05

10

minus05 00 05 10minus10

x1

(a) Phase diagram of the axial modal

minus10

minus05

00

05

10

x1

1020 1040 1060 10801000t

(b) Oscillogram of the axial modal

minus2

minus1

0

1

2

x4

15 20 25 3010x3

(c) Phase diagram of the torsional modal

10

15

20

25

30x3

1020 1040 1060 10801000t

(d) Oscillogram of the torsional modal

1

0 0

11

2

3

minus1minus1

x3

x2 x 1

(e) Three-dimensional phase diagram

minus15

minus10

minus05

00

05

10

15

x2

0500 10 15minus10 minus05minus15

x1

(f) Poincare section

Figure 9 Period responses of the system (1199022= 05025)

Shock and Vibration 13x2

minus05 00 05 10minus10

x1

minus10

minus05

00

05

10

(a) Phase diagram of the axial modal

1020 1040 1060 10801000t

minus10

minus05

00

05

10

x1

(b) Oscillogram of the axial modal

minus20

minus15

minus10

minus05

00

05

10

15

20

x4

15 20 25 3010x3

(c) Phase diagram of the torsional modal

10

15

20

25

30x3

1020 1040 1060 10801000t

(d) Oscillogram of the torsional modal

1

0 0

11

2

3

minus1 minus1

x3

x2 x 1

(e) Three-dimensional phase diagram

minus10 minus05minus15 05 10 1500x1

minus15

minus10

minus05

00

05

10

15

x2

(f) Poincare section

Figure 10 Period responses of the system (1199022= 0635)

14 Shock and Vibration

minus2

minus1

0

1

2

x2

10 2 3minus1minus2

x1

(a) Phase diagram of the axial modal

1050 1100 1150 12001000t

minus2

minus1

0

1

2

3

x1

(b) Oscillogram of the axial modal

2 4minus2 0minus4

x3

minus4

minus2

0

2

4

x4

(c) Phase diagram of the torsional modal

minus4

minus2

0

2

4x3

1050 1100 1150 12001000t

(d) Oscillogram of the torsional modal

2

00

2

4

0

5

minus2

x3

minus5

x2

minus2

x 1

(e) Three-dimensional phase diagram

minus15

minus10

minus05

00

05

10

15

x2

0500 10 15minus10 minus05minus15

x1

(f) Poincare section

Figure 11 Period responses of the system (1199022= 07725)

Shock and Vibration 15x2

minus05 00 05 10minus10

x1

minus10

minus05

00

05

10

(a) Phase diagram of the axial modal

1020 1040 1060 10801000t

minus10

minus05

00

05

10

x1

(b) Oscillogram of the axial modal

15 20 25 3010x3

minus2

minus1

0

1

2

x4

(c) Phase diagram of the torsional modal

10

15

20

25

30x3

1020 1040 1060 10801000t

(d) Oscillogram of the torsional modal

1

0 0

11

2

3

minus1minus1

x3

x2

x 1

(e) Three-dimensional phase diagram

minus10 minus05minus15 05 10 1500x1

minus15

minus10

minus05

00

05

10

15

x2

(f) Poincare section

Figure 12 Period-doubled responses of the system (1199022= 079)

16 Shock and Vibration

+

1

24

1205872ℎ31198642

11987121198773

+

9

2

1205874119877ℎ1198641

1198714

)

1198869=

1198621

2120587119877radic120588119864

11988610

=

1205872ℎ31198641120581

411987121205881198772

minus

ℎ1198641

41205881198772

minus

1205874ℎ1198771198641

161198714120588

minus

1205872ℎ1198641120581

161198712120588119877

11988611

= minus

91205874ℎ1198771198641

5121198714120588

minus

3ℎ31198641

321205881198775

minus

1205872ℎ1198642

321198712120588119877

minus

1205872ℎ31198642

38411987121205881198773

minus

9ℎ1198641

321205881198773

minus

1205872ℎ1198641120581

641198712120588119877

11988612

=

1198622

2120587119877radic120588119864

11988613

= minus

41205872119877ℎ1198642

1198712120588

minus

ℎ1198641

119877120588

minus

1205872ℎ31198642

31198712120588119877

11988614

=

1205872ℎ31198641120581

411987121205881198773

minus

ℎ1198641

41205881198773

minus

1205874ℎ1198771198641

161198714120588

minus

131205872ℎ1198641120581

161198712120588119877

11988615

= minus

1205872ℎ31198642

2411987121205881198773

minus

1205872ℎ1198642

21198712120588119877

minus

91205874ℎ1198641120581

21198714120588

minus

3ℎ1198641

41205881198773

minus

1205872ℎ1198641120581

1198712120588119877

minus

31205874ℎ31198641

21198714120588119877

(A1)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to acknowledge financial support byNatural Science Foundation of China Project 11372071 andPostdoctoral Fund of China Project 2013M541339

References

[1] P D Spanos A M Chevallier N Politis and B Payne ldquoOilwell drilling a vibration perspectiverdquo The Shock and VibrationDigest vol 35 pp 81ndash99 2013

[2] W D Aldred andM C Sheppard ldquoDrillstring vibrations a newgeneration mechanism and control strategiesrdquo in Proceedingsof the SPE Annual Technical Conference and Exhibition SPE-24582-MS pp 353ndash363 Washington DC USA October 1992

[3] T V Aarrestand H A Tonnesen and A Kyllingstad ldquoDrill-string vibrations theory and experiments on full-scale researchdrilling rigrdquo inProceedings of the IADCSPEDrilling ConferenceSPE Paper No 14760 pp 311ndash321 Dallas Tex USA 1986

[4] J D Jansen ldquoNon-linear rotor dynamics as applied to oilwelldrillstring vibrationsrdquo Journal of Sound and Vibration vol 147no 1 pp 115ndash135 1991

[5] A S Yigit and A P Christoforou ldquoCoupled axial and transversevibrations of oilwell drillstringsrdquo Journal of Sound and Vibra-tion vol 195 no 4 pp 617ndash627 1996

[6] A S Yigit and A P Christoforou ldquoCoupled torsional and bend-ing vibration of drill-strings subject to impact with frictionrdquoJournal of Sound and Vibration vol 215 no 1 pp 167ndash181 1998

[7] K Z Khan Longitudinal and torsional vibration of drill-strings [MS thesis] Massachusetts Institute of TechnologyCambridge Mass USA 1986

[8] R J Shyu Bending of rotating drill-strings [PhD thesis] Mas-sachusetts Institute of Technology 1989

[9] A P Christoforou and A S Yigit ldquoFully coupled vibrations ofactively controlled drillstringsrdquo Journal of Sound and Vibrationvol 267 no 5 pp 1029ndash1045 2003

[10] M A Trindade C Wolter and R Sampaio ldquoKarhunen-Loevedecomposition of coupled axialbending vibrations of beamssubject to impactsrdquo Journal of Sound and Vibration vol 279 no3ndash5 pp 1015ndash1036 2005

[11] R Sampaio M T Piovan and G Venero Lozano ldquoCoupledaxialtorsional vibrations of drill-strings bymeans of non-linearmodelrdquo Mechanics Research Communications vol 34 no 5-6pp 497ndash502 2007

[12] H Hakimi and S Moradi ldquoDrillstring vibration analysis usingdifferential quadrature methodrdquo Journal of Petroleum Scienceand Engineering vol 70 no 3-4 pp 235ndash242 2010

[13] F S Ren S Chen and Z G Yao ldquoDynamics analysis andvibration suppression of a flexible rotation beamrdquo AppliedMechanics and Materials vol 214 pp 165ndash172 2012

[14] F-S Ren and Z-G Yao ldquoStudy on nonlinear dynamics andbifurcations in rotating compressive-drill stringrdquo EngineeringMechanics vol 30 no 10 pp 251ndash256 2013

[15] Z-F Li Y-G Zhang X-T Hou W-D Liu and G-Q XuldquoAnalysis of longitudinal and torsion vibration of drill-stringsrdquoEngineering Mechanics vol 21 no 6 pp 203ndash210 2004 (Chi-nese)

[16] G J Sheu and S M Yang ldquoDynamic analysis of a spinningRayleigh beamrdquo International Journal of Mechanical Sciencesvol 47 no 2 pp 157ndash169 2005

[17] X Zhou X Qichong D Hu et al ldquoThe nonlinear dynam-ics analysis of bottom drillstring of air drillingrdquo Journal ofChongqing University of Science and Technology vol 4 pp 119ndash121 2013 (Chinese)

[18] Y-J Jia P Jiang X-H Zhu and Y-F Zhang ldquoNonlinearbuckling simulation of drillstring system during pilot holedrilling in horizontal directional drillingrdquo Journal of SystemSimulation vol 25 no 4 pp 821ndash825 2013 (Chinese)

[19] W-S Xiao Z-Y Liu H-Y Wang X-F Wang L Fu and Y-C Yin ldquoTensional vibration analysis of drill-string by FSIrdquo OilField Equipment vol 42 pp 23ndash26 2013 (Chinese)

[20] D Shao Z Guan X Wen and Y Shi ldquoExperiment on lateralvibration characteristics of horizontal rotary drilling stringrdquoJournal of China University of Petroleum vol 37 pp 100ndash1052013 (Chinese)

[21] W Zhang X Zhu Z Zhou and W Ma ldquoEffect of rotationalspeeds mutation on the dynamic characteristics of the drill-string system in a vertical wellrdquo Journal of PetrochemicalUniversities vol 26 no 2 pp 47ndash51 2013 (Chinese)

[22] Y Tang ldquoNonlinear vibrations of axially accelerating viscoelas-tic Timoshenko beamsrdquo Chinese Journal of Theoretical andApplied Mechanics vol 45 no 6 pp 965ndash973 2013 (Chinese)

Shock and Vibration 17

[23] P B Goncalves and Z J G N Del Prado ldquoNonlinear oscilla-tions and stability of parametrically excited cylindrical shellsrdquoMeccanica vol 37 no 6 pp 569ndash597 2002

[24] M Strozzi and F Pellicano ldquoNonlinear vibrations of function-ally graded cylindrical shellsrdquo Thin-Walled Structures vol 67pp 63ndash77 2013

[25] M Amabili Nonlinear Vibrations and Stability of Shells andPlates Cambridge University Press Cambridge CambridgeUK 2008

[26] A W Leissa Vibrations of Shells Government Printing OfficeWashington DC USA 1973

[27] N Yamaki Elastic Stability of Circular Cylindrical Shells North-Holland Amsterdam The Netherlands 1984

[28] J N ReddyMechanics of LaminatedComposite Plates and ShellsCRC Press Boca Raton Fla USA 2004

[29] A H Nayfeh and D T Mook Nonlinear Oscillations Wiley-VCH Press Weinheim Germany 2004

[30] P D Spanos A K Sengupta R A Cunningham and P RPaslay ldquoModeling of roller cone bit lift-off dynamics in rotarydrillingrdquo Journal of Energy Resources Technology vol 117 no 3pp 197ndash207 1995

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Shock and Vibration

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DistributedSensor Networks

International Journal of

4 Shock and Vibration

120590120579120579119909=0

= 1205901205791205790

which account for the boundary conditions ofthe force

Using the Galerkin method to disperse (5a) and (5b)substituting (6) into (5a) and (5b) both sides of (5a) and (5b)are multiplied by the mode of the right side of (6) integratedin the circumferential and thickness direction Making useof modal orthogonality the motion equation of the middlesurface displacement is obtained as follows

1= 119886111986211+

119877ℎ1205901199091199090

2119871120588

minus

ℎ1205901205791205790

2120588119877

1199081minus 11988621199081minus 11988631199082

+ 119886411990811199082

2minus 11988651199083

1

(7a)

2= 119886111986222minus 11988661199081+

1

120588

ℎ1205901205791205790

119877

1199082+ 119886711990821199082

1minus 11988681199083

2 (7b)

where 1198861 1198862 1198863 1198864 1198865 1198866 1198867 and 119886

8are the constant

coefficients as listed in the Appendix From (7a) and (7b)we know that the vibration is a nonlinear coupled equationof axial and torsional vibration and that unilaterally takinginto account the single direction of the vibration is obviouslymuch too simplified to describe the actual vibrationThe axialstress as external excitation directly affects the dynamicsresponse The torsional stress 120590

1205791205790 as parameter excitation

affects the vibration by changing the axial stiffness therebyaffecting the natural frequency of the drill string Assumingthe axial and torsional excitation frequencies are integer timesof the rotation frequency of the drill string the equationitself will have two linear natural frequencies Due to theexistence of the coupling and nonlinear terms there isa great difference between the actual vibration frequencyand the linear natural frequency In addition subharmonicresonance superharmonic resonance bifurcation and chaosphenomena are possible in the vibration system with theparameters changing and the energy translating between themodals

3 Nonlinear Dynamics Average Equation

To facilitate the analysis replace the coefficients in (7a) and(7b) by 119886

119894 119887119895and use the dimensionless transformation as

follows (in the following dimensionless equation the asteriskwas removed for the convenience of expression)

119909lowast

1=

1199091

119871

119909lowast

2=

1199092

119871

119905lowast

=

119877

1198712radic

119864

120588

119905

120590lowast

1199091199090=

1205901199091199090

119864

120590lowast

1205791205790=

1205901205791205790

119864

(8)

The multiscale transformation is employed as follows

119886119894997888rarr 120576119886

119894

119887119895997888rarr 120576119887

119895

119894 = 1 119895 = 2

(9)

where 120576 is a small parameter used as a perturbation parameterfor determining the approximate solution for the motion ofthe drill string [22]

Substituting (9) into (7a) and (7b) one can obtain amotion equation that includes the small parameter 120576 Hereassume the form of the equation as follows

119909119894(119905 120576) = 119910

1198940(1198790 1198791) + 120576119910

119894(1198790 1198791) + sdot sdot sdot (119894 = 1 2) (10)

where 1198790= 119905 and 119879

1= 120576119905

Here the differential operators are employed as in thefollowing form

119889

119889119905

=

120597

1205971198790

1205971198790

120597119905

+

120597

1205971198791

1205971198791

120597119905

+ sdot sdot sdot = 1198630+ 1205761198631+ sdot sdot sdot (11a)

1198892

1198891199052

= (1198630+ 1205761198631+ sdot sdot sdot)2

= 1198632

0+ 2120576119863

01198631+ sdot sdot sdot (11b)

where 1198630= 120597120597119879

0and 119863

1= 120597120597119879

1

Taking into account the principal parameter resonanceand 2 1 internal resonance the resonance relationship can beexpressed as follows

1205962

1= 41205962+ 1205761205901

1205962

2= 1205962+ 1205761205902

(12)

where 1205961and 120596

2are the axial and torsional linear natural

frequencies respectivelyTaking into account the vibration response caused by

the axial and torsional excitation in the case of the aboveresonance the dimensionless axial and torsional excitationin the frequency domain can be expanded into the followingform including the two resonance frequencies [29]

1205901199091199090

= 1199021119890119894119905

+ 1199021119890minus119894119905

+ 11990221198902119894119905

+ 1199022119890minus119894119905

(13a)

1205901205791205790

= 1199023119890119894119905

+ 1199023119890minus119894119905

+ 11990241198902119894119905

+ 1199024119890minus119894119905

(13b)

where 1199021and 119902

2are the amplitude of the low frequency and

high frequency of the initial axial stress respectively and 1199023

and 1199024are the amplitude of the low and high frequencies of

the initial torsional stress respectivelySubstituting (8)ndash(12) into (7a) and (7b) and comparing

the same step coefficient of the small parameter 120576 on bothsides of the equations one obtains a differential equation inthe following form

For 1205760

1198632

011991010

+ 41199101= 0 (14a)

Shock and Vibration 5

For 1205761

1198632

011991020

+ 1199102= 0 (14b)

The plural form solutions of (14a) and (14b) can be expressedas

11991010

= 1198601(1198791) 1198902119894119905

+ 1198601(1198791) 119890minus2119894119905

(15a)

11991020

= 1198602(1198791) 119890119894119905

+ 1198602(1198792) 119890minus119894119905

(15b)

where 1198601and 119860

2are the conjugation of 119860

1and 119860

2 respec-

tively The plural solutions of 1198601and 119860

2are formed as

follows

1198601=

1

2

1199091+

1

2

1198941199092

1198602=

1

2

1199093+

1

2

1198941199094

(16)

where 1199091 1199092 1199093 and 119909

4are the projections of the vibration

vector on the complex plane which is equal to the vibrationvector described as the amplitude and the phase

Substituting (15a) and (15b) into (13a) and (13b) oneobtains themotion equationwith the small parameter 120576 thenmaking the long term equal to zero one obtains the averagingequation in the rectangular coordinate as follows

1= minus

1

2

11988691199091minus

1

8

11988610

1199092(1199092

3+ 1199092

4)

+

3

16

11988611

1199092(1199092

1+ 1199092

2)

(17a)

2= minus

1

2

11988691199092minus

1

8

11988610

1199091(1199092

3+ 1199092

4)

minus

3

16

11988611

1199091(1199092

1+ 1199092

2) minus

1

2

1199022

(17b)

3= minus

1

2

11988612

1199093+

1

2

11988613

1199094+

1

4

11988614

1199094(1199092

1+ 1199092

2)

+

3

8

11988615

1199094(1199092

3+ 1199092

4)

(17c)

4= minus

1

2

11988612

1199094minus

1

2

11988613

1199093minus

1

4

11988614

1199093(1199092

1+ 1199092

2)

minus

3

8

11988615

1199093(1199092

3+ 1199092

4) minus 1199023

(17d)

where 1198869 11988610 11988611 11988612 11988613 11988614 and 119886

15are the constant

coefficients listed in the AppendixThe change rules for the amplitude and phase angles in

the motion equation are found in (17a) (17b) (17c) and (17d)using the form of the differential equation We find that 119902

1

and 1199024have disappeared in (17a) (17b) (17c) and (17d) and

the parameters that can affect the dynamics response are only1199022and 1199023 We also know that 119902

2is the high-frequency item of

the initial axial excitation and 1199023is the low-frequency item of

the initial torsional excitation corresponding to the dynamicresponse of the drill string in low- and high-rotation speedareas respectively So we try to reveal relationships betweenthe two parameters with the coupling vibration of the drillstring by numerical simulation

4 Numerical Simulation andQualitative Analysis

In the present section the numerical simulation resultsobtained from using the proposed model are discussed forthe upper portion of the BHA The Runge-Kutta-Fehlbergmethod with adaptive steps is employed to perform thesimulations aiming at obtaining the dynamic response of thecoupled axial and torsional vibration in the case of resonanceThe geometric properties of the upper segment are the length119871 = 1050m themiddle diameter119877 = 01057m the thicknessℎ = 00171m Poissonrsquos ratio 120581 = 026 the elastic modulus119864 = 210GPa the density 120588 = 7850 kgm3 and the dampingcoefficients 119862

1and 119862

2 calculated from the considerations of

Spanos et al [30] The representative simulation results areshown along with the changes in 119902

2and 1199023

41 Torsional Excitation The Runge-Kutta-Fehlberg methodis employed to analyze the average of (17a) (17b) (17c) and(17d) and bifurcations of the system under exciting forces areobtained as shown in Figure 3

From the bifurcation along with the change in tor-sional excitation 119902

3 the response of the coupled axial

and torsional vibrations progresses through a cycle fromperiodic motion to doubling periodic motion to period-multiplying motion to quasiperiodic motion while exhibit-ing this unique phenomenon of nonlinear dynamics bifur-cation When the coupled vibration response is quasiperi-odic motion the amplitude of the drill string is obviouslyhigher than the period motion With increasing amplitudeof the excitation the responses change from quasiperiodicmotion to periodic motion the amplitude of the vibrationdoes not increase Instead it decreases to a certain extentwhich is different from the results of the linear analysismethod

By only increasing the torsional excitation until 1199023

=

724 (see Figure 3) and leaving other initial conditions andparameters the same this creates the periodic responsesshown in Figure 4

When 1199023

= 726 the amplitude of the system increasescorrespondingly accordinglyThe period-doubling responsesof system are shown in Figure 5

When 1199023

= 75 the phase diagram of the coupled vibra-tion changes the jumping phenomenon is more obviouspresent period-multiplying responses are shown in Figure 6

By increasing the value of 1199023so that 119902

3= 768 the qual-

itative nature of the system response changes presentingcorresponding quasiperiodic motions as shown in Figure 7This shows that new similar trajectories are derived from theoriginal trajectories of the phase diagram and they representmultifrequency vibration characteristics in the oscillogramand vibration frequencies that are close to the state ofcontinuous change Then the system vibration is in a statebetween the period vibration and the chaotic vibration qual-itatively the frequency bandwidth of the coupled vibrationalso increases the chaotic motions of the drill string aremorelikely to occur which leads to violent vibration of the drillstring

6 Shock and Vibration

x1

7 8 9 1110q3

12

14

16

18

(a) Bifurcation diagram of the axial modal

x3

10 118 97

q3

minus05

00

05

10

(b) Bifurcation diagram of the torsional modal

Figure 3 Bifurcations of the system under torsional excitation

It is well-known that the torsional vibration of the drillstring is mainly generated from the rotational speed changeof the bit when breaking rock intermittently The excitationfrequency is related to the rotation speed stiffness of thedrill string and characteristics of the rock The higher therock hardness the bigger the torsional stress and the greaterthe torsional amplitude We know from (17a) (17b) (17c)and (17d) that torsional stress strongly affects the couplingvibration when in low-frequency excitation but when inhigh-frequency excitation the torsional stress does not effec-tively contribute to the coupling vibration So in the practicaldrilling process when the drill string is in a high-rotationspeed zone the torsional stress will not lead to couplingvibration however when the drill string is in a low-rotationspeed zone more attention should be paid to the torsionalstress and appropriate drilling process parameters should beadopted to reduce the torsional stress especially when stick-slip vibration occurs This is necessary because at this timethe drill string is in low torsional excitation and large torsionalstress where it is more prone to produce a coupling vibrationof the drill string as suggested by Christoforou and Yigit [9]

Based on the above simulation analysis we also foundthat torsional excitation will affect both torsional and axialvibration simultaneously Torsional and axial vibration aresimilar qualitatively and in the nature of the vibration andare synchronous in form proving that energy can transferbetween the torsional and the axial vibration modal

42 Axial Excitation The above analysis accounts for theimpact of torsional excitation on the coupling of the axial andtorsional vibrations Here the impact of axial excitation willbe investigated

Thebifurcation diagram (Figure 8) shows the preliminaryresonance tendency The system undergoes a loop fromchaos to periodic motion then to period-three motionthen to period-multiplying and back to chaos showing

the nonlinear dynamic phenomenon of the period-doublingbifurcation and chaos

Increasing axial excitation until 1199022

= 05025 and keep-ing the remaining other initial conditions and parametersunchanged the system prevents period-doubling responsesas shown in Figure 9

When 1199022

= 0635 the system presents period-threeresponses as shown in Figure 10

When 1199022= 07725 the phase diagram changes drastically

and presents a jumping phenomenon then the systemresponds chaotically as shown in Figure 11

When 1199022

= 079 the qualitative nature of the couplingvibration of the drill string changes the present period-doubled responses are shown in Figure 12 Here the originalphase trajectory of the chaotic motion (Figure 11) contractsfor two trajectories and presents vibration characteristicsof frequency-doubling in the phase diagram The systemvibration turns from chaotic vibration into doubling periodicmotion and the stability of the system increases

As can be seen from the above qualitative analysis of theaxial excitation of the coupling vibration we find the samephenomenon as in the torsional excitation analysis for thecoupling vibration namely that axial excitation affects bothtorsional and axial vibration simultaneously and is similarqualitatively and also synchronous in form

Along with the increase in the axial excitation the systemcoupling vibration turns from period-doubling to period-three to chaotic and finally to period-doubling So the axialexcitation parameter leads to a coupling vibration in somespecial zone In the process of practical drilling the axialjump mainly contributes to the axial excitation The higherthe rotation speed the higher the excitation frequency Thelarger the WOB becomes the larger both the axial amplitudeand the axial stress will be We know from (17a) (17b) (17c)and (17d) that the axial excitation contributes to the couplingvibration when the drill string is in a high-rotation speedzone Therefore in the process of automation drilling with

Shock and Vibration 7

x2

minus3

minus2

minus1

1

0

2

3

minus2minus4 20x1

(a) Phase diagram of the axial modal

1020 10301010 1040 10501000t

minus4

minus3

minus2

minus1

0

1

2

x1

(b) Oscillogram of the axial modal

00 05 10minus10 minus05minus15

x3

minus15

minus10

minus05

00

05

10

x4

(c) Phase diagram of the torsional modal

1010 1020 1030 1040 10501000t

minus15

minus10

minus05

00

05

10x3

(d) Oscillogram of the torsional modal

5

0 02

0

1

minus1

minus2

x3

minus5

x2

minus4

minus2 x1

(e) Three-dimensional phase diagram

minus15

minus10

minus05

00

05

10

15

x2

minus10 minus05 00 05 10 15minus15

x1

(f) Poincare section

Figure 4 Period responses of the system (1199023= 724)

8 Shock and Vibration

x2

minus3

minus2

minus1

1

0

2

3

0minus2 2minus4

x1

(a) Phase diagram of the axial modal

1010 1020 1030 1040 10501000t

minus4

minus3

minus2

minus1

0

1

2

x1

(b) Oscillogram of the axial modal

minus10 minus05 00 05 10minus15

x3

minus15

minus10

minus05

00

05

10

x4

(c) Phase diagram of the torsional modal

1010 1020 1030 1040 10501000t

minus15

minus10

minus05

00

05

10x3

(d) Oscillogram of the torsional modal

5

00

2

0

1

minus1

minus2

x3

minus5

x2

minus4

minus2 x 1

(e) Three-dimensional phase diagram

0500 10 15minus10 minus05minus15

x1

minus15

minus10

minus05

00

05

10

15

x2

(f) Poincare section

Figure 5 Period responses of the system (1199023= 726)

Shock and Vibration 9

x2

minus2 0 2minus4

x1

minus3

minus2

minus1

1

0

2

3

(a) Phase diagram of the axial modal

minus4

minus3

minus2

minus1

0

1

2

x1

1010 1020 1030 1040 10501000t

(b) Oscillogram of the axial modal

minus10 minus05 00 05 10minus15

x3

minus15

minus10

minus05

00

05

10

x4

(c) Phase diagram of the torsional modal

minus15

minus10

minus05

00

05

10x3

1010 1020 1030 1040 10501000t

(d) Oscillogram of the torsional modal

5

0 02

0

1

minus1

minus2

x3

minus5

x2

minus4

minus2 x 1

(e) Three-dimensional phase diagram

minus15

minus10

minus05

00

05

10

15

x2

0500 10 15minus10 minus05minus15

x1

(f) Poincare section

Figure 6 Period responses of the system (1199023= 75)

10 Shock and Vibration

x2

minus2 0 2minus4

x1

minus3

minus2

minus1

1

0

2

3

(a) Phase diagram of the axial modal

1020 1040 1060 10801000t

minus4

minus3

minus2

minus1

0

1

2

x1

(b) Oscillogram of the axial modal

minus10 minus05 00 05 10minus15

x3

minus15

minus10

minus05

00

05

10

x4

(c) Phase diagram of the torsional modal

1020 1040 1060 10801000t

minus15

minus10

minus05

00

05

10x3

(d) Oscillogram of the torsional modal

5

00

2

0

1

minus1

minus2

x3

minus5

x2

minus4

minus2 x 1

(e) Three-dimensional phase diagram

minus10 minus05 00 05 10 15minus15

x1

minus15

minus10

minus05

00

05

10

15

x2

(f) Poincare section

Figure 7 The almost-periodic responses of the system (1199023= 768)

Shock and Vibration 11

06 07 08 09 1005q2

x2

minus10

minus05

00

(a) Bifurcation diagram of the axial modal

x3

06 07 0805 1009q2

10

15

20

(b) Bifurcation diagram of the torsional modal

Figure 8 Bifurcation of the resonance frequency-multiplier

a constant WOB in the case of a lower-rotation speed andlower rock hardness high drill efficiency can be attained byproperly increasing the WOB In the case of high-rotationspeed the coupling vibration is sensitive to the axial stressso the proper WOB should be adopted to avoid the couplingvibration of the drill string based on geological parameters

5 Conclusions

In this paper wemodeled and analyzed drill-string vibrationsby focusing on the coupled axialtorsional vibrations bymeans of nonlinear dynamics and qualitative analysis aimingat revealing the key effective factors influencing the coupledvibrations Here the drill string was described as a simplifiedequivalent flexible shell under axial rotation in which theexcitation loads and boundary conditions of the drill stringwere simplified After dimensionless processing we built thedynamics motion equation and the average equation

We found that the low-frequency amplitude expressionof the torsional excitation and the high-frequency amplitudeexpression of the axial excitation in the average equation arethe factors that determine the coupling vibration Based onthis procedure numerical simulations were carried out withadaptive steps using the Runge-Kutta-Fehlberg method todiscover the response of the system vibrations Further wefound that a change in the torsional or the axial excitationaffects both the torsional and the axial vibration simultane-ously and that it is similar qualitatively and synchronous inform

The results of the simulation analysis show that when thedrill string is in a lower-speed rotation zone the torsionalexcitation mainly contributes to the coupling vibrationincreasing the torsional stress of the drill string that moreeasily leads to the coupling vibrationWhen in a higher-speedrotating zone the axial excitation mainly contributes to thecoupling vibration so in a particular interval it is more likelyto cause the coupling vibration of the drill string

Appendix

Consider

1198861= minus

1

2119871120588120587

1198862=

1

120588

(

1

12

ℎ31198642

1198773

minus

ℎ1198642

119877

minus

1

4

1205872119877ℎ1198641

1198712

)

1198863=

1

120588

(

8

45

ℎ31198642

1198711198772

minus

32

15

ℎ1198642

119871

minus

32

15

ℎ1205811198641

119871

)

1198864=

1

120588

(

1

4

1205872ℎ31205811198641

11987131198773

minus

13

16

1205872ℎ1205811198641

1198712119877

minus

1

4

ℎ1198642

1198773

minus

1

16

1205874119877ℎ1198641

1198714

)

1198865=

1

120588

(

1

384

1205872ℎ31198642

11987121198773

minus

9

512

1205874119877ℎ1198641

1198714

minus

3

32

ℎ31198642

1198775

minus

9

32

ℎ1198642

1198773

minus

1

32

1205872ℎ1198641

1198712119877

minus

1

64

1205872ℎ1205811198641

1198712119877

)

1198866=

13ℎ1205811198641

15120588119871

1198867=

1

120588

(

1

4

1205872ℎ31205811198641

11987121198773

+

13

16

1205872ℎ1205811198641

1198712119877

minus

1

4

ℎ1198641

1198773

+

1

16

1205874119877ℎ1198641

1198714

+

3

4

ℎ1198641

1198713

+

1

24

1205872ℎ31198642

11987121198773

+

9

2

1205874119877ℎ1198641

1198714

)

1198868=

1

120588

(

1

2

1205872ℎ1198642

1198712119877

+

3

2

1205874ℎ31198641

1198714119877

+

1205872ℎ1205811198641

1198712119877

+

3

4

ℎ1198641

1198713

12 Shock and Vibrationx2

minus10

minus05

00

05

10

minus05 00 05 10minus10

x1

(a) Phase diagram of the axial modal

minus10

minus05

00

05

10

x1

1020 1040 1060 10801000t

(b) Oscillogram of the axial modal

minus2

minus1

0

1

2

x4

15 20 25 3010x3

(c) Phase diagram of the torsional modal

10

15

20

25

30x3

1020 1040 1060 10801000t

(d) Oscillogram of the torsional modal

1

0 0

11

2

3

minus1minus1

x3

x2 x 1

(e) Three-dimensional phase diagram

minus15

minus10

minus05

00

05

10

15

x2

0500 10 15minus10 minus05minus15

x1

(f) Poincare section

Figure 9 Period responses of the system (1199022= 05025)

Shock and Vibration 13x2

minus05 00 05 10minus10

x1

minus10

minus05

00

05

10

(a) Phase diagram of the axial modal

1020 1040 1060 10801000t

minus10

minus05

00

05

10

x1

(b) Oscillogram of the axial modal

minus20

minus15

minus10

minus05

00

05

10

15

20

x4

15 20 25 3010x3

(c) Phase diagram of the torsional modal

10

15

20

25

30x3

1020 1040 1060 10801000t

(d) Oscillogram of the torsional modal

1

0 0

11

2

3

minus1 minus1

x3

x2 x 1

(e) Three-dimensional phase diagram

minus10 minus05minus15 05 10 1500x1

minus15

minus10

minus05

00

05

10

15

x2

(f) Poincare section

Figure 10 Period responses of the system (1199022= 0635)

14 Shock and Vibration

minus2

minus1

0

1

2

x2

10 2 3minus1minus2

x1

(a) Phase diagram of the axial modal

1050 1100 1150 12001000t

minus2

minus1

0

1

2

3

x1

(b) Oscillogram of the axial modal

2 4minus2 0minus4

x3

minus4

minus2

0

2

4

x4

(c) Phase diagram of the torsional modal

minus4

minus2

0

2

4x3

1050 1100 1150 12001000t

(d) Oscillogram of the torsional modal

2

00

2

4

0

5

minus2

x3

minus5

x2

minus2

x 1

(e) Three-dimensional phase diagram

minus15

minus10

minus05

00

05

10

15

x2

0500 10 15minus10 minus05minus15

x1

(f) Poincare section

Figure 11 Period responses of the system (1199022= 07725)

Shock and Vibration 15x2

minus05 00 05 10minus10

x1

minus10

minus05

00

05

10

(a) Phase diagram of the axial modal

1020 1040 1060 10801000t

minus10

minus05

00

05

10

x1

(b) Oscillogram of the axial modal

15 20 25 3010x3

minus2

minus1

0

1

2

x4

(c) Phase diagram of the torsional modal

10

15

20

25

30x3

1020 1040 1060 10801000t

(d) Oscillogram of the torsional modal

1

0 0

11

2

3

minus1minus1

x3

x2

x 1

(e) Three-dimensional phase diagram

minus10 minus05minus15 05 10 1500x1

minus15

minus10

minus05

00

05

10

15

x2

(f) Poincare section

Figure 12 Period-doubled responses of the system (1199022= 079)

16 Shock and Vibration

+

1

24

1205872ℎ31198642

11987121198773

+

9

2

1205874119877ℎ1198641

1198714

)

1198869=

1198621

2120587119877radic120588119864

11988610

=

1205872ℎ31198641120581

411987121205881198772

minus

ℎ1198641

41205881198772

minus

1205874ℎ1198771198641

161198714120588

minus

1205872ℎ1198641120581

161198712120588119877

11988611

= minus

91205874ℎ1198771198641

5121198714120588

minus

3ℎ31198641

321205881198775

minus

1205872ℎ1198642

321198712120588119877

minus

1205872ℎ31198642

38411987121205881198773

minus

9ℎ1198641

321205881198773

minus

1205872ℎ1198641120581

641198712120588119877

11988612

=

1198622

2120587119877radic120588119864

11988613

= minus

41205872119877ℎ1198642

1198712120588

minus

ℎ1198641

119877120588

minus

1205872ℎ31198642

31198712120588119877

11988614

=

1205872ℎ31198641120581

411987121205881198773

minus

ℎ1198641

41205881198773

minus

1205874ℎ1198771198641

161198714120588

minus

131205872ℎ1198641120581

161198712120588119877

11988615

= minus

1205872ℎ31198642

2411987121205881198773

minus

1205872ℎ1198642

21198712120588119877

minus

91205874ℎ1198641120581

21198714120588

minus

3ℎ1198641

41205881198773

minus

1205872ℎ1198641120581

1198712120588119877

minus

31205874ℎ31198641

21198714120588119877

(A1)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to acknowledge financial support byNatural Science Foundation of China Project 11372071 andPostdoctoral Fund of China Project 2013M541339

References

[1] P D Spanos A M Chevallier N Politis and B Payne ldquoOilwell drilling a vibration perspectiverdquo The Shock and VibrationDigest vol 35 pp 81ndash99 2013

[2] W D Aldred andM C Sheppard ldquoDrillstring vibrations a newgeneration mechanism and control strategiesrdquo in Proceedingsof the SPE Annual Technical Conference and Exhibition SPE-24582-MS pp 353ndash363 Washington DC USA October 1992

[3] T V Aarrestand H A Tonnesen and A Kyllingstad ldquoDrill-string vibrations theory and experiments on full-scale researchdrilling rigrdquo inProceedings of the IADCSPEDrilling ConferenceSPE Paper No 14760 pp 311ndash321 Dallas Tex USA 1986

[4] J D Jansen ldquoNon-linear rotor dynamics as applied to oilwelldrillstring vibrationsrdquo Journal of Sound and Vibration vol 147no 1 pp 115ndash135 1991

[5] A S Yigit and A P Christoforou ldquoCoupled axial and transversevibrations of oilwell drillstringsrdquo Journal of Sound and Vibra-tion vol 195 no 4 pp 617ndash627 1996

[6] A S Yigit and A P Christoforou ldquoCoupled torsional and bend-ing vibration of drill-strings subject to impact with frictionrdquoJournal of Sound and Vibration vol 215 no 1 pp 167ndash181 1998

[7] K Z Khan Longitudinal and torsional vibration of drill-strings [MS thesis] Massachusetts Institute of TechnologyCambridge Mass USA 1986

[8] R J Shyu Bending of rotating drill-strings [PhD thesis] Mas-sachusetts Institute of Technology 1989

[9] A P Christoforou and A S Yigit ldquoFully coupled vibrations ofactively controlled drillstringsrdquo Journal of Sound and Vibrationvol 267 no 5 pp 1029ndash1045 2003

[10] M A Trindade C Wolter and R Sampaio ldquoKarhunen-Loevedecomposition of coupled axialbending vibrations of beamssubject to impactsrdquo Journal of Sound and Vibration vol 279 no3ndash5 pp 1015ndash1036 2005

[11] R Sampaio M T Piovan and G Venero Lozano ldquoCoupledaxialtorsional vibrations of drill-strings bymeans of non-linearmodelrdquo Mechanics Research Communications vol 34 no 5-6pp 497ndash502 2007

[12] H Hakimi and S Moradi ldquoDrillstring vibration analysis usingdifferential quadrature methodrdquo Journal of Petroleum Scienceand Engineering vol 70 no 3-4 pp 235ndash242 2010

[13] F S Ren S Chen and Z G Yao ldquoDynamics analysis andvibration suppression of a flexible rotation beamrdquo AppliedMechanics and Materials vol 214 pp 165ndash172 2012

[14] F-S Ren and Z-G Yao ldquoStudy on nonlinear dynamics andbifurcations in rotating compressive-drill stringrdquo EngineeringMechanics vol 30 no 10 pp 251ndash256 2013

[15] Z-F Li Y-G Zhang X-T Hou W-D Liu and G-Q XuldquoAnalysis of longitudinal and torsion vibration of drill-stringsrdquoEngineering Mechanics vol 21 no 6 pp 203ndash210 2004 (Chi-nese)

[16] G J Sheu and S M Yang ldquoDynamic analysis of a spinningRayleigh beamrdquo International Journal of Mechanical Sciencesvol 47 no 2 pp 157ndash169 2005

[17] X Zhou X Qichong D Hu et al ldquoThe nonlinear dynam-ics analysis of bottom drillstring of air drillingrdquo Journal ofChongqing University of Science and Technology vol 4 pp 119ndash121 2013 (Chinese)

[18] Y-J Jia P Jiang X-H Zhu and Y-F Zhang ldquoNonlinearbuckling simulation of drillstring system during pilot holedrilling in horizontal directional drillingrdquo Journal of SystemSimulation vol 25 no 4 pp 821ndash825 2013 (Chinese)

[19] W-S Xiao Z-Y Liu H-Y Wang X-F Wang L Fu and Y-C Yin ldquoTensional vibration analysis of drill-string by FSIrdquo OilField Equipment vol 42 pp 23ndash26 2013 (Chinese)

[20] D Shao Z Guan X Wen and Y Shi ldquoExperiment on lateralvibration characteristics of horizontal rotary drilling stringrdquoJournal of China University of Petroleum vol 37 pp 100ndash1052013 (Chinese)

[21] W Zhang X Zhu Z Zhou and W Ma ldquoEffect of rotationalspeeds mutation on the dynamic characteristics of the drill-string system in a vertical wellrdquo Journal of PetrochemicalUniversities vol 26 no 2 pp 47ndash51 2013 (Chinese)

[22] Y Tang ldquoNonlinear vibrations of axially accelerating viscoelas-tic Timoshenko beamsrdquo Chinese Journal of Theoretical andApplied Mechanics vol 45 no 6 pp 965ndash973 2013 (Chinese)

Shock and Vibration 17

[23] P B Goncalves and Z J G N Del Prado ldquoNonlinear oscilla-tions and stability of parametrically excited cylindrical shellsrdquoMeccanica vol 37 no 6 pp 569ndash597 2002

[24] M Strozzi and F Pellicano ldquoNonlinear vibrations of function-ally graded cylindrical shellsrdquo Thin-Walled Structures vol 67pp 63ndash77 2013

[25] M Amabili Nonlinear Vibrations and Stability of Shells andPlates Cambridge University Press Cambridge CambridgeUK 2008

[26] A W Leissa Vibrations of Shells Government Printing OfficeWashington DC USA 1973

[27] N Yamaki Elastic Stability of Circular Cylindrical Shells North-Holland Amsterdam The Netherlands 1984

[28] J N ReddyMechanics of LaminatedComposite Plates and ShellsCRC Press Boca Raton Fla USA 2004

[29] A H Nayfeh and D T Mook Nonlinear Oscillations Wiley-VCH Press Weinheim Germany 2004

[30] P D Spanos A K Sengupta R A Cunningham and P RPaslay ldquoModeling of roller cone bit lift-off dynamics in rotarydrillingrdquo Journal of Energy Resources Technology vol 117 no 3pp 197ndash207 1995

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Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

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DistributedSensor Networks

International Journal of

Shock and Vibration 5

For 1205761

1198632

011991020

+ 1199102= 0 (14b)

The plural form solutions of (14a) and (14b) can be expressedas

11991010

= 1198601(1198791) 1198902119894119905

+ 1198601(1198791) 119890minus2119894119905

(15a)

11991020

= 1198602(1198791) 119890119894119905

+ 1198602(1198792) 119890minus119894119905

(15b)

where 1198601and 119860

2are the conjugation of 119860

1and 119860

2 respec-

tively The plural solutions of 1198601and 119860

2are formed as

follows

1198601=

1

2

1199091+

1

2

1198941199092

1198602=

1

2

1199093+

1

2

1198941199094

(16)

where 1199091 1199092 1199093 and 119909

4are the projections of the vibration

vector on the complex plane which is equal to the vibrationvector described as the amplitude and the phase

Substituting (15a) and (15b) into (13a) and (13b) oneobtains themotion equationwith the small parameter 120576 thenmaking the long term equal to zero one obtains the averagingequation in the rectangular coordinate as follows

1= minus

1

2

11988691199091minus

1

8

11988610

1199092(1199092

3+ 1199092

4)

+

3

16

11988611

1199092(1199092

1+ 1199092

2)

(17a)

2= minus

1

2

11988691199092minus

1

8

11988610

1199091(1199092

3+ 1199092

4)

minus

3

16

11988611

1199091(1199092

1+ 1199092

2) minus

1

2

1199022

(17b)

3= minus

1

2

11988612

1199093+

1

2

11988613

1199094+

1

4

11988614

1199094(1199092

1+ 1199092

2)

+

3

8

11988615

1199094(1199092

3+ 1199092

4)

(17c)

4= minus

1

2

11988612

1199094minus

1

2

11988613

1199093minus

1

4

11988614

1199093(1199092

1+ 1199092

2)

minus

3

8

11988615

1199093(1199092

3+ 1199092

4) minus 1199023

(17d)

where 1198869 11988610 11988611 11988612 11988613 11988614 and 119886

15are the constant

coefficients listed in the AppendixThe change rules for the amplitude and phase angles in

the motion equation are found in (17a) (17b) (17c) and (17d)using the form of the differential equation We find that 119902

1

and 1199024have disappeared in (17a) (17b) (17c) and (17d) and

the parameters that can affect the dynamics response are only1199022and 1199023 We also know that 119902

2is the high-frequency item of

the initial axial excitation and 1199023is the low-frequency item of

the initial torsional excitation corresponding to the dynamicresponse of the drill string in low- and high-rotation speedareas respectively So we try to reveal relationships betweenthe two parameters with the coupling vibration of the drillstring by numerical simulation

4 Numerical Simulation andQualitative Analysis

In the present section the numerical simulation resultsobtained from using the proposed model are discussed forthe upper portion of the BHA The Runge-Kutta-Fehlbergmethod with adaptive steps is employed to perform thesimulations aiming at obtaining the dynamic response of thecoupled axial and torsional vibration in the case of resonanceThe geometric properties of the upper segment are the length119871 = 1050m themiddle diameter119877 = 01057m the thicknessℎ = 00171m Poissonrsquos ratio 120581 = 026 the elastic modulus119864 = 210GPa the density 120588 = 7850 kgm3 and the dampingcoefficients 119862

1and 119862

2 calculated from the considerations of

Spanos et al [30] The representative simulation results areshown along with the changes in 119902

2and 1199023

41 Torsional Excitation The Runge-Kutta-Fehlberg methodis employed to analyze the average of (17a) (17b) (17c) and(17d) and bifurcations of the system under exciting forces areobtained as shown in Figure 3

From the bifurcation along with the change in tor-sional excitation 119902

3 the response of the coupled axial

and torsional vibrations progresses through a cycle fromperiodic motion to doubling periodic motion to period-multiplying motion to quasiperiodic motion while exhibit-ing this unique phenomenon of nonlinear dynamics bifur-cation When the coupled vibration response is quasiperi-odic motion the amplitude of the drill string is obviouslyhigher than the period motion With increasing amplitudeof the excitation the responses change from quasiperiodicmotion to periodic motion the amplitude of the vibrationdoes not increase Instead it decreases to a certain extentwhich is different from the results of the linear analysismethod

By only increasing the torsional excitation until 1199023

=

724 (see Figure 3) and leaving other initial conditions andparameters the same this creates the periodic responsesshown in Figure 4

When 1199023

= 726 the amplitude of the system increasescorrespondingly accordinglyThe period-doubling responsesof system are shown in Figure 5

When 1199023

= 75 the phase diagram of the coupled vibra-tion changes the jumping phenomenon is more obviouspresent period-multiplying responses are shown in Figure 6

By increasing the value of 1199023so that 119902

3= 768 the qual-

itative nature of the system response changes presentingcorresponding quasiperiodic motions as shown in Figure 7This shows that new similar trajectories are derived from theoriginal trajectories of the phase diagram and they representmultifrequency vibration characteristics in the oscillogramand vibration frequencies that are close to the state ofcontinuous change Then the system vibration is in a statebetween the period vibration and the chaotic vibration qual-itatively the frequency bandwidth of the coupled vibrationalso increases the chaotic motions of the drill string aremorelikely to occur which leads to violent vibration of the drillstring

6 Shock and Vibration

x1

7 8 9 1110q3

12

14

16

18

(a) Bifurcation diagram of the axial modal

x3

10 118 97

q3

minus05

00

05

10

(b) Bifurcation diagram of the torsional modal

Figure 3 Bifurcations of the system under torsional excitation

It is well-known that the torsional vibration of the drillstring is mainly generated from the rotational speed changeof the bit when breaking rock intermittently The excitationfrequency is related to the rotation speed stiffness of thedrill string and characteristics of the rock The higher therock hardness the bigger the torsional stress and the greaterthe torsional amplitude We know from (17a) (17b) (17c)and (17d) that torsional stress strongly affects the couplingvibration when in low-frequency excitation but when inhigh-frequency excitation the torsional stress does not effec-tively contribute to the coupling vibration So in the practicaldrilling process when the drill string is in a high-rotationspeed zone the torsional stress will not lead to couplingvibration however when the drill string is in a low-rotationspeed zone more attention should be paid to the torsionalstress and appropriate drilling process parameters should beadopted to reduce the torsional stress especially when stick-slip vibration occurs This is necessary because at this timethe drill string is in low torsional excitation and large torsionalstress where it is more prone to produce a coupling vibrationof the drill string as suggested by Christoforou and Yigit [9]

Based on the above simulation analysis we also foundthat torsional excitation will affect both torsional and axialvibration simultaneously Torsional and axial vibration aresimilar qualitatively and in the nature of the vibration andare synchronous in form proving that energy can transferbetween the torsional and the axial vibration modal

42 Axial Excitation The above analysis accounts for theimpact of torsional excitation on the coupling of the axial andtorsional vibrations Here the impact of axial excitation willbe investigated

Thebifurcation diagram (Figure 8) shows the preliminaryresonance tendency The system undergoes a loop fromchaos to periodic motion then to period-three motionthen to period-multiplying and back to chaos showing

the nonlinear dynamic phenomenon of the period-doublingbifurcation and chaos

Increasing axial excitation until 1199022

= 05025 and keep-ing the remaining other initial conditions and parametersunchanged the system prevents period-doubling responsesas shown in Figure 9

When 1199022

= 0635 the system presents period-threeresponses as shown in Figure 10

When 1199022= 07725 the phase diagram changes drastically

and presents a jumping phenomenon then the systemresponds chaotically as shown in Figure 11

When 1199022

= 079 the qualitative nature of the couplingvibration of the drill string changes the present period-doubled responses are shown in Figure 12 Here the originalphase trajectory of the chaotic motion (Figure 11) contractsfor two trajectories and presents vibration characteristicsof frequency-doubling in the phase diagram The systemvibration turns from chaotic vibration into doubling periodicmotion and the stability of the system increases

As can be seen from the above qualitative analysis of theaxial excitation of the coupling vibration we find the samephenomenon as in the torsional excitation analysis for thecoupling vibration namely that axial excitation affects bothtorsional and axial vibration simultaneously and is similarqualitatively and also synchronous in form

Along with the increase in the axial excitation the systemcoupling vibration turns from period-doubling to period-three to chaotic and finally to period-doubling So the axialexcitation parameter leads to a coupling vibration in somespecial zone In the process of practical drilling the axialjump mainly contributes to the axial excitation The higherthe rotation speed the higher the excitation frequency Thelarger the WOB becomes the larger both the axial amplitudeand the axial stress will be We know from (17a) (17b) (17c)and (17d) that the axial excitation contributes to the couplingvibration when the drill string is in a high-rotation speedzone Therefore in the process of automation drilling with

Shock and Vibration 7

x2

minus3

minus2

minus1

1

0

2

3

minus2minus4 20x1

(a) Phase diagram of the axial modal

1020 10301010 1040 10501000t

minus4

minus3

minus2

minus1

0

1

2

x1

(b) Oscillogram of the axial modal

00 05 10minus10 minus05minus15

x3

minus15

minus10

minus05

00

05

10

x4

(c) Phase diagram of the torsional modal

1010 1020 1030 1040 10501000t

minus15

minus10

minus05

00

05

10x3

(d) Oscillogram of the torsional modal

5

0 02

0

1

minus1

minus2

x3

minus5

x2

minus4

minus2 x1

(e) Three-dimensional phase diagram

minus15

minus10

minus05

00

05

10

15

x2

minus10 minus05 00 05 10 15minus15

x1

(f) Poincare section

Figure 4 Period responses of the system (1199023= 724)

8 Shock and Vibration

x2

minus3

minus2

minus1

1

0

2

3

0minus2 2minus4

x1

(a) Phase diagram of the axial modal

1010 1020 1030 1040 10501000t

minus4

minus3

minus2

minus1

0

1

2

x1

(b) Oscillogram of the axial modal

minus10 minus05 00 05 10minus15

x3

minus15

minus10

minus05

00

05

10

x4

(c) Phase diagram of the torsional modal

1010 1020 1030 1040 10501000t

minus15

minus10

minus05

00

05

10x3

(d) Oscillogram of the torsional modal

5

00

2

0

1

minus1

minus2

x3

minus5

x2

minus4

minus2 x 1

(e) Three-dimensional phase diagram

0500 10 15minus10 minus05minus15

x1

minus15

minus10

minus05

00

05

10

15

x2

(f) Poincare section

Figure 5 Period responses of the system (1199023= 726)

Shock and Vibration 9

x2

minus2 0 2minus4

x1

minus3

minus2

minus1

1

0

2

3

(a) Phase diagram of the axial modal

minus4

minus3

minus2

minus1

0

1

2

x1

1010 1020 1030 1040 10501000t

(b) Oscillogram of the axial modal

minus10 minus05 00 05 10minus15

x3

minus15

minus10

minus05

00

05

10

x4

(c) Phase diagram of the torsional modal

minus15

minus10

minus05

00

05

10x3

1010 1020 1030 1040 10501000t

(d) Oscillogram of the torsional modal

5

0 02

0

1

minus1

minus2

x3

minus5

x2

minus4

minus2 x 1

(e) Three-dimensional phase diagram

minus15

minus10

minus05

00

05

10

15

x2

0500 10 15minus10 minus05minus15

x1

(f) Poincare section

Figure 6 Period responses of the system (1199023= 75)

10 Shock and Vibration

x2

minus2 0 2minus4

x1

minus3

minus2

minus1

1

0

2

3

(a) Phase diagram of the axial modal

1020 1040 1060 10801000t

minus4

minus3

minus2

minus1

0

1

2

x1

(b) Oscillogram of the axial modal

minus10 minus05 00 05 10minus15

x3

minus15

minus10

minus05

00

05

10

x4

(c) Phase diagram of the torsional modal

1020 1040 1060 10801000t

minus15

minus10

minus05

00

05

10x3

(d) Oscillogram of the torsional modal

5

00

2

0

1

minus1

minus2

x3

minus5

x2

minus4

minus2 x 1

(e) Three-dimensional phase diagram

minus10 minus05 00 05 10 15minus15

x1

minus15

minus10

minus05

00

05

10

15

x2

(f) Poincare section

Figure 7 The almost-periodic responses of the system (1199023= 768)

Shock and Vibration 11

06 07 08 09 1005q2

x2

minus10

minus05

00

(a) Bifurcation diagram of the axial modal

x3

06 07 0805 1009q2

10

15

20

(b) Bifurcation diagram of the torsional modal

Figure 8 Bifurcation of the resonance frequency-multiplier

a constant WOB in the case of a lower-rotation speed andlower rock hardness high drill efficiency can be attained byproperly increasing the WOB In the case of high-rotationspeed the coupling vibration is sensitive to the axial stressso the proper WOB should be adopted to avoid the couplingvibration of the drill string based on geological parameters

5 Conclusions

In this paper wemodeled and analyzed drill-string vibrationsby focusing on the coupled axialtorsional vibrations bymeans of nonlinear dynamics and qualitative analysis aimingat revealing the key effective factors influencing the coupledvibrations Here the drill string was described as a simplifiedequivalent flexible shell under axial rotation in which theexcitation loads and boundary conditions of the drill stringwere simplified After dimensionless processing we built thedynamics motion equation and the average equation

We found that the low-frequency amplitude expressionof the torsional excitation and the high-frequency amplitudeexpression of the axial excitation in the average equation arethe factors that determine the coupling vibration Based onthis procedure numerical simulations were carried out withadaptive steps using the Runge-Kutta-Fehlberg method todiscover the response of the system vibrations Further wefound that a change in the torsional or the axial excitationaffects both the torsional and the axial vibration simultane-ously and that it is similar qualitatively and synchronous inform

The results of the simulation analysis show that when thedrill string is in a lower-speed rotation zone the torsionalexcitation mainly contributes to the coupling vibrationincreasing the torsional stress of the drill string that moreeasily leads to the coupling vibrationWhen in a higher-speedrotating zone the axial excitation mainly contributes to thecoupling vibration so in a particular interval it is more likelyto cause the coupling vibration of the drill string

Appendix

Consider

1198861= minus

1

2119871120588120587

1198862=

1

120588

(

1

12

ℎ31198642

1198773

minus

ℎ1198642

119877

minus

1

4

1205872119877ℎ1198641

1198712

)

1198863=

1

120588

(

8

45

ℎ31198642

1198711198772

minus

32

15

ℎ1198642

119871

minus

32

15

ℎ1205811198641

119871

)

1198864=

1

120588

(

1

4

1205872ℎ31205811198641

11987131198773

minus

13

16

1205872ℎ1205811198641

1198712119877

minus

1

4

ℎ1198642

1198773

minus

1

16

1205874119877ℎ1198641

1198714

)

1198865=

1

120588

(

1

384

1205872ℎ31198642

11987121198773

minus

9

512

1205874119877ℎ1198641

1198714

minus

3

32

ℎ31198642

1198775

minus

9

32

ℎ1198642

1198773

minus

1

32

1205872ℎ1198641

1198712119877

minus

1

64

1205872ℎ1205811198641

1198712119877

)

1198866=

13ℎ1205811198641

15120588119871

1198867=

1

120588

(

1

4

1205872ℎ31205811198641

11987121198773

+

13

16

1205872ℎ1205811198641

1198712119877

minus

1

4

ℎ1198641

1198773

+

1

16

1205874119877ℎ1198641

1198714

+

3

4

ℎ1198641

1198713

+

1

24

1205872ℎ31198642

11987121198773

+

9

2

1205874119877ℎ1198641

1198714

)

1198868=

1

120588

(

1

2

1205872ℎ1198642

1198712119877

+

3

2

1205874ℎ31198641

1198714119877

+

1205872ℎ1205811198641

1198712119877

+

3

4

ℎ1198641

1198713

12 Shock and Vibrationx2

minus10

minus05

00

05

10

minus05 00 05 10minus10

x1

(a) Phase diagram of the axial modal

minus10

minus05

00

05

10

x1

1020 1040 1060 10801000t

(b) Oscillogram of the axial modal

minus2

minus1

0

1

2

x4

15 20 25 3010x3

(c) Phase diagram of the torsional modal

10

15

20

25

30x3

1020 1040 1060 10801000t

(d) Oscillogram of the torsional modal

1

0 0

11

2

3

minus1minus1

x3

x2 x 1

(e) Three-dimensional phase diagram

minus15

minus10

minus05

00

05

10

15

x2

0500 10 15minus10 minus05minus15

x1

(f) Poincare section

Figure 9 Period responses of the system (1199022= 05025)

Shock and Vibration 13x2

minus05 00 05 10minus10

x1

minus10

minus05

00

05

10

(a) Phase diagram of the axial modal

1020 1040 1060 10801000t

minus10

minus05

00

05

10

x1

(b) Oscillogram of the axial modal

minus20

minus15

minus10

minus05

00

05

10

15

20

x4

15 20 25 3010x3

(c) Phase diagram of the torsional modal

10

15

20

25

30x3

1020 1040 1060 10801000t

(d) Oscillogram of the torsional modal

1

0 0

11

2

3

minus1 minus1

x3

x2 x 1

(e) Three-dimensional phase diagram

minus10 minus05minus15 05 10 1500x1

minus15

minus10

minus05

00

05

10

15

x2

(f) Poincare section

Figure 10 Period responses of the system (1199022= 0635)

14 Shock and Vibration

minus2

minus1

0

1

2

x2

10 2 3minus1minus2

x1

(a) Phase diagram of the axial modal

1050 1100 1150 12001000t

minus2

minus1

0

1

2

3

x1

(b) Oscillogram of the axial modal

2 4minus2 0minus4

x3

minus4

minus2

0

2

4

x4

(c) Phase diagram of the torsional modal

minus4

minus2

0

2

4x3

1050 1100 1150 12001000t

(d) Oscillogram of the torsional modal

2

00

2

4

0

5

minus2

x3

minus5

x2

minus2

x 1

(e) Three-dimensional phase diagram

minus15

minus10

minus05

00

05

10

15

x2

0500 10 15minus10 minus05minus15

x1

(f) Poincare section

Figure 11 Period responses of the system (1199022= 07725)

Shock and Vibration 15x2

minus05 00 05 10minus10

x1

minus10

minus05

00

05

10

(a) Phase diagram of the axial modal

1020 1040 1060 10801000t

minus10

minus05

00

05

10

x1

(b) Oscillogram of the axial modal

15 20 25 3010x3

minus2

minus1

0

1

2

x4

(c) Phase diagram of the torsional modal

10

15

20

25

30x3

1020 1040 1060 10801000t

(d) Oscillogram of the torsional modal

1

0 0

11

2

3

minus1minus1

x3

x2

x 1

(e) Three-dimensional phase diagram

minus10 minus05minus15 05 10 1500x1

minus15

minus10

minus05

00

05

10

15

x2

(f) Poincare section

Figure 12 Period-doubled responses of the system (1199022= 079)

16 Shock and Vibration

+

1

24

1205872ℎ31198642

11987121198773

+

9

2

1205874119877ℎ1198641

1198714

)

1198869=

1198621

2120587119877radic120588119864

11988610

=

1205872ℎ31198641120581

411987121205881198772

minus

ℎ1198641

41205881198772

minus

1205874ℎ1198771198641

161198714120588

minus

1205872ℎ1198641120581

161198712120588119877

11988611

= minus

91205874ℎ1198771198641

5121198714120588

minus

3ℎ31198641

321205881198775

minus

1205872ℎ1198642

321198712120588119877

minus

1205872ℎ31198642

38411987121205881198773

minus

9ℎ1198641

321205881198773

minus

1205872ℎ1198641120581

641198712120588119877

11988612

=

1198622

2120587119877radic120588119864

11988613

= minus

41205872119877ℎ1198642

1198712120588

minus

ℎ1198641

119877120588

minus

1205872ℎ31198642

31198712120588119877

11988614

=

1205872ℎ31198641120581

411987121205881198773

minus

ℎ1198641

41205881198773

minus

1205874ℎ1198771198641

161198714120588

minus

131205872ℎ1198641120581

161198712120588119877

11988615

= minus

1205872ℎ31198642

2411987121205881198773

minus

1205872ℎ1198642

21198712120588119877

minus

91205874ℎ1198641120581

21198714120588

minus

3ℎ1198641

41205881198773

minus

1205872ℎ1198641120581

1198712120588119877

minus

31205874ℎ31198641

21198714120588119877

(A1)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to acknowledge financial support byNatural Science Foundation of China Project 11372071 andPostdoctoral Fund of China Project 2013M541339

References

[1] P D Spanos A M Chevallier N Politis and B Payne ldquoOilwell drilling a vibration perspectiverdquo The Shock and VibrationDigest vol 35 pp 81ndash99 2013

[2] W D Aldred andM C Sheppard ldquoDrillstring vibrations a newgeneration mechanism and control strategiesrdquo in Proceedingsof the SPE Annual Technical Conference and Exhibition SPE-24582-MS pp 353ndash363 Washington DC USA October 1992

[3] T V Aarrestand H A Tonnesen and A Kyllingstad ldquoDrill-string vibrations theory and experiments on full-scale researchdrilling rigrdquo inProceedings of the IADCSPEDrilling ConferenceSPE Paper No 14760 pp 311ndash321 Dallas Tex USA 1986

[4] J D Jansen ldquoNon-linear rotor dynamics as applied to oilwelldrillstring vibrationsrdquo Journal of Sound and Vibration vol 147no 1 pp 115ndash135 1991

[5] A S Yigit and A P Christoforou ldquoCoupled axial and transversevibrations of oilwell drillstringsrdquo Journal of Sound and Vibra-tion vol 195 no 4 pp 617ndash627 1996

[6] A S Yigit and A P Christoforou ldquoCoupled torsional and bend-ing vibration of drill-strings subject to impact with frictionrdquoJournal of Sound and Vibration vol 215 no 1 pp 167ndash181 1998

[7] K Z Khan Longitudinal and torsional vibration of drill-strings [MS thesis] Massachusetts Institute of TechnologyCambridge Mass USA 1986

[8] R J Shyu Bending of rotating drill-strings [PhD thesis] Mas-sachusetts Institute of Technology 1989

[9] A P Christoforou and A S Yigit ldquoFully coupled vibrations ofactively controlled drillstringsrdquo Journal of Sound and Vibrationvol 267 no 5 pp 1029ndash1045 2003

[10] M A Trindade C Wolter and R Sampaio ldquoKarhunen-Loevedecomposition of coupled axialbending vibrations of beamssubject to impactsrdquo Journal of Sound and Vibration vol 279 no3ndash5 pp 1015ndash1036 2005

[11] R Sampaio M T Piovan and G Venero Lozano ldquoCoupledaxialtorsional vibrations of drill-strings bymeans of non-linearmodelrdquo Mechanics Research Communications vol 34 no 5-6pp 497ndash502 2007

[12] H Hakimi and S Moradi ldquoDrillstring vibration analysis usingdifferential quadrature methodrdquo Journal of Petroleum Scienceand Engineering vol 70 no 3-4 pp 235ndash242 2010

[13] F S Ren S Chen and Z G Yao ldquoDynamics analysis andvibration suppression of a flexible rotation beamrdquo AppliedMechanics and Materials vol 214 pp 165ndash172 2012

[14] F-S Ren and Z-G Yao ldquoStudy on nonlinear dynamics andbifurcations in rotating compressive-drill stringrdquo EngineeringMechanics vol 30 no 10 pp 251ndash256 2013

[15] Z-F Li Y-G Zhang X-T Hou W-D Liu and G-Q XuldquoAnalysis of longitudinal and torsion vibration of drill-stringsrdquoEngineering Mechanics vol 21 no 6 pp 203ndash210 2004 (Chi-nese)

[16] G J Sheu and S M Yang ldquoDynamic analysis of a spinningRayleigh beamrdquo International Journal of Mechanical Sciencesvol 47 no 2 pp 157ndash169 2005

[17] X Zhou X Qichong D Hu et al ldquoThe nonlinear dynam-ics analysis of bottom drillstring of air drillingrdquo Journal ofChongqing University of Science and Technology vol 4 pp 119ndash121 2013 (Chinese)

[18] Y-J Jia P Jiang X-H Zhu and Y-F Zhang ldquoNonlinearbuckling simulation of drillstring system during pilot holedrilling in horizontal directional drillingrdquo Journal of SystemSimulation vol 25 no 4 pp 821ndash825 2013 (Chinese)

[19] W-S Xiao Z-Y Liu H-Y Wang X-F Wang L Fu and Y-C Yin ldquoTensional vibration analysis of drill-string by FSIrdquo OilField Equipment vol 42 pp 23ndash26 2013 (Chinese)

[20] D Shao Z Guan X Wen and Y Shi ldquoExperiment on lateralvibration characteristics of horizontal rotary drilling stringrdquoJournal of China University of Petroleum vol 37 pp 100ndash1052013 (Chinese)

[21] W Zhang X Zhu Z Zhou and W Ma ldquoEffect of rotationalspeeds mutation on the dynamic characteristics of the drill-string system in a vertical wellrdquo Journal of PetrochemicalUniversities vol 26 no 2 pp 47ndash51 2013 (Chinese)

[22] Y Tang ldquoNonlinear vibrations of axially accelerating viscoelas-tic Timoshenko beamsrdquo Chinese Journal of Theoretical andApplied Mechanics vol 45 no 6 pp 965ndash973 2013 (Chinese)

Shock and Vibration 17

[23] P B Goncalves and Z J G N Del Prado ldquoNonlinear oscilla-tions and stability of parametrically excited cylindrical shellsrdquoMeccanica vol 37 no 6 pp 569ndash597 2002

[24] M Strozzi and F Pellicano ldquoNonlinear vibrations of function-ally graded cylindrical shellsrdquo Thin-Walled Structures vol 67pp 63ndash77 2013

[25] M Amabili Nonlinear Vibrations and Stability of Shells andPlates Cambridge University Press Cambridge CambridgeUK 2008

[26] A W Leissa Vibrations of Shells Government Printing OfficeWashington DC USA 1973

[27] N Yamaki Elastic Stability of Circular Cylindrical Shells North-Holland Amsterdam The Netherlands 1984

[28] J N ReddyMechanics of LaminatedComposite Plates and ShellsCRC Press Boca Raton Fla USA 2004

[29] A H Nayfeh and D T Mook Nonlinear Oscillations Wiley-VCH Press Weinheim Germany 2004

[30] P D Spanos A K Sengupta R A Cunningham and P RPaslay ldquoModeling of roller cone bit lift-off dynamics in rotarydrillingrdquo Journal of Energy Resources Technology vol 117 no 3pp 197ndash207 1995

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DistributedSensor Networks

International Journal of

6 Shock and Vibration

x1

7 8 9 1110q3

12

14

16

18

(a) Bifurcation diagram of the axial modal

x3

10 118 97

q3

minus05

00

05

10

(b) Bifurcation diagram of the torsional modal

Figure 3 Bifurcations of the system under torsional excitation

It is well-known that the torsional vibration of the drillstring is mainly generated from the rotational speed changeof the bit when breaking rock intermittently The excitationfrequency is related to the rotation speed stiffness of thedrill string and characteristics of the rock The higher therock hardness the bigger the torsional stress and the greaterthe torsional amplitude We know from (17a) (17b) (17c)and (17d) that torsional stress strongly affects the couplingvibration when in low-frequency excitation but when inhigh-frequency excitation the torsional stress does not effec-tively contribute to the coupling vibration So in the practicaldrilling process when the drill string is in a high-rotationspeed zone the torsional stress will not lead to couplingvibration however when the drill string is in a low-rotationspeed zone more attention should be paid to the torsionalstress and appropriate drilling process parameters should beadopted to reduce the torsional stress especially when stick-slip vibration occurs This is necessary because at this timethe drill string is in low torsional excitation and large torsionalstress where it is more prone to produce a coupling vibrationof the drill string as suggested by Christoforou and Yigit [9]

Based on the above simulation analysis we also foundthat torsional excitation will affect both torsional and axialvibration simultaneously Torsional and axial vibration aresimilar qualitatively and in the nature of the vibration andare synchronous in form proving that energy can transferbetween the torsional and the axial vibration modal

42 Axial Excitation The above analysis accounts for theimpact of torsional excitation on the coupling of the axial andtorsional vibrations Here the impact of axial excitation willbe investigated

Thebifurcation diagram (Figure 8) shows the preliminaryresonance tendency The system undergoes a loop fromchaos to periodic motion then to period-three motionthen to period-multiplying and back to chaos showing

the nonlinear dynamic phenomenon of the period-doublingbifurcation and chaos

Increasing axial excitation until 1199022

= 05025 and keep-ing the remaining other initial conditions and parametersunchanged the system prevents period-doubling responsesas shown in Figure 9

When 1199022

= 0635 the system presents period-threeresponses as shown in Figure 10

When 1199022= 07725 the phase diagram changes drastically

and presents a jumping phenomenon then the systemresponds chaotically as shown in Figure 11

When 1199022

= 079 the qualitative nature of the couplingvibration of the drill string changes the present period-doubled responses are shown in Figure 12 Here the originalphase trajectory of the chaotic motion (Figure 11) contractsfor two trajectories and presents vibration characteristicsof frequency-doubling in the phase diagram The systemvibration turns from chaotic vibration into doubling periodicmotion and the stability of the system increases

As can be seen from the above qualitative analysis of theaxial excitation of the coupling vibration we find the samephenomenon as in the torsional excitation analysis for thecoupling vibration namely that axial excitation affects bothtorsional and axial vibration simultaneously and is similarqualitatively and also synchronous in form

Along with the increase in the axial excitation the systemcoupling vibration turns from period-doubling to period-three to chaotic and finally to period-doubling So the axialexcitation parameter leads to a coupling vibration in somespecial zone In the process of practical drilling the axialjump mainly contributes to the axial excitation The higherthe rotation speed the higher the excitation frequency Thelarger the WOB becomes the larger both the axial amplitudeand the axial stress will be We know from (17a) (17b) (17c)and (17d) that the axial excitation contributes to the couplingvibration when the drill string is in a high-rotation speedzone Therefore in the process of automation drilling with

Shock and Vibration 7

x2

minus3

minus2

minus1

1

0

2

3

minus2minus4 20x1

(a) Phase diagram of the axial modal

1020 10301010 1040 10501000t

minus4

minus3

minus2

minus1

0

1

2

x1

(b) Oscillogram of the axial modal

00 05 10minus10 minus05minus15

x3

minus15

minus10

minus05

00

05

10

x4

(c) Phase diagram of the torsional modal

1010 1020 1030 1040 10501000t

minus15

minus10

minus05

00

05

10x3

(d) Oscillogram of the torsional modal

5

0 02

0

1

minus1

minus2

x3

minus5

x2

minus4

minus2 x1

(e) Three-dimensional phase diagram

minus15

minus10

minus05

00

05

10

15

x2

minus10 minus05 00 05 10 15minus15

x1

(f) Poincare section

Figure 4 Period responses of the system (1199023= 724)

8 Shock and Vibration

x2

minus3

minus2

minus1

1

0

2

3

0minus2 2minus4

x1

(a) Phase diagram of the axial modal

1010 1020 1030 1040 10501000t

minus4

minus3

minus2

minus1

0

1

2

x1

(b) Oscillogram of the axial modal

minus10 minus05 00 05 10minus15

x3

minus15

minus10

minus05

00

05

10

x4

(c) Phase diagram of the torsional modal

1010 1020 1030 1040 10501000t

minus15

minus10

minus05

00

05

10x3

(d) Oscillogram of the torsional modal

5

00

2

0

1

minus1

minus2

x3

minus5

x2

minus4

minus2 x 1

(e) Three-dimensional phase diagram

0500 10 15minus10 minus05minus15

x1

minus15

minus10

minus05

00

05

10

15

x2

(f) Poincare section

Figure 5 Period responses of the system (1199023= 726)

Shock and Vibration 9

x2

minus2 0 2minus4

x1

minus3

minus2

minus1

1

0

2

3

(a) Phase diagram of the axial modal

minus4

minus3

minus2

minus1

0

1

2

x1

1010 1020 1030 1040 10501000t

(b) Oscillogram of the axial modal

minus10 minus05 00 05 10minus15

x3

minus15

minus10

minus05

00

05

10

x4

(c) Phase diagram of the torsional modal

minus15

minus10

minus05

00

05

10x3

1010 1020 1030 1040 10501000t

(d) Oscillogram of the torsional modal

5

0 02

0

1

minus1

minus2

x3

minus5

x2

minus4

minus2 x 1

(e) Three-dimensional phase diagram

minus15

minus10

minus05

00

05

10

15

x2

0500 10 15minus10 minus05minus15

x1

(f) Poincare section

Figure 6 Period responses of the system (1199023= 75)

10 Shock and Vibration

x2

minus2 0 2minus4

x1

minus3

minus2

minus1

1

0

2

3

(a) Phase diagram of the axial modal

1020 1040 1060 10801000t

minus4

minus3

minus2

minus1

0

1

2

x1

(b) Oscillogram of the axial modal

minus10 minus05 00 05 10minus15

x3

minus15

minus10

minus05

00

05

10

x4

(c) Phase diagram of the torsional modal

1020 1040 1060 10801000t

minus15

minus10

minus05

00

05

10x3

(d) Oscillogram of the torsional modal

5

00

2

0

1

minus1

minus2

x3

minus5

x2

minus4

minus2 x 1

(e) Three-dimensional phase diagram

minus10 minus05 00 05 10 15minus15

x1

minus15

minus10

minus05

00

05

10

15

x2

(f) Poincare section

Figure 7 The almost-periodic responses of the system (1199023= 768)

Shock and Vibration 11

06 07 08 09 1005q2

x2

minus10

minus05

00

(a) Bifurcation diagram of the axial modal

x3

06 07 0805 1009q2

10

15

20

(b) Bifurcation diagram of the torsional modal

Figure 8 Bifurcation of the resonance frequency-multiplier

a constant WOB in the case of a lower-rotation speed andlower rock hardness high drill efficiency can be attained byproperly increasing the WOB In the case of high-rotationspeed the coupling vibration is sensitive to the axial stressso the proper WOB should be adopted to avoid the couplingvibration of the drill string based on geological parameters

5 Conclusions

In this paper wemodeled and analyzed drill-string vibrationsby focusing on the coupled axialtorsional vibrations bymeans of nonlinear dynamics and qualitative analysis aimingat revealing the key effective factors influencing the coupledvibrations Here the drill string was described as a simplifiedequivalent flexible shell under axial rotation in which theexcitation loads and boundary conditions of the drill stringwere simplified After dimensionless processing we built thedynamics motion equation and the average equation

We found that the low-frequency amplitude expressionof the torsional excitation and the high-frequency amplitudeexpression of the axial excitation in the average equation arethe factors that determine the coupling vibration Based onthis procedure numerical simulations were carried out withadaptive steps using the Runge-Kutta-Fehlberg method todiscover the response of the system vibrations Further wefound that a change in the torsional or the axial excitationaffects both the torsional and the axial vibration simultane-ously and that it is similar qualitatively and synchronous inform

The results of the simulation analysis show that when thedrill string is in a lower-speed rotation zone the torsionalexcitation mainly contributes to the coupling vibrationincreasing the torsional stress of the drill string that moreeasily leads to the coupling vibrationWhen in a higher-speedrotating zone the axial excitation mainly contributes to thecoupling vibration so in a particular interval it is more likelyto cause the coupling vibration of the drill string

Appendix

Consider

1198861= minus

1

2119871120588120587

1198862=

1

120588

(

1

12

ℎ31198642

1198773

minus

ℎ1198642

119877

minus

1

4

1205872119877ℎ1198641

1198712

)

1198863=

1

120588

(

8

45

ℎ31198642

1198711198772

minus

32

15

ℎ1198642

119871

minus

32

15

ℎ1205811198641

119871

)

1198864=

1

120588

(

1

4

1205872ℎ31205811198641

11987131198773

minus

13

16

1205872ℎ1205811198641

1198712119877

minus

1

4

ℎ1198642

1198773

minus

1

16

1205874119877ℎ1198641

1198714

)

1198865=

1

120588

(

1

384

1205872ℎ31198642

11987121198773

minus

9

512

1205874119877ℎ1198641

1198714

minus

3

32

ℎ31198642

1198775

minus

9

32

ℎ1198642

1198773

minus

1

32

1205872ℎ1198641

1198712119877

minus

1

64

1205872ℎ1205811198641

1198712119877

)

1198866=

13ℎ1205811198641

15120588119871

1198867=

1

120588

(

1

4

1205872ℎ31205811198641

11987121198773

+

13

16

1205872ℎ1205811198641

1198712119877

minus

1

4

ℎ1198641

1198773

+

1

16

1205874119877ℎ1198641

1198714

+

3

4

ℎ1198641

1198713

+

1

24

1205872ℎ31198642

11987121198773

+

9

2

1205874119877ℎ1198641

1198714

)

1198868=

1

120588

(

1

2

1205872ℎ1198642

1198712119877

+

3

2

1205874ℎ31198641

1198714119877

+

1205872ℎ1205811198641

1198712119877

+

3

4

ℎ1198641

1198713

12 Shock and Vibrationx2

minus10

minus05

00

05

10

minus05 00 05 10minus10

x1

(a) Phase diagram of the axial modal

minus10

minus05

00

05

10

x1

1020 1040 1060 10801000t

(b) Oscillogram of the axial modal

minus2

minus1

0

1

2

x4

15 20 25 3010x3

(c) Phase diagram of the torsional modal

10

15

20

25

30x3

1020 1040 1060 10801000t

(d) Oscillogram of the torsional modal

1

0 0

11

2

3

minus1minus1

x3

x2 x 1

(e) Three-dimensional phase diagram

minus15

minus10

minus05

00

05

10

15

x2

0500 10 15minus10 minus05minus15

x1

(f) Poincare section

Figure 9 Period responses of the system (1199022= 05025)

Shock and Vibration 13x2

minus05 00 05 10minus10

x1

minus10

minus05

00

05

10

(a) Phase diagram of the axial modal

1020 1040 1060 10801000t

minus10

minus05

00

05

10

x1

(b) Oscillogram of the axial modal

minus20

minus15

minus10

minus05

00

05

10

15

20

x4

15 20 25 3010x3

(c) Phase diagram of the torsional modal

10

15

20

25

30x3

1020 1040 1060 10801000t

(d) Oscillogram of the torsional modal

1

0 0

11

2

3

minus1 minus1

x3

x2 x 1

(e) Three-dimensional phase diagram

minus10 minus05minus15 05 10 1500x1

minus15

minus10

minus05

00

05

10

15

x2

(f) Poincare section

Figure 10 Period responses of the system (1199022= 0635)

14 Shock and Vibration

minus2

minus1

0

1

2

x2

10 2 3minus1minus2

x1

(a) Phase diagram of the axial modal

1050 1100 1150 12001000t

minus2

minus1

0

1

2

3

x1

(b) Oscillogram of the axial modal

2 4minus2 0minus4

x3

minus4

minus2

0

2

4

x4

(c) Phase diagram of the torsional modal

minus4

minus2

0

2

4x3

1050 1100 1150 12001000t

(d) Oscillogram of the torsional modal

2

00

2

4

0

5

minus2

x3

minus5

x2

minus2

x 1

(e) Three-dimensional phase diagram

minus15

minus10

minus05

00

05

10

15

x2

0500 10 15minus10 minus05minus15

x1

(f) Poincare section

Figure 11 Period responses of the system (1199022= 07725)

Shock and Vibration 15x2

minus05 00 05 10minus10

x1

minus10

minus05

00

05

10

(a) Phase diagram of the axial modal

1020 1040 1060 10801000t

minus10

minus05

00

05

10

x1

(b) Oscillogram of the axial modal

15 20 25 3010x3

minus2

minus1

0

1

2

x4

(c) Phase diagram of the torsional modal

10

15

20

25

30x3

1020 1040 1060 10801000t

(d) Oscillogram of the torsional modal

1

0 0

11

2

3

minus1minus1

x3

x2

x 1

(e) Three-dimensional phase diagram

minus10 minus05minus15 05 10 1500x1

minus15

minus10

minus05

00

05

10

15

x2

(f) Poincare section

Figure 12 Period-doubled responses of the system (1199022= 079)

16 Shock and Vibration

+

1

24

1205872ℎ31198642

11987121198773

+

9

2

1205874119877ℎ1198641

1198714

)

1198869=

1198621

2120587119877radic120588119864

11988610

=

1205872ℎ31198641120581

411987121205881198772

minus

ℎ1198641

41205881198772

minus

1205874ℎ1198771198641

161198714120588

minus

1205872ℎ1198641120581

161198712120588119877

11988611

= minus

91205874ℎ1198771198641

5121198714120588

minus

3ℎ31198641

321205881198775

minus

1205872ℎ1198642

321198712120588119877

minus

1205872ℎ31198642

38411987121205881198773

minus

9ℎ1198641

321205881198773

minus

1205872ℎ1198641120581

641198712120588119877

11988612

=

1198622

2120587119877radic120588119864

11988613

= minus

41205872119877ℎ1198642

1198712120588

minus

ℎ1198641

119877120588

minus

1205872ℎ31198642

31198712120588119877

11988614

=

1205872ℎ31198641120581

411987121205881198773

minus

ℎ1198641

41205881198773

minus

1205874ℎ1198771198641

161198714120588

minus

131205872ℎ1198641120581

161198712120588119877

11988615

= minus

1205872ℎ31198642

2411987121205881198773

minus

1205872ℎ1198642

21198712120588119877

minus

91205874ℎ1198641120581

21198714120588

minus

3ℎ1198641

41205881198773

minus

1205872ℎ1198641120581

1198712120588119877

minus

31205874ℎ31198641

21198714120588119877

(A1)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to acknowledge financial support byNatural Science Foundation of China Project 11372071 andPostdoctoral Fund of China Project 2013M541339

References

[1] P D Spanos A M Chevallier N Politis and B Payne ldquoOilwell drilling a vibration perspectiverdquo The Shock and VibrationDigest vol 35 pp 81ndash99 2013

[2] W D Aldred andM C Sheppard ldquoDrillstring vibrations a newgeneration mechanism and control strategiesrdquo in Proceedingsof the SPE Annual Technical Conference and Exhibition SPE-24582-MS pp 353ndash363 Washington DC USA October 1992

[3] T V Aarrestand H A Tonnesen and A Kyllingstad ldquoDrill-string vibrations theory and experiments on full-scale researchdrilling rigrdquo inProceedings of the IADCSPEDrilling ConferenceSPE Paper No 14760 pp 311ndash321 Dallas Tex USA 1986

[4] J D Jansen ldquoNon-linear rotor dynamics as applied to oilwelldrillstring vibrationsrdquo Journal of Sound and Vibration vol 147no 1 pp 115ndash135 1991

[5] A S Yigit and A P Christoforou ldquoCoupled axial and transversevibrations of oilwell drillstringsrdquo Journal of Sound and Vibra-tion vol 195 no 4 pp 617ndash627 1996

[6] A S Yigit and A P Christoforou ldquoCoupled torsional and bend-ing vibration of drill-strings subject to impact with frictionrdquoJournal of Sound and Vibration vol 215 no 1 pp 167ndash181 1998

[7] K Z Khan Longitudinal and torsional vibration of drill-strings [MS thesis] Massachusetts Institute of TechnologyCambridge Mass USA 1986

[8] R J Shyu Bending of rotating drill-strings [PhD thesis] Mas-sachusetts Institute of Technology 1989

[9] A P Christoforou and A S Yigit ldquoFully coupled vibrations ofactively controlled drillstringsrdquo Journal of Sound and Vibrationvol 267 no 5 pp 1029ndash1045 2003

[10] M A Trindade C Wolter and R Sampaio ldquoKarhunen-Loevedecomposition of coupled axialbending vibrations of beamssubject to impactsrdquo Journal of Sound and Vibration vol 279 no3ndash5 pp 1015ndash1036 2005

[11] R Sampaio M T Piovan and G Venero Lozano ldquoCoupledaxialtorsional vibrations of drill-strings bymeans of non-linearmodelrdquo Mechanics Research Communications vol 34 no 5-6pp 497ndash502 2007

[12] H Hakimi and S Moradi ldquoDrillstring vibration analysis usingdifferential quadrature methodrdquo Journal of Petroleum Scienceand Engineering vol 70 no 3-4 pp 235ndash242 2010

[13] F S Ren S Chen and Z G Yao ldquoDynamics analysis andvibration suppression of a flexible rotation beamrdquo AppliedMechanics and Materials vol 214 pp 165ndash172 2012

[14] F-S Ren and Z-G Yao ldquoStudy on nonlinear dynamics andbifurcations in rotating compressive-drill stringrdquo EngineeringMechanics vol 30 no 10 pp 251ndash256 2013

[15] Z-F Li Y-G Zhang X-T Hou W-D Liu and G-Q XuldquoAnalysis of longitudinal and torsion vibration of drill-stringsrdquoEngineering Mechanics vol 21 no 6 pp 203ndash210 2004 (Chi-nese)

[16] G J Sheu and S M Yang ldquoDynamic analysis of a spinningRayleigh beamrdquo International Journal of Mechanical Sciencesvol 47 no 2 pp 157ndash169 2005

[17] X Zhou X Qichong D Hu et al ldquoThe nonlinear dynam-ics analysis of bottom drillstring of air drillingrdquo Journal ofChongqing University of Science and Technology vol 4 pp 119ndash121 2013 (Chinese)

[18] Y-J Jia P Jiang X-H Zhu and Y-F Zhang ldquoNonlinearbuckling simulation of drillstring system during pilot holedrilling in horizontal directional drillingrdquo Journal of SystemSimulation vol 25 no 4 pp 821ndash825 2013 (Chinese)

[19] W-S Xiao Z-Y Liu H-Y Wang X-F Wang L Fu and Y-C Yin ldquoTensional vibration analysis of drill-string by FSIrdquo OilField Equipment vol 42 pp 23ndash26 2013 (Chinese)

[20] D Shao Z Guan X Wen and Y Shi ldquoExperiment on lateralvibration characteristics of horizontal rotary drilling stringrdquoJournal of China University of Petroleum vol 37 pp 100ndash1052013 (Chinese)

[21] W Zhang X Zhu Z Zhou and W Ma ldquoEffect of rotationalspeeds mutation on the dynamic characteristics of the drill-string system in a vertical wellrdquo Journal of PetrochemicalUniversities vol 26 no 2 pp 47ndash51 2013 (Chinese)

[22] Y Tang ldquoNonlinear vibrations of axially accelerating viscoelas-tic Timoshenko beamsrdquo Chinese Journal of Theoretical andApplied Mechanics vol 45 no 6 pp 965ndash973 2013 (Chinese)

Shock and Vibration 17

[23] P B Goncalves and Z J G N Del Prado ldquoNonlinear oscilla-tions and stability of parametrically excited cylindrical shellsrdquoMeccanica vol 37 no 6 pp 569ndash597 2002

[24] M Strozzi and F Pellicano ldquoNonlinear vibrations of function-ally graded cylindrical shellsrdquo Thin-Walled Structures vol 67pp 63ndash77 2013

[25] M Amabili Nonlinear Vibrations and Stability of Shells andPlates Cambridge University Press Cambridge CambridgeUK 2008

[26] A W Leissa Vibrations of Shells Government Printing OfficeWashington DC USA 1973

[27] N Yamaki Elastic Stability of Circular Cylindrical Shells North-Holland Amsterdam The Netherlands 1984

[28] J N ReddyMechanics of LaminatedComposite Plates and ShellsCRC Press Boca Raton Fla USA 2004

[29] A H Nayfeh and D T Mook Nonlinear Oscillations Wiley-VCH Press Weinheim Germany 2004

[30] P D Spanos A K Sengupta R A Cunningham and P RPaslay ldquoModeling of roller cone bit lift-off dynamics in rotarydrillingrdquo Journal of Energy Resources Technology vol 117 no 3pp 197ndash207 1995

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Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Navigation and Observation

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DistributedSensor Networks

International Journal of

Shock and Vibration 7

x2

minus3

minus2

minus1

1

0

2

3

minus2minus4 20x1

(a) Phase diagram of the axial modal

1020 10301010 1040 10501000t

minus4

minus3

minus2

minus1

0

1

2

x1

(b) Oscillogram of the axial modal

00 05 10minus10 minus05minus15

x3

minus15

minus10

minus05

00

05

10

x4

(c) Phase diagram of the torsional modal

1010 1020 1030 1040 10501000t

minus15

minus10

minus05

00

05

10x3

(d) Oscillogram of the torsional modal

5

0 02

0

1

minus1

minus2

x3

minus5

x2

minus4

minus2 x1

(e) Three-dimensional phase diagram

minus15

minus10

minus05

00

05

10

15

x2

minus10 minus05 00 05 10 15minus15

x1

(f) Poincare section

Figure 4 Period responses of the system (1199023= 724)

8 Shock and Vibration

x2

minus3

minus2

minus1

1

0

2

3

0minus2 2minus4

x1

(a) Phase diagram of the axial modal

1010 1020 1030 1040 10501000t

minus4

minus3

minus2

minus1

0

1

2

x1

(b) Oscillogram of the axial modal

minus10 minus05 00 05 10minus15

x3

minus15

minus10

minus05

00

05

10

x4

(c) Phase diagram of the torsional modal

1010 1020 1030 1040 10501000t

minus15

minus10

minus05

00

05

10x3

(d) Oscillogram of the torsional modal

5

00

2

0

1

minus1

minus2

x3

minus5

x2

minus4

minus2 x 1

(e) Three-dimensional phase diagram

0500 10 15minus10 minus05minus15

x1

minus15

minus10

minus05

00

05

10

15

x2

(f) Poincare section

Figure 5 Period responses of the system (1199023= 726)

Shock and Vibration 9

x2

minus2 0 2minus4

x1

minus3

minus2

minus1

1

0

2

3

(a) Phase diagram of the axial modal

minus4

minus3

minus2

minus1

0

1

2

x1

1010 1020 1030 1040 10501000t

(b) Oscillogram of the axial modal

minus10 minus05 00 05 10minus15

x3

minus15

minus10

minus05

00

05

10

x4

(c) Phase diagram of the torsional modal

minus15

minus10

minus05

00

05

10x3

1010 1020 1030 1040 10501000t

(d) Oscillogram of the torsional modal

5

0 02

0

1

minus1

minus2

x3

minus5

x2

minus4

minus2 x 1

(e) Three-dimensional phase diagram

minus15

minus10

minus05

00

05

10

15

x2

0500 10 15minus10 minus05minus15

x1

(f) Poincare section

Figure 6 Period responses of the system (1199023= 75)

10 Shock and Vibration

x2

minus2 0 2minus4

x1

minus3

minus2

minus1

1

0

2

3

(a) Phase diagram of the axial modal

1020 1040 1060 10801000t

minus4

minus3

minus2

minus1

0

1

2

x1

(b) Oscillogram of the axial modal

minus10 minus05 00 05 10minus15

x3

minus15

minus10

minus05

00

05

10

x4

(c) Phase diagram of the torsional modal

1020 1040 1060 10801000t

minus15

minus10

minus05

00

05

10x3

(d) Oscillogram of the torsional modal

5

00

2

0

1

minus1

minus2

x3

minus5

x2

minus4

minus2 x 1

(e) Three-dimensional phase diagram

minus10 minus05 00 05 10 15minus15

x1

minus15

minus10

minus05

00

05

10

15

x2

(f) Poincare section

Figure 7 The almost-periodic responses of the system (1199023= 768)

Shock and Vibration 11

06 07 08 09 1005q2

x2

minus10

minus05

00

(a) Bifurcation diagram of the axial modal

x3

06 07 0805 1009q2

10

15

20

(b) Bifurcation diagram of the torsional modal

Figure 8 Bifurcation of the resonance frequency-multiplier

a constant WOB in the case of a lower-rotation speed andlower rock hardness high drill efficiency can be attained byproperly increasing the WOB In the case of high-rotationspeed the coupling vibration is sensitive to the axial stressso the proper WOB should be adopted to avoid the couplingvibration of the drill string based on geological parameters

5 Conclusions

In this paper wemodeled and analyzed drill-string vibrationsby focusing on the coupled axialtorsional vibrations bymeans of nonlinear dynamics and qualitative analysis aimingat revealing the key effective factors influencing the coupledvibrations Here the drill string was described as a simplifiedequivalent flexible shell under axial rotation in which theexcitation loads and boundary conditions of the drill stringwere simplified After dimensionless processing we built thedynamics motion equation and the average equation

We found that the low-frequency amplitude expressionof the torsional excitation and the high-frequency amplitudeexpression of the axial excitation in the average equation arethe factors that determine the coupling vibration Based onthis procedure numerical simulations were carried out withadaptive steps using the Runge-Kutta-Fehlberg method todiscover the response of the system vibrations Further wefound that a change in the torsional or the axial excitationaffects both the torsional and the axial vibration simultane-ously and that it is similar qualitatively and synchronous inform

The results of the simulation analysis show that when thedrill string is in a lower-speed rotation zone the torsionalexcitation mainly contributes to the coupling vibrationincreasing the torsional stress of the drill string that moreeasily leads to the coupling vibrationWhen in a higher-speedrotating zone the axial excitation mainly contributes to thecoupling vibration so in a particular interval it is more likelyto cause the coupling vibration of the drill string

Appendix

Consider

1198861= minus

1

2119871120588120587

1198862=

1

120588

(

1

12

ℎ31198642

1198773

minus

ℎ1198642

119877

minus

1

4

1205872119877ℎ1198641

1198712

)

1198863=

1

120588

(

8

45

ℎ31198642

1198711198772

minus

32

15

ℎ1198642

119871

minus

32

15

ℎ1205811198641

119871

)

1198864=

1

120588

(

1

4

1205872ℎ31205811198641

11987131198773

minus

13

16

1205872ℎ1205811198641

1198712119877

minus

1

4

ℎ1198642

1198773

minus

1

16

1205874119877ℎ1198641

1198714

)

1198865=

1

120588

(

1

384

1205872ℎ31198642

11987121198773

minus

9

512

1205874119877ℎ1198641

1198714

minus

3

32

ℎ31198642

1198775

minus

9

32

ℎ1198642

1198773

minus

1

32

1205872ℎ1198641

1198712119877

minus

1

64

1205872ℎ1205811198641

1198712119877

)

1198866=

13ℎ1205811198641

15120588119871

1198867=

1

120588

(

1

4

1205872ℎ31205811198641

11987121198773

+

13

16

1205872ℎ1205811198641

1198712119877

minus

1

4

ℎ1198641

1198773

+

1

16

1205874119877ℎ1198641

1198714

+

3

4

ℎ1198641

1198713

+

1

24

1205872ℎ31198642

11987121198773

+

9

2

1205874119877ℎ1198641

1198714

)

1198868=

1

120588

(

1

2

1205872ℎ1198642

1198712119877

+

3

2

1205874ℎ31198641

1198714119877

+

1205872ℎ1205811198641

1198712119877

+

3

4

ℎ1198641

1198713

12 Shock and Vibrationx2

minus10

minus05

00

05

10

minus05 00 05 10minus10

x1

(a) Phase diagram of the axial modal

minus10

minus05

00

05

10

x1

1020 1040 1060 10801000t

(b) Oscillogram of the axial modal

minus2

minus1

0

1

2

x4

15 20 25 3010x3

(c) Phase diagram of the torsional modal

10

15

20

25

30x3

1020 1040 1060 10801000t

(d) Oscillogram of the torsional modal

1

0 0

11

2

3

minus1minus1

x3

x2 x 1

(e) Three-dimensional phase diagram

minus15

minus10

minus05

00

05

10

15

x2

0500 10 15minus10 minus05minus15

x1

(f) Poincare section

Figure 9 Period responses of the system (1199022= 05025)

Shock and Vibration 13x2

minus05 00 05 10minus10

x1

minus10

minus05

00

05

10

(a) Phase diagram of the axial modal

1020 1040 1060 10801000t

minus10

minus05

00

05

10

x1

(b) Oscillogram of the axial modal

minus20

minus15

minus10

minus05

00

05

10

15

20

x4

15 20 25 3010x3

(c) Phase diagram of the torsional modal

10

15

20

25

30x3

1020 1040 1060 10801000t

(d) Oscillogram of the torsional modal

1

0 0

11

2

3

minus1 minus1

x3

x2 x 1

(e) Three-dimensional phase diagram

minus10 minus05minus15 05 10 1500x1

minus15

minus10

minus05

00

05

10

15

x2

(f) Poincare section

Figure 10 Period responses of the system (1199022= 0635)

14 Shock and Vibration

minus2

minus1

0

1

2

x2

10 2 3minus1minus2

x1

(a) Phase diagram of the axial modal

1050 1100 1150 12001000t

minus2

minus1

0

1

2

3

x1

(b) Oscillogram of the axial modal

2 4minus2 0minus4

x3

minus4

minus2

0

2

4

x4

(c) Phase diagram of the torsional modal

minus4

minus2

0

2

4x3

1050 1100 1150 12001000t

(d) Oscillogram of the torsional modal

2

00

2

4

0

5

minus2

x3

minus5

x2

minus2

x 1

(e) Three-dimensional phase diagram

minus15

minus10

minus05

00

05

10

15

x2

0500 10 15minus10 minus05minus15

x1

(f) Poincare section

Figure 11 Period responses of the system (1199022= 07725)

Shock and Vibration 15x2

minus05 00 05 10minus10

x1

minus10

minus05

00

05

10

(a) Phase diagram of the axial modal

1020 1040 1060 10801000t

minus10

minus05

00

05

10

x1

(b) Oscillogram of the axial modal

15 20 25 3010x3

minus2

minus1

0

1

2

x4

(c) Phase diagram of the torsional modal

10

15

20

25

30x3

1020 1040 1060 10801000t

(d) Oscillogram of the torsional modal

1

0 0

11

2

3

minus1minus1

x3

x2

x 1

(e) Three-dimensional phase diagram

minus10 minus05minus15 05 10 1500x1

minus15

minus10

minus05

00

05

10

15

x2

(f) Poincare section

Figure 12 Period-doubled responses of the system (1199022= 079)

16 Shock and Vibration

+

1

24

1205872ℎ31198642

11987121198773

+

9

2

1205874119877ℎ1198641

1198714

)

1198869=

1198621

2120587119877radic120588119864

11988610

=

1205872ℎ31198641120581

411987121205881198772

minus

ℎ1198641

41205881198772

minus

1205874ℎ1198771198641

161198714120588

minus

1205872ℎ1198641120581

161198712120588119877

11988611

= minus

91205874ℎ1198771198641

5121198714120588

minus

3ℎ31198641

321205881198775

minus

1205872ℎ1198642

321198712120588119877

minus

1205872ℎ31198642

38411987121205881198773

minus

9ℎ1198641

321205881198773

minus

1205872ℎ1198641120581

641198712120588119877

11988612

=

1198622

2120587119877radic120588119864

11988613

= minus

41205872119877ℎ1198642

1198712120588

minus

ℎ1198641

119877120588

minus

1205872ℎ31198642

31198712120588119877

11988614

=

1205872ℎ31198641120581

411987121205881198773

minus

ℎ1198641

41205881198773

minus

1205874ℎ1198771198641

161198714120588

minus

131205872ℎ1198641120581

161198712120588119877

11988615

= minus

1205872ℎ31198642

2411987121205881198773

minus

1205872ℎ1198642

21198712120588119877

minus

91205874ℎ1198641120581

21198714120588

minus

3ℎ1198641

41205881198773

minus

1205872ℎ1198641120581

1198712120588119877

minus

31205874ℎ31198641

21198714120588119877

(A1)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to acknowledge financial support byNatural Science Foundation of China Project 11372071 andPostdoctoral Fund of China Project 2013M541339

References

[1] P D Spanos A M Chevallier N Politis and B Payne ldquoOilwell drilling a vibration perspectiverdquo The Shock and VibrationDigest vol 35 pp 81ndash99 2013

[2] W D Aldred andM C Sheppard ldquoDrillstring vibrations a newgeneration mechanism and control strategiesrdquo in Proceedingsof the SPE Annual Technical Conference and Exhibition SPE-24582-MS pp 353ndash363 Washington DC USA October 1992

[3] T V Aarrestand H A Tonnesen and A Kyllingstad ldquoDrill-string vibrations theory and experiments on full-scale researchdrilling rigrdquo inProceedings of the IADCSPEDrilling ConferenceSPE Paper No 14760 pp 311ndash321 Dallas Tex USA 1986

[4] J D Jansen ldquoNon-linear rotor dynamics as applied to oilwelldrillstring vibrationsrdquo Journal of Sound and Vibration vol 147no 1 pp 115ndash135 1991

[5] A S Yigit and A P Christoforou ldquoCoupled axial and transversevibrations of oilwell drillstringsrdquo Journal of Sound and Vibra-tion vol 195 no 4 pp 617ndash627 1996

[6] A S Yigit and A P Christoforou ldquoCoupled torsional and bend-ing vibration of drill-strings subject to impact with frictionrdquoJournal of Sound and Vibration vol 215 no 1 pp 167ndash181 1998

[7] K Z Khan Longitudinal and torsional vibration of drill-strings [MS thesis] Massachusetts Institute of TechnologyCambridge Mass USA 1986

[8] R J Shyu Bending of rotating drill-strings [PhD thesis] Mas-sachusetts Institute of Technology 1989

[9] A P Christoforou and A S Yigit ldquoFully coupled vibrations ofactively controlled drillstringsrdquo Journal of Sound and Vibrationvol 267 no 5 pp 1029ndash1045 2003

[10] M A Trindade C Wolter and R Sampaio ldquoKarhunen-Loevedecomposition of coupled axialbending vibrations of beamssubject to impactsrdquo Journal of Sound and Vibration vol 279 no3ndash5 pp 1015ndash1036 2005

[11] R Sampaio M T Piovan and G Venero Lozano ldquoCoupledaxialtorsional vibrations of drill-strings bymeans of non-linearmodelrdquo Mechanics Research Communications vol 34 no 5-6pp 497ndash502 2007

[12] H Hakimi and S Moradi ldquoDrillstring vibration analysis usingdifferential quadrature methodrdquo Journal of Petroleum Scienceand Engineering vol 70 no 3-4 pp 235ndash242 2010

[13] F S Ren S Chen and Z G Yao ldquoDynamics analysis andvibration suppression of a flexible rotation beamrdquo AppliedMechanics and Materials vol 214 pp 165ndash172 2012

[14] F-S Ren and Z-G Yao ldquoStudy on nonlinear dynamics andbifurcations in rotating compressive-drill stringrdquo EngineeringMechanics vol 30 no 10 pp 251ndash256 2013

[15] Z-F Li Y-G Zhang X-T Hou W-D Liu and G-Q XuldquoAnalysis of longitudinal and torsion vibration of drill-stringsrdquoEngineering Mechanics vol 21 no 6 pp 203ndash210 2004 (Chi-nese)

[16] G J Sheu and S M Yang ldquoDynamic analysis of a spinningRayleigh beamrdquo International Journal of Mechanical Sciencesvol 47 no 2 pp 157ndash169 2005

[17] X Zhou X Qichong D Hu et al ldquoThe nonlinear dynam-ics analysis of bottom drillstring of air drillingrdquo Journal ofChongqing University of Science and Technology vol 4 pp 119ndash121 2013 (Chinese)

[18] Y-J Jia P Jiang X-H Zhu and Y-F Zhang ldquoNonlinearbuckling simulation of drillstring system during pilot holedrilling in horizontal directional drillingrdquo Journal of SystemSimulation vol 25 no 4 pp 821ndash825 2013 (Chinese)

[19] W-S Xiao Z-Y Liu H-Y Wang X-F Wang L Fu and Y-C Yin ldquoTensional vibration analysis of drill-string by FSIrdquo OilField Equipment vol 42 pp 23ndash26 2013 (Chinese)

[20] D Shao Z Guan X Wen and Y Shi ldquoExperiment on lateralvibration characteristics of horizontal rotary drilling stringrdquoJournal of China University of Petroleum vol 37 pp 100ndash1052013 (Chinese)

[21] W Zhang X Zhu Z Zhou and W Ma ldquoEffect of rotationalspeeds mutation on the dynamic characteristics of the drill-string system in a vertical wellrdquo Journal of PetrochemicalUniversities vol 26 no 2 pp 47ndash51 2013 (Chinese)

[22] Y Tang ldquoNonlinear vibrations of axially accelerating viscoelas-tic Timoshenko beamsrdquo Chinese Journal of Theoretical andApplied Mechanics vol 45 no 6 pp 965ndash973 2013 (Chinese)

Shock and Vibration 17

[23] P B Goncalves and Z J G N Del Prado ldquoNonlinear oscilla-tions and stability of parametrically excited cylindrical shellsrdquoMeccanica vol 37 no 6 pp 569ndash597 2002

[24] M Strozzi and F Pellicano ldquoNonlinear vibrations of function-ally graded cylindrical shellsrdquo Thin-Walled Structures vol 67pp 63ndash77 2013

[25] M Amabili Nonlinear Vibrations and Stability of Shells andPlates Cambridge University Press Cambridge CambridgeUK 2008

[26] A W Leissa Vibrations of Shells Government Printing OfficeWashington DC USA 1973

[27] N Yamaki Elastic Stability of Circular Cylindrical Shells North-Holland Amsterdam The Netherlands 1984

[28] J N ReddyMechanics of LaminatedComposite Plates and ShellsCRC Press Boca Raton Fla USA 2004

[29] A H Nayfeh and D T Mook Nonlinear Oscillations Wiley-VCH Press Weinheim Germany 2004

[30] P D Spanos A K Sengupta R A Cunningham and P RPaslay ldquoModeling of roller cone bit lift-off dynamics in rotarydrillingrdquo Journal of Energy Resources Technology vol 117 no 3pp 197ndash207 1995

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Submit your manuscripts athttpwwwhindawicom

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Navigation and Observation

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DistributedSensor Networks

International Journal of

8 Shock and Vibration

x2

minus3

minus2

minus1

1

0

2

3

0minus2 2minus4

x1

(a) Phase diagram of the axial modal

1010 1020 1030 1040 10501000t

minus4

minus3

minus2

minus1

0

1

2

x1

(b) Oscillogram of the axial modal

minus10 minus05 00 05 10minus15

x3

minus15

minus10

minus05

00

05

10

x4

(c) Phase diagram of the torsional modal

1010 1020 1030 1040 10501000t

minus15

minus10

minus05

00

05

10x3

(d) Oscillogram of the torsional modal

5

00

2

0

1

minus1

minus2

x3

minus5

x2

minus4

minus2 x 1

(e) Three-dimensional phase diagram

0500 10 15minus10 minus05minus15

x1

minus15

minus10

minus05

00

05

10

15

x2

(f) Poincare section

Figure 5 Period responses of the system (1199023= 726)

Shock and Vibration 9

x2

minus2 0 2minus4

x1

minus3

minus2

minus1

1

0

2

3

(a) Phase diagram of the axial modal

minus4

minus3

minus2

minus1

0

1

2

x1

1010 1020 1030 1040 10501000t

(b) Oscillogram of the axial modal

minus10 minus05 00 05 10minus15

x3

minus15

minus10

minus05

00

05

10

x4

(c) Phase diagram of the torsional modal

minus15

minus10

minus05

00

05

10x3

1010 1020 1030 1040 10501000t

(d) Oscillogram of the torsional modal

5

0 02

0

1

minus1

minus2

x3

minus5

x2

minus4

minus2 x 1

(e) Three-dimensional phase diagram

minus15

minus10

minus05

00

05

10

15

x2

0500 10 15minus10 minus05minus15

x1

(f) Poincare section

Figure 6 Period responses of the system (1199023= 75)

10 Shock and Vibration

x2

minus2 0 2minus4

x1

minus3

minus2

minus1

1

0

2

3

(a) Phase diagram of the axial modal

1020 1040 1060 10801000t

minus4

minus3

minus2

minus1

0

1

2

x1

(b) Oscillogram of the axial modal

minus10 minus05 00 05 10minus15

x3

minus15

minus10

minus05

00

05

10

x4

(c) Phase diagram of the torsional modal

1020 1040 1060 10801000t

minus15

minus10

minus05

00

05

10x3

(d) Oscillogram of the torsional modal

5

00

2

0

1

minus1

minus2

x3

minus5

x2

minus4

minus2 x 1

(e) Three-dimensional phase diagram

minus10 minus05 00 05 10 15minus15

x1

minus15

minus10

minus05

00

05

10

15

x2

(f) Poincare section

Figure 7 The almost-periodic responses of the system (1199023= 768)

Shock and Vibration 11

06 07 08 09 1005q2

x2

minus10

minus05

00

(a) Bifurcation diagram of the axial modal

x3

06 07 0805 1009q2

10

15

20

(b) Bifurcation diagram of the torsional modal

Figure 8 Bifurcation of the resonance frequency-multiplier

a constant WOB in the case of a lower-rotation speed andlower rock hardness high drill efficiency can be attained byproperly increasing the WOB In the case of high-rotationspeed the coupling vibration is sensitive to the axial stressso the proper WOB should be adopted to avoid the couplingvibration of the drill string based on geological parameters

5 Conclusions

In this paper wemodeled and analyzed drill-string vibrationsby focusing on the coupled axialtorsional vibrations bymeans of nonlinear dynamics and qualitative analysis aimingat revealing the key effective factors influencing the coupledvibrations Here the drill string was described as a simplifiedequivalent flexible shell under axial rotation in which theexcitation loads and boundary conditions of the drill stringwere simplified After dimensionless processing we built thedynamics motion equation and the average equation

We found that the low-frequency amplitude expressionof the torsional excitation and the high-frequency amplitudeexpression of the axial excitation in the average equation arethe factors that determine the coupling vibration Based onthis procedure numerical simulations were carried out withadaptive steps using the Runge-Kutta-Fehlberg method todiscover the response of the system vibrations Further wefound that a change in the torsional or the axial excitationaffects both the torsional and the axial vibration simultane-ously and that it is similar qualitatively and synchronous inform

The results of the simulation analysis show that when thedrill string is in a lower-speed rotation zone the torsionalexcitation mainly contributes to the coupling vibrationincreasing the torsional stress of the drill string that moreeasily leads to the coupling vibrationWhen in a higher-speedrotating zone the axial excitation mainly contributes to thecoupling vibration so in a particular interval it is more likelyto cause the coupling vibration of the drill string

Appendix

Consider

1198861= minus

1

2119871120588120587

1198862=

1

120588

(

1

12

ℎ31198642

1198773

minus

ℎ1198642

119877

minus

1

4

1205872119877ℎ1198641

1198712

)

1198863=

1

120588

(

8

45

ℎ31198642

1198711198772

minus

32

15

ℎ1198642

119871

minus

32

15

ℎ1205811198641

119871

)

1198864=

1

120588

(

1

4

1205872ℎ31205811198641

11987131198773

minus

13

16

1205872ℎ1205811198641

1198712119877

minus

1

4

ℎ1198642

1198773

minus

1

16

1205874119877ℎ1198641

1198714

)

1198865=

1

120588

(

1

384

1205872ℎ31198642

11987121198773

minus

9

512

1205874119877ℎ1198641

1198714

minus

3

32

ℎ31198642

1198775

minus

9

32

ℎ1198642

1198773

minus

1

32

1205872ℎ1198641

1198712119877

minus

1

64

1205872ℎ1205811198641

1198712119877

)

1198866=

13ℎ1205811198641

15120588119871

1198867=

1

120588

(

1

4

1205872ℎ31205811198641

11987121198773

+

13

16

1205872ℎ1205811198641

1198712119877

minus

1

4

ℎ1198641

1198773

+

1

16

1205874119877ℎ1198641

1198714

+

3

4

ℎ1198641

1198713

+

1

24

1205872ℎ31198642

11987121198773

+

9

2

1205874119877ℎ1198641

1198714

)

1198868=

1

120588

(

1

2

1205872ℎ1198642

1198712119877

+

3

2

1205874ℎ31198641

1198714119877

+

1205872ℎ1205811198641

1198712119877

+

3

4

ℎ1198641

1198713

12 Shock and Vibrationx2

minus10

minus05

00

05

10

minus05 00 05 10minus10

x1

(a) Phase diagram of the axial modal

minus10

minus05

00

05

10

x1

1020 1040 1060 10801000t

(b) Oscillogram of the axial modal

minus2

minus1

0

1

2

x4

15 20 25 3010x3

(c) Phase diagram of the torsional modal

10

15

20

25

30x3

1020 1040 1060 10801000t

(d) Oscillogram of the torsional modal

1

0 0

11

2

3

minus1minus1

x3

x2 x 1

(e) Three-dimensional phase diagram

minus15

minus10

minus05

00

05

10

15

x2

0500 10 15minus10 minus05minus15

x1

(f) Poincare section

Figure 9 Period responses of the system (1199022= 05025)

Shock and Vibration 13x2

minus05 00 05 10minus10

x1

minus10

minus05

00

05

10

(a) Phase diagram of the axial modal

1020 1040 1060 10801000t

minus10

minus05

00

05

10

x1

(b) Oscillogram of the axial modal

minus20

minus15

minus10

minus05

00

05

10

15

20

x4

15 20 25 3010x3

(c) Phase diagram of the torsional modal

10

15

20

25

30x3

1020 1040 1060 10801000t

(d) Oscillogram of the torsional modal

1

0 0

11

2

3

minus1 minus1

x3

x2 x 1

(e) Three-dimensional phase diagram

minus10 minus05minus15 05 10 1500x1

minus15

minus10

minus05

00

05

10

15

x2

(f) Poincare section

Figure 10 Period responses of the system (1199022= 0635)

14 Shock and Vibration

minus2

minus1

0

1

2

x2

10 2 3minus1minus2

x1

(a) Phase diagram of the axial modal

1050 1100 1150 12001000t

minus2

minus1

0

1

2

3

x1

(b) Oscillogram of the axial modal

2 4minus2 0minus4

x3

minus4

minus2

0

2

4

x4

(c) Phase diagram of the torsional modal

minus4

minus2

0

2

4x3

1050 1100 1150 12001000t

(d) Oscillogram of the torsional modal

2

00

2

4

0

5

minus2

x3

minus5

x2

minus2

x 1

(e) Three-dimensional phase diagram

minus15

minus10

minus05

00

05

10

15

x2

0500 10 15minus10 minus05minus15

x1

(f) Poincare section

Figure 11 Period responses of the system (1199022= 07725)

Shock and Vibration 15x2

minus05 00 05 10minus10

x1

minus10

minus05

00

05

10

(a) Phase diagram of the axial modal

1020 1040 1060 10801000t

minus10

minus05

00

05

10

x1

(b) Oscillogram of the axial modal

15 20 25 3010x3

minus2

minus1

0

1

2

x4

(c) Phase diagram of the torsional modal

10

15

20

25

30x3

1020 1040 1060 10801000t

(d) Oscillogram of the torsional modal

1

0 0

11

2

3

minus1minus1

x3

x2

x 1

(e) Three-dimensional phase diagram

minus10 minus05minus15 05 10 1500x1

minus15

minus10

minus05

00

05

10

15

x2

(f) Poincare section

Figure 12 Period-doubled responses of the system (1199022= 079)

16 Shock and Vibration

+

1

24

1205872ℎ31198642

11987121198773

+

9

2

1205874119877ℎ1198641

1198714

)

1198869=

1198621

2120587119877radic120588119864

11988610

=

1205872ℎ31198641120581

411987121205881198772

minus

ℎ1198641

41205881198772

minus

1205874ℎ1198771198641

161198714120588

minus

1205872ℎ1198641120581

161198712120588119877

11988611

= minus

91205874ℎ1198771198641

5121198714120588

minus

3ℎ31198641

321205881198775

minus

1205872ℎ1198642

321198712120588119877

minus

1205872ℎ31198642

38411987121205881198773

minus

9ℎ1198641

321205881198773

minus

1205872ℎ1198641120581

641198712120588119877

11988612

=

1198622

2120587119877radic120588119864

11988613

= minus

41205872119877ℎ1198642

1198712120588

minus

ℎ1198641

119877120588

minus

1205872ℎ31198642

31198712120588119877

11988614

=

1205872ℎ31198641120581

411987121205881198773

minus

ℎ1198641

41205881198773

minus

1205874ℎ1198771198641

161198714120588

minus

131205872ℎ1198641120581

161198712120588119877

11988615

= minus

1205872ℎ31198642

2411987121205881198773

minus

1205872ℎ1198642

21198712120588119877

minus

91205874ℎ1198641120581

21198714120588

minus

3ℎ1198641

41205881198773

minus

1205872ℎ1198641120581

1198712120588119877

minus

31205874ℎ31198641

21198714120588119877

(A1)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to acknowledge financial support byNatural Science Foundation of China Project 11372071 andPostdoctoral Fund of China Project 2013M541339

References

[1] P D Spanos A M Chevallier N Politis and B Payne ldquoOilwell drilling a vibration perspectiverdquo The Shock and VibrationDigest vol 35 pp 81ndash99 2013

[2] W D Aldred andM C Sheppard ldquoDrillstring vibrations a newgeneration mechanism and control strategiesrdquo in Proceedingsof the SPE Annual Technical Conference and Exhibition SPE-24582-MS pp 353ndash363 Washington DC USA October 1992

[3] T V Aarrestand H A Tonnesen and A Kyllingstad ldquoDrill-string vibrations theory and experiments on full-scale researchdrilling rigrdquo inProceedings of the IADCSPEDrilling ConferenceSPE Paper No 14760 pp 311ndash321 Dallas Tex USA 1986

[4] J D Jansen ldquoNon-linear rotor dynamics as applied to oilwelldrillstring vibrationsrdquo Journal of Sound and Vibration vol 147no 1 pp 115ndash135 1991

[5] A S Yigit and A P Christoforou ldquoCoupled axial and transversevibrations of oilwell drillstringsrdquo Journal of Sound and Vibra-tion vol 195 no 4 pp 617ndash627 1996

[6] A S Yigit and A P Christoforou ldquoCoupled torsional and bend-ing vibration of drill-strings subject to impact with frictionrdquoJournal of Sound and Vibration vol 215 no 1 pp 167ndash181 1998

[7] K Z Khan Longitudinal and torsional vibration of drill-strings [MS thesis] Massachusetts Institute of TechnologyCambridge Mass USA 1986

[8] R J Shyu Bending of rotating drill-strings [PhD thesis] Mas-sachusetts Institute of Technology 1989

[9] A P Christoforou and A S Yigit ldquoFully coupled vibrations ofactively controlled drillstringsrdquo Journal of Sound and Vibrationvol 267 no 5 pp 1029ndash1045 2003

[10] M A Trindade C Wolter and R Sampaio ldquoKarhunen-Loevedecomposition of coupled axialbending vibrations of beamssubject to impactsrdquo Journal of Sound and Vibration vol 279 no3ndash5 pp 1015ndash1036 2005

[11] R Sampaio M T Piovan and G Venero Lozano ldquoCoupledaxialtorsional vibrations of drill-strings bymeans of non-linearmodelrdquo Mechanics Research Communications vol 34 no 5-6pp 497ndash502 2007

[12] H Hakimi and S Moradi ldquoDrillstring vibration analysis usingdifferential quadrature methodrdquo Journal of Petroleum Scienceand Engineering vol 70 no 3-4 pp 235ndash242 2010

[13] F S Ren S Chen and Z G Yao ldquoDynamics analysis andvibration suppression of a flexible rotation beamrdquo AppliedMechanics and Materials vol 214 pp 165ndash172 2012

[14] F-S Ren and Z-G Yao ldquoStudy on nonlinear dynamics andbifurcations in rotating compressive-drill stringrdquo EngineeringMechanics vol 30 no 10 pp 251ndash256 2013

[15] Z-F Li Y-G Zhang X-T Hou W-D Liu and G-Q XuldquoAnalysis of longitudinal and torsion vibration of drill-stringsrdquoEngineering Mechanics vol 21 no 6 pp 203ndash210 2004 (Chi-nese)

[16] G J Sheu and S M Yang ldquoDynamic analysis of a spinningRayleigh beamrdquo International Journal of Mechanical Sciencesvol 47 no 2 pp 157ndash169 2005

[17] X Zhou X Qichong D Hu et al ldquoThe nonlinear dynam-ics analysis of bottom drillstring of air drillingrdquo Journal ofChongqing University of Science and Technology vol 4 pp 119ndash121 2013 (Chinese)

[18] Y-J Jia P Jiang X-H Zhu and Y-F Zhang ldquoNonlinearbuckling simulation of drillstring system during pilot holedrilling in horizontal directional drillingrdquo Journal of SystemSimulation vol 25 no 4 pp 821ndash825 2013 (Chinese)

[19] W-S Xiao Z-Y Liu H-Y Wang X-F Wang L Fu and Y-C Yin ldquoTensional vibration analysis of drill-string by FSIrdquo OilField Equipment vol 42 pp 23ndash26 2013 (Chinese)

[20] D Shao Z Guan X Wen and Y Shi ldquoExperiment on lateralvibration characteristics of horizontal rotary drilling stringrdquoJournal of China University of Petroleum vol 37 pp 100ndash1052013 (Chinese)

[21] W Zhang X Zhu Z Zhou and W Ma ldquoEffect of rotationalspeeds mutation on the dynamic characteristics of the drill-string system in a vertical wellrdquo Journal of PetrochemicalUniversities vol 26 no 2 pp 47ndash51 2013 (Chinese)

[22] Y Tang ldquoNonlinear vibrations of axially accelerating viscoelas-tic Timoshenko beamsrdquo Chinese Journal of Theoretical andApplied Mechanics vol 45 no 6 pp 965ndash973 2013 (Chinese)

Shock and Vibration 17

[23] P B Goncalves and Z J G N Del Prado ldquoNonlinear oscilla-tions and stability of parametrically excited cylindrical shellsrdquoMeccanica vol 37 no 6 pp 569ndash597 2002

[24] M Strozzi and F Pellicano ldquoNonlinear vibrations of function-ally graded cylindrical shellsrdquo Thin-Walled Structures vol 67pp 63ndash77 2013

[25] M Amabili Nonlinear Vibrations and Stability of Shells andPlates Cambridge University Press Cambridge CambridgeUK 2008

[26] A W Leissa Vibrations of Shells Government Printing OfficeWashington DC USA 1973

[27] N Yamaki Elastic Stability of Circular Cylindrical Shells North-Holland Amsterdam The Netherlands 1984

[28] J N ReddyMechanics of LaminatedComposite Plates and ShellsCRC Press Boca Raton Fla USA 2004

[29] A H Nayfeh and D T Mook Nonlinear Oscillations Wiley-VCH Press Weinheim Germany 2004

[30] P D Spanos A K Sengupta R A Cunningham and P RPaslay ldquoModeling of roller cone bit lift-off dynamics in rotarydrillingrdquo Journal of Energy Resources Technology vol 117 no 3pp 197ndash207 1995

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 9

x2

minus2 0 2minus4

x1

minus3

minus2

minus1

1

0

2

3

(a) Phase diagram of the axial modal

minus4

minus3

minus2

minus1

0

1

2

x1

1010 1020 1030 1040 10501000t

(b) Oscillogram of the axial modal

minus10 minus05 00 05 10minus15

x3

minus15

minus10

minus05

00

05

10

x4

(c) Phase diagram of the torsional modal

minus15

minus10

minus05

00

05

10x3

1010 1020 1030 1040 10501000t

(d) Oscillogram of the torsional modal

5

0 02

0

1

minus1

minus2

x3

minus5

x2

minus4

minus2 x 1

(e) Three-dimensional phase diagram

minus15

minus10

minus05

00

05

10

15

x2

0500 10 15minus10 minus05minus15

x1

(f) Poincare section

Figure 6 Period responses of the system (1199023= 75)

10 Shock and Vibration

x2

minus2 0 2minus4

x1

minus3

minus2

minus1

1

0

2

3

(a) Phase diagram of the axial modal

1020 1040 1060 10801000t

minus4

minus3

minus2

minus1

0

1

2

x1

(b) Oscillogram of the axial modal

minus10 minus05 00 05 10minus15

x3

minus15

minus10

minus05

00

05

10

x4

(c) Phase diagram of the torsional modal

1020 1040 1060 10801000t

minus15

minus10

minus05

00

05

10x3

(d) Oscillogram of the torsional modal

5

00

2

0

1

minus1

minus2

x3

minus5

x2

minus4

minus2 x 1

(e) Three-dimensional phase diagram

minus10 minus05 00 05 10 15minus15

x1

minus15

minus10

minus05

00

05

10

15

x2

(f) Poincare section

Figure 7 The almost-periodic responses of the system (1199023= 768)

Shock and Vibration 11

06 07 08 09 1005q2

x2

minus10

minus05

00

(a) Bifurcation diagram of the axial modal

x3

06 07 0805 1009q2

10

15

20

(b) Bifurcation diagram of the torsional modal

Figure 8 Bifurcation of the resonance frequency-multiplier

a constant WOB in the case of a lower-rotation speed andlower rock hardness high drill efficiency can be attained byproperly increasing the WOB In the case of high-rotationspeed the coupling vibration is sensitive to the axial stressso the proper WOB should be adopted to avoid the couplingvibration of the drill string based on geological parameters

5 Conclusions

In this paper wemodeled and analyzed drill-string vibrationsby focusing on the coupled axialtorsional vibrations bymeans of nonlinear dynamics and qualitative analysis aimingat revealing the key effective factors influencing the coupledvibrations Here the drill string was described as a simplifiedequivalent flexible shell under axial rotation in which theexcitation loads and boundary conditions of the drill stringwere simplified After dimensionless processing we built thedynamics motion equation and the average equation

We found that the low-frequency amplitude expressionof the torsional excitation and the high-frequency amplitudeexpression of the axial excitation in the average equation arethe factors that determine the coupling vibration Based onthis procedure numerical simulations were carried out withadaptive steps using the Runge-Kutta-Fehlberg method todiscover the response of the system vibrations Further wefound that a change in the torsional or the axial excitationaffects both the torsional and the axial vibration simultane-ously and that it is similar qualitatively and synchronous inform

The results of the simulation analysis show that when thedrill string is in a lower-speed rotation zone the torsionalexcitation mainly contributes to the coupling vibrationincreasing the torsional stress of the drill string that moreeasily leads to the coupling vibrationWhen in a higher-speedrotating zone the axial excitation mainly contributes to thecoupling vibration so in a particular interval it is more likelyto cause the coupling vibration of the drill string

Appendix

Consider

1198861= minus

1

2119871120588120587

1198862=

1

120588

(

1

12

ℎ31198642

1198773

minus

ℎ1198642

119877

minus

1

4

1205872119877ℎ1198641

1198712

)

1198863=

1

120588

(

8

45

ℎ31198642

1198711198772

minus

32

15

ℎ1198642

119871

minus

32

15

ℎ1205811198641

119871

)

1198864=

1

120588

(

1

4

1205872ℎ31205811198641

11987131198773

minus

13

16

1205872ℎ1205811198641

1198712119877

minus

1

4

ℎ1198642

1198773

minus

1

16

1205874119877ℎ1198641

1198714

)

1198865=

1

120588

(

1

384

1205872ℎ31198642

11987121198773

minus

9

512

1205874119877ℎ1198641

1198714

minus

3

32

ℎ31198642

1198775

minus

9

32

ℎ1198642

1198773

minus

1

32

1205872ℎ1198641

1198712119877

minus

1

64

1205872ℎ1205811198641

1198712119877

)

1198866=

13ℎ1205811198641

15120588119871

1198867=

1

120588

(

1

4

1205872ℎ31205811198641

11987121198773

+

13

16

1205872ℎ1205811198641

1198712119877

minus

1

4

ℎ1198641

1198773

+

1

16

1205874119877ℎ1198641

1198714

+

3

4

ℎ1198641

1198713

+

1

24

1205872ℎ31198642

11987121198773

+

9

2

1205874119877ℎ1198641

1198714

)

1198868=

1

120588

(

1

2

1205872ℎ1198642

1198712119877

+

3

2

1205874ℎ31198641

1198714119877

+

1205872ℎ1205811198641

1198712119877

+

3

4

ℎ1198641

1198713

12 Shock and Vibrationx2

minus10

minus05

00

05

10

minus05 00 05 10minus10

x1

(a) Phase diagram of the axial modal

minus10

minus05

00

05

10

x1

1020 1040 1060 10801000t

(b) Oscillogram of the axial modal

minus2

minus1

0

1

2

x4

15 20 25 3010x3

(c) Phase diagram of the torsional modal

10

15

20

25

30x3

1020 1040 1060 10801000t

(d) Oscillogram of the torsional modal

1

0 0

11

2

3

minus1minus1

x3

x2 x 1

(e) Three-dimensional phase diagram

minus15

minus10

minus05

00

05

10

15

x2

0500 10 15minus10 minus05minus15

x1

(f) Poincare section

Figure 9 Period responses of the system (1199022= 05025)

Shock and Vibration 13x2

minus05 00 05 10minus10

x1

minus10

minus05

00

05

10

(a) Phase diagram of the axial modal

1020 1040 1060 10801000t

minus10

minus05

00

05

10

x1

(b) Oscillogram of the axial modal

minus20

minus15

minus10

minus05

00

05

10

15

20

x4

15 20 25 3010x3

(c) Phase diagram of the torsional modal

10

15

20

25

30x3

1020 1040 1060 10801000t

(d) Oscillogram of the torsional modal

1

0 0

11

2

3

minus1 minus1

x3

x2 x 1

(e) Three-dimensional phase diagram

minus10 minus05minus15 05 10 1500x1

minus15

minus10

minus05

00

05

10

15

x2

(f) Poincare section

Figure 10 Period responses of the system (1199022= 0635)

14 Shock and Vibration

minus2

minus1

0

1

2

x2

10 2 3minus1minus2

x1

(a) Phase diagram of the axial modal

1050 1100 1150 12001000t

minus2

minus1

0

1

2

3

x1

(b) Oscillogram of the axial modal

2 4minus2 0minus4

x3

minus4

minus2

0

2

4

x4

(c) Phase diagram of the torsional modal

minus4

minus2

0

2

4x3

1050 1100 1150 12001000t

(d) Oscillogram of the torsional modal

2

00

2

4

0

5

minus2

x3

minus5

x2

minus2

x 1

(e) Three-dimensional phase diagram

minus15

minus10

minus05

00

05

10

15

x2

0500 10 15minus10 minus05minus15

x1

(f) Poincare section

Figure 11 Period responses of the system (1199022= 07725)

Shock and Vibration 15x2

minus05 00 05 10minus10

x1

minus10

minus05

00

05

10

(a) Phase diagram of the axial modal

1020 1040 1060 10801000t

minus10

minus05

00

05

10

x1

(b) Oscillogram of the axial modal

15 20 25 3010x3

minus2

minus1

0

1

2

x4

(c) Phase diagram of the torsional modal

10

15

20

25

30x3

1020 1040 1060 10801000t

(d) Oscillogram of the torsional modal

1

0 0

11

2

3

minus1minus1

x3

x2

x 1

(e) Three-dimensional phase diagram

minus10 minus05minus15 05 10 1500x1

minus15

minus10

minus05

00

05

10

15

x2

(f) Poincare section

Figure 12 Period-doubled responses of the system (1199022= 079)

16 Shock and Vibration

+

1

24

1205872ℎ31198642

11987121198773

+

9

2

1205874119877ℎ1198641

1198714

)

1198869=

1198621

2120587119877radic120588119864

11988610

=

1205872ℎ31198641120581

411987121205881198772

minus

ℎ1198641

41205881198772

minus

1205874ℎ1198771198641

161198714120588

minus

1205872ℎ1198641120581

161198712120588119877

11988611

= minus

91205874ℎ1198771198641

5121198714120588

minus

3ℎ31198641

321205881198775

minus

1205872ℎ1198642

321198712120588119877

minus

1205872ℎ31198642

38411987121205881198773

minus

9ℎ1198641

321205881198773

minus

1205872ℎ1198641120581

641198712120588119877

11988612

=

1198622

2120587119877radic120588119864

11988613

= minus

41205872119877ℎ1198642

1198712120588

minus

ℎ1198641

119877120588

minus

1205872ℎ31198642

31198712120588119877

11988614

=

1205872ℎ31198641120581

411987121205881198773

minus

ℎ1198641

41205881198773

minus

1205874ℎ1198771198641

161198714120588

minus

131205872ℎ1198641120581

161198712120588119877

11988615

= minus

1205872ℎ31198642

2411987121205881198773

minus

1205872ℎ1198642

21198712120588119877

minus

91205874ℎ1198641120581

21198714120588

minus

3ℎ1198641

41205881198773

minus

1205872ℎ1198641120581

1198712120588119877

minus

31205874ℎ31198641

21198714120588119877

(A1)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to acknowledge financial support byNatural Science Foundation of China Project 11372071 andPostdoctoral Fund of China Project 2013M541339

References

[1] P D Spanos A M Chevallier N Politis and B Payne ldquoOilwell drilling a vibration perspectiverdquo The Shock and VibrationDigest vol 35 pp 81ndash99 2013

[2] W D Aldred andM C Sheppard ldquoDrillstring vibrations a newgeneration mechanism and control strategiesrdquo in Proceedingsof the SPE Annual Technical Conference and Exhibition SPE-24582-MS pp 353ndash363 Washington DC USA October 1992

[3] T V Aarrestand H A Tonnesen and A Kyllingstad ldquoDrill-string vibrations theory and experiments on full-scale researchdrilling rigrdquo inProceedings of the IADCSPEDrilling ConferenceSPE Paper No 14760 pp 311ndash321 Dallas Tex USA 1986

[4] J D Jansen ldquoNon-linear rotor dynamics as applied to oilwelldrillstring vibrationsrdquo Journal of Sound and Vibration vol 147no 1 pp 115ndash135 1991

[5] A S Yigit and A P Christoforou ldquoCoupled axial and transversevibrations of oilwell drillstringsrdquo Journal of Sound and Vibra-tion vol 195 no 4 pp 617ndash627 1996

[6] A S Yigit and A P Christoforou ldquoCoupled torsional and bend-ing vibration of drill-strings subject to impact with frictionrdquoJournal of Sound and Vibration vol 215 no 1 pp 167ndash181 1998

[7] K Z Khan Longitudinal and torsional vibration of drill-strings [MS thesis] Massachusetts Institute of TechnologyCambridge Mass USA 1986

[8] R J Shyu Bending of rotating drill-strings [PhD thesis] Mas-sachusetts Institute of Technology 1989

[9] A P Christoforou and A S Yigit ldquoFully coupled vibrations ofactively controlled drillstringsrdquo Journal of Sound and Vibrationvol 267 no 5 pp 1029ndash1045 2003

[10] M A Trindade C Wolter and R Sampaio ldquoKarhunen-Loevedecomposition of coupled axialbending vibrations of beamssubject to impactsrdquo Journal of Sound and Vibration vol 279 no3ndash5 pp 1015ndash1036 2005

[11] R Sampaio M T Piovan and G Venero Lozano ldquoCoupledaxialtorsional vibrations of drill-strings bymeans of non-linearmodelrdquo Mechanics Research Communications vol 34 no 5-6pp 497ndash502 2007

[12] H Hakimi and S Moradi ldquoDrillstring vibration analysis usingdifferential quadrature methodrdquo Journal of Petroleum Scienceand Engineering vol 70 no 3-4 pp 235ndash242 2010

[13] F S Ren S Chen and Z G Yao ldquoDynamics analysis andvibration suppression of a flexible rotation beamrdquo AppliedMechanics and Materials vol 214 pp 165ndash172 2012

[14] F-S Ren and Z-G Yao ldquoStudy on nonlinear dynamics andbifurcations in rotating compressive-drill stringrdquo EngineeringMechanics vol 30 no 10 pp 251ndash256 2013

[15] Z-F Li Y-G Zhang X-T Hou W-D Liu and G-Q XuldquoAnalysis of longitudinal and torsion vibration of drill-stringsrdquoEngineering Mechanics vol 21 no 6 pp 203ndash210 2004 (Chi-nese)

[16] G J Sheu and S M Yang ldquoDynamic analysis of a spinningRayleigh beamrdquo International Journal of Mechanical Sciencesvol 47 no 2 pp 157ndash169 2005

[17] X Zhou X Qichong D Hu et al ldquoThe nonlinear dynam-ics analysis of bottom drillstring of air drillingrdquo Journal ofChongqing University of Science and Technology vol 4 pp 119ndash121 2013 (Chinese)

[18] Y-J Jia P Jiang X-H Zhu and Y-F Zhang ldquoNonlinearbuckling simulation of drillstring system during pilot holedrilling in horizontal directional drillingrdquo Journal of SystemSimulation vol 25 no 4 pp 821ndash825 2013 (Chinese)

[19] W-S Xiao Z-Y Liu H-Y Wang X-F Wang L Fu and Y-C Yin ldquoTensional vibration analysis of drill-string by FSIrdquo OilField Equipment vol 42 pp 23ndash26 2013 (Chinese)

[20] D Shao Z Guan X Wen and Y Shi ldquoExperiment on lateralvibration characteristics of horizontal rotary drilling stringrdquoJournal of China University of Petroleum vol 37 pp 100ndash1052013 (Chinese)

[21] W Zhang X Zhu Z Zhou and W Ma ldquoEffect of rotationalspeeds mutation on the dynamic characteristics of the drill-string system in a vertical wellrdquo Journal of PetrochemicalUniversities vol 26 no 2 pp 47ndash51 2013 (Chinese)

[22] Y Tang ldquoNonlinear vibrations of axially accelerating viscoelas-tic Timoshenko beamsrdquo Chinese Journal of Theoretical andApplied Mechanics vol 45 no 6 pp 965ndash973 2013 (Chinese)

Shock and Vibration 17

[23] P B Goncalves and Z J G N Del Prado ldquoNonlinear oscilla-tions and stability of parametrically excited cylindrical shellsrdquoMeccanica vol 37 no 6 pp 569ndash597 2002

[24] M Strozzi and F Pellicano ldquoNonlinear vibrations of function-ally graded cylindrical shellsrdquo Thin-Walled Structures vol 67pp 63ndash77 2013

[25] M Amabili Nonlinear Vibrations and Stability of Shells andPlates Cambridge University Press Cambridge CambridgeUK 2008

[26] A W Leissa Vibrations of Shells Government Printing OfficeWashington DC USA 1973

[27] N Yamaki Elastic Stability of Circular Cylindrical Shells North-Holland Amsterdam The Netherlands 1984

[28] J N ReddyMechanics of LaminatedComposite Plates and ShellsCRC Press Boca Raton Fla USA 2004

[29] A H Nayfeh and D T Mook Nonlinear Oscillations Wiley-VCH Press Weinheim Germany 2004

[30] P D Spanos A K Sengupta R A Cunningham and P RPaslay ldquoModeling of roller cone bit lift-off dynamics in rotarydrillingrdquo Journal of Energy Resources Technology vol 117 no 3pp 197ndash207 1995

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

10 Shock and Vibration

x2

minus2 0 2minus4

x1

minus3

minus2

minus1

1

0

2

3

(a) Phase diagram of the axial modal

1020 1040 1060 10801000t

minus4

minus3

minus2

minus1

0

1

2

x1

(b) Oscillogram of the axial modal

minus10 minus05 00 05 10minus15

x3

minus15

minus10

minus05

00

05

10

x4

(c) Phase diagram of the torsional modal

1020 1040 1060 10801000t

minus15

minus10

minus05

00

05

10x3

(d) Oscillogram of the torsional modal

5

00

2

0

1

minus1

minus2

x3

minus5

x2

minus4

minus2 x 1

(e) Three-dimensional phase diagram

minus10 minus05 00 05 10 15minus15

x1

minus15

minus10

minus05

00

05

10

15

x2

(f) Poincare section

Figure 7 The almost-periodic responses of the system (1199023= 768)

Shock and Vibration 11

06 07 08 09 1005q2

x2

minus10

minus05

00

(a) Bifurcation diagram of the axial modal

x3

06 07 0805 1009q2

10

15

20

(b) Bifurcation diagram of the torsional modal

Figure 8 Bifurcation of the resonance frequency-multiplier

a constant WOB in the case of a lower-rotation speed andlower rock hardness high drill efficiency can be attained byproperly increasing the WOB In the case of high-rotationspeed the coupling vibration is sensitive to the axial stressso the proper WOB should be adopted to avoid the couplingvibration of the drill string based on geological parameters

5 Conclusions

In this paper wemodeled and analyzed drill-string vibrationsby focusing on the coupled axialtorsional vibrations bymeans of nonlinear dynamics and qualitative analysis aimingat revealing the key effective factors influencing the coupledvibrations Here the drill string was described as a simplifiedequivalent flexible shell under axial rotation in which theexcitation loads and boundary conditions of the drill stringwere simplified After dimensionless processing we built thedynamics motion equation and the average equation

We found that the low-frequency amplitude expressionof the torsional excitation and the high-frequency amplitudeexpression of the axial excitation in the average equation arethe factors that determine the coupling vibration Based onthis procedure numerical simulations were carried out withadaptive steps using the Runge-Kutta-Fehlberg method todiscover the response of the system vibrations Further wefound that a change in the torsional or the axial excitationaffects both the torsional and the axial vibration simultane-ously and that it is similar qualitatively and synchronous inform

The results of the simulation analysis show that when thedrill string is in a lower-speed rotation zone the torsionalexcitation mainly contributes to the coupling vibrationincreasing the torsional stress of the drill string that moreeasily leads to the coupling vibrationWhen in a higher-speedrotating zone the axial excitation mainly contributes to thecoupling vibration so in a particular interval it is more likelyto cause the coupling vibration of the drill string

Appendix

Consider

1198861= minus

1

2119871120588120587

1198862=

1

120588

(

1

12

ℎ31198642

1198773

minus

ℎ1198642

119877

minus

1

4

1205872119877ℎ1198641

1198712

)

1198863=

1

120588

(

8

45

ℎ31198642

1198711198772

minus

32

15

ℎ1198642

119871

minus

32

15

ℎ1205811198641

119871

)

1198864=

1

120588

(

1

4

1205872ℎ31205811198641

11987131198773

minus

13

16

1205872ℎ1205811198641

1198712119877

minus

1

4

ℎ1198642

1198773

minus

1

16

1205874119877ℎ1198641

1198714

)

1198865=

1

120588

(

1

384

1205872ℎ31198642

11987121198773

minus

9

512

1205874119877ℎ1198641

1198714

minus

3

32

ℎ31198642

1198775

minus

9

32

ℎ1198642

1198773

minus

1

32

1205872ℎ1198641

1198712119877

minus

1

64

1205872ℎ1205811198641

1198712119877

)

1198866=

13ℎ1205811198641

15120588119871

1198867=

1

120588

(

1

4

1205872ℎ31205811198641

11987121198773

+

13

16

1205872ℎ1205811198641

1198712119877

minus

1

4

ℎ1198641

1198773

+

1

16

1205874119877ℎ1198641

1198714

+

3

4

ℎ1198641

1198713

+

1

24

1205872ℎ31198642

11987121198773

+

9

2

1205874119877ℎ1198641

1198714

)

1198868=

1

120588

(

1

2

1205872ℎ1198642

1198712119877

+

3

2

1205874ℎ31198641

1198714119877

+

1205872ℎ1205811198641

1198712119877

+

3

4

ℎ1198641

1198713

12 Shock and Vibrationx2

minus10

minus05

00

05

10

minus05 00 05 10minus10

x1

(a) Phase diagram of the axial modal

minus10

minus05

00

05

10

x1

1020 1040 1060 10801000t

(b) Oscillogram of the axial modal

minus2

minus1

0

1

2

x4

15 20 25 3010x3

(c) Phase diagram of the torsional modal

10

15

20

25

30x3

1020 1040 1060 10801000t

(d) Oscillogram of the torsional modal

1

0 0

11

2

3

minus1minus1

x3

x2 x 1

(e) Three-dimensional phase diagram

minus15

minus10

minus05

00

05

10

15

x2

0500 10 15minus10 minus05minus15

x1

(f) Poincare section

Figure 9 Period responses of the system (1199022= 05025)

Shock and Vibration 13x2

minus05 00 05 10minus10

x1

minus10

minus05

00

05

10

(a) Phase diagram of the axial modal

1020 1040 1060 10801000t

minus10

minus05

00

05

10

x1

(b) Oscillogram of the axial modal

minus20

minus15

minus10

minus05

00

05

10

15

20

x4

15 20 25 3010x3

(c) Phase diagram of the torsional modal

10

15

20

25

30x3

1020 1040 1060 10801000t

(d) Oscillogram of the torsional modal

1

0 0

11

2

3

minus1 minus1

x3

x2 x 1

(e) Three-dimensional phase diagram

minus10 minus05minus15 05 10 1500x1

minus15

minus10

minus05

00

05

10

15

x2

(f) Poincare section

Figure 10 Period responses of the system (1199022= 0635)

14 Shock and Vibration

minus2

minus1

0

1

2

x2

10 2 3minus1minus2

x1

(a) Phase diagram of the axial modal

1050 1100 1150 12001000t

minus2

minus1

0

1

2

3

x1

(b) Oscillogram of the axial modal

2 4minus2 0minus4

x3

minus4

minus2

0

2

4

x4

(c) Phase diagram of the torsional modal

minus4

minus2

0

2

4x3

1050 1100 1150 12001000t

(d) Oscillogram of the torsional modal

2

00

2

4

0

5

minus2

x3

minus5

x2

minus2

x 1

(e) Three-dimensional phase diagram

minus15

minus10

minus05

00

05

10

15

x2

0500 10 15minus10 minus05minus15

x1

(f) Poincare section

Figure 11 Period responses of the system (1199022= 07725)

Shock and Vibration 15x2

minus05 00 05 10minus10

x1

minus10

minus05

00

05

10

(a) Phase diagram of the axial modal

1020 1040 1060 10801000t

minus10

minus05

00

05

10

x1

(b) Oscillogram of the axial modal

15 20 25 3010x3

minus2

minus1

0

1

2

x4

(c) Phase diagram of the torsional modal

10

15

20

25

30x3

1020 1040 1060 10801000t

(d) Oscillogram of the torsional modal

1

0 0

11

2

3

minus1minus1

x3

x2

x 1

(e) Three-dimensional phase diagram

minus10 minus05minus15 05 10 1500x1

minus15

minus10

minus05

00

05

10

15

x2

(f) Poincare section

Figure 12 Period-doubled responses of the system (1199022= 079)

16 Shock and Vibration

+

1

24

1205872ℎ31198642

11987121198773

+

9

2

1205874119877ℎ1198641

1198714

)

1198869=

1198621

2120587119877radic120588119864

11988610

=

1205872ℎ31198641120581

411987121205881198772

minus

ℎ1198641

41205881198772

minus

1205874ℎ1198771198641

161198714120588

minus

1205872ℎ1198641120581

161198712120588119877

11988611

= minus

91205874ℎ1198771198641

5121198714120588

minus

3ℎ31198641

321205881198775

minus

1205872ℎ1198642

321198712120588119877

minus

1205872ℎ31198642

38411987121205881198773

minus

9ℎ1198641

321205881198773

minus

1205872ℎ1198641120581

641198712120588119877

11988612

=

1198622

2120587119877radic120588119864

11988613

= minus

41205872119877ℎ1198642

1198712120588

minus

ℎ1198641

119877120588

minus

1205872ℎ31198642

31198712120588119877

11988614

=

1205872ℎ31198641120581

411987121205881198773

minus

ℎ1198641

41205881198773

minus

1205874ℎ1198771198641

161198714120588

minus

131205872ℎ1198641120581

161198712120588119877

11988615

= minus

1205872ℎ31198642

2411987121205881198773

minus

1205872ℎ1198642

21198712120588119877

minus

91205874ℎ1198641120581

21198714120588

minus

3ℎ1198641

41205881198773

minus

1205872ℎ1198641120581

1198712120588119877

minus

31205874ℎ31198641

21198714120588119877

(A1)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to acknowledge financial support byNatural Science Foundation of China Project 11372071 andPostdoctoral Fund of China Project 2013M541339

References

[1] P D Spanos A M Chevallier N Politis and B Payne ldquoOilwell drilling a vibration perspectiverdquo The Shock and VibrationDigest vol 35 pp 81ndash99 2013

[2] W D Aldred andM C Sheppard ldquoDrillstring vibrations a newgeneration mechanism and control strategiesrdquo in Proceedingsof the SPE Annual Technical Conference and Exhibition SPE-24582-MS pp 353ndash363 Washington DC USA October 1992

[3] T V Aarrestand H A Tonnesen and A Kyllingstad ldquoDrill-string vibrations theory and experiments on full-scale researchdrilling rigrdquo inProceedings of the IADCSPEDrilling ConferenceSPE Paper No 14760 pp 311ndash321 Dallas Tex USA 1986

[4] J D Jansen ldquoNon-linear rotor dynamics as applied to oilwelldrillstring vibrationsrdquo Journal of Sound and Vibration vol 147no 1 pp 115ndash135 1991

[5] A S Yigit and A P Christoforou ldquoCoupled axial and transversevibrations of oilwell drillstringsrdquo Journal of Sound and Vibra-tion vol 195 no 4 pp 617ndash627 1996

[6] A S Yigit and A P Christoforou ldquoCoupled torsional and bend-ing vibration of drill-strings subject to impact with frictionrdquoJournal of Sound and Vibration vol 215 no 1 pp 167ndash181 1998

[7] K Z Khan Longitudinal and torsional vibration of drill-strings [MS thesis] Massachusetts Institute of TechnologyCambridge Mass USA 1986

[8] R J Shyu Bending of rotating drill-strings [PhD thesis] Mas-sachusetts Institute of Technology 1989

[9] A P Christoforou and A S Yigit ldquoFully coupled vibrations ofactively controlled drillstringsrdquo Journal of Sound and Vibrationvol 267 no 5 pp 1029ndash1045 2003

[10] M A Trindade C Wolter and R Sampaio ldquoKarhunen-Loevedecomposition of coupled axialbending vibrations of beamssubject to impactsrdquo Journal of Sound and Vibration vol 279 no3ndash5 pp 1015ndash1036 2005

[11] R Sampaio M T Piovan and G Venero Lozano ldquoCoupledaxialtorsional vibrations of drill-strings bymeans of non-linearmodelrdquo Mechanics Research Communications vol 34 no 5-6pp 497ndash502 2007

[12] H Hakimi and S Moradi ldquoDrillstring vibration analysis usingdifferential quadrature methodrdquo Journal of Petroleum Scienceand Engineering vol 70 no 3-4 pp 235ndash242 2010

[13] F S Ren S Chen and Z G Yao ldquoDynamics analysis andvibration suppression of a flexible rotation beamrdquo AppliedMechanics and Materials vol 214 pp 165ndash172 2012

[14] F-S Ren and Z-G Yao ldquoStudy on nonlinear dynamics andbifurcations in rotating compressive-drill stringrdquo EngineeringMechanics vol 30 no 10 pp 251ndash256 2013

[15] Z-F Li Y-G Zhang X-T Hou W-D Liu and G-Q XuldquoAnalysis of longitudinal and torsion vibration of drill-stringsrdquoEngineering Mechanics vol 21 no 6 pp 203ndash210 2004 (Chi-nese)

[16] G J Sheu and S M Yang ldquoDynamic analysis of a spinningRayleigh beamrdquo International Journal of Mechanical Sciencesvol 47 no 2 pp 157ndash169 2005

[17] X Zhou X Qichong D Hu et al ldquoThe nonlinear dynam-ics analysis of bottom drillstring of air drillingrdquo Journal ofChongqing University of Science and Technology vol 4 pp 119ndash121 2013 (Chinese)

[18] Y-J Jia P Jiang X-H Zhu and Y-F Zhang ldquoNonlinearbuckling simulation of drillstring system during pilot holedrilling in horizontal directional drillingrdquo Journal of SystemSimulation vol 25 no 4 pp 821ndash825 2013 (Chinese)

[19] W-S Xiao Z-Y Liu H-Y Wang X-F Wang L Fu and Y-C Yin ldquoTensional vibration analysis of drill-string by FSIrdquo OilField Equipment vol 42 pp 23ndash26 2013 (Chinese)

[20] D Shao Z Guan X Wen and Y Shi ldquoExperiment on lateralvibration characteristics of horizontal rotary drilling stringrdquoJournal of China University of Petroleum vol 37 pp 100ndash1052013 (Chinese)

[21] W Zhang X Zhu Z Zhou and W Ma ldquoEffect of rotationalspeeds mutation on the dynamic characteristics of the drill-string system in a vertical wellrdquo Journal of PetrochemicalUniversities vol 26 no 2 pp 47ndash51 2013 (Chinese)

[22] Y Tang ldquoNonlinear vibrations of axially accelerating viscoelas-tic Timoshenko beamsrdquo Chinese Journal of Theoretical andApplied Mechanics vol 45 no 6 pp 965ndash973 2013 (Chinese)

Shock and Vibration 17

[23] P B Goncalves and Z J G N Del Prado ldquoNonlinear oscilla-tions and stability of parametrically excited cylindrical shellsrdquoMeccanica vol 37 no 6 pp 569ndash597 2002

[24] M Strozzi and F Pellicano ldquoNonlinear vibrations of function-ally graded cylindrical shellsrdquo Thin-Walled Structures vol 67pp 63ndash77 2013

[25] M Amabili Nonlinear Vibrations and Stability of Shells andPlates Cambridge University Press Cambridge CambridgeUK 2008

[26] A W Leissa Vibrations of Shells Government Printing OfficeWashington DC USA 1973

[27] N Yamaki Elastic Stability of Circular Cylindrical Shells North-Holland Amsterdam The Netherlands 1984

[28] J N ReddyMechanics of LaminatedComposite Plates and ShellsCRC Press Boca Raton Fla USA 2004

[29] A H Nayfeh and D T Mook Nonlinear Oscillations Wiley-VCH Press Weinheim Germany 2004

[30] P D Spanos A K Sengupta R A Cunningham and P RPaslay ldquoModeling of roller cone bit lift-off dynamics in rotarydrillingrdquo Journal of Energy Resources Technology vol 117 no 3pp 197ndash207 1995

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

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Volume 2014

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 11

06 07 08 09 1005q2

x2

minus10

minus05

00

(a) Bifurcation diagram of the axial modal

x3

06 07 0805 1009q2

10

15

20

(b) Bifurcation diagram of the torsional modal

Figure 8 Bifurcation of the resonance frequency-multiplier

a constant WOB in the case of a lower-rotation speed andlower rock hardness high drill efficiency can be attained byproperly increasing the WOB In the case of high-rotationspeed the coupling vibration is sensitive to the axial stressso the proper WOB should be adopted to avoid the couplingvibration of the drill string based on geological parameters

5 Conclusions

In this paper wemodeled and analyzed drill-string vibrationsby focusing on the coupled axialtorsional vibrations bymeans of nonlinear dynamics and qualitative analysis aimingat revealing the key effective factors influencing the coupledvibrations Here the drill string was described as a simplifiedequivalent flexible shell under axial rotation in which theexcitation loads and boundary conditions of the drill stringwere simplified After dimensionless processing we built thedynamics motion equation and the average equation

We found that the low-frequency amplitude expressionof the torsional excitation and the high-frequency amplitudeexpression of the axial excitation in the average equation arethe factors that determine the coupling vibration Based onthis procedure numerical simulations were carried out withadaptive steps using the Runge-Kutta-Fehlberg method todiscover the response of the system vibrations Further wefound that a change in the torsional or the axial excitationaffects both the torsional and the axial vibration simultane-ously and that it is similar qualitatively and synchronous inform

The results of the simulation analysis show that when thedrill string is in a lower-speed rotation zone the torsionalexcitation mainly contributes to the coupling vibrationincreasing the torsional stress of the drill string that moreeasily leads to the coupling vibrationWhen in a higher-speedrotating zone the axial excitation mainly contributes to thecoupling vibration so in a particular interval it is more likelyto cause the coupling vibration of the drill string

Appendix

Consider

1198861= minus

1

2119871120588120587

1198862=

1

120588

(

1

12

ℎ31198642

1198773

minus

ℎ1198642

119877

minus

1

4

1205872119877ℎ1198641

1198712

)

1198863=

1

120588

(

8

45

ℎ31198642

1198711198772

minus

32

15

ℎ1198642

119871

minus

32

15

ℎ1205811198641

119871

)

1198864=

1

120588

(

1

4

1205872ℎ31205811198641

11987131198773

minus

13

16

1205872ℎ1205811198641

1198712119877

minus

1

4

ℎ1198642

1198773

minus

1

16

1205874119877ℎ1198641

1198714

)

1198865=

1

120588

(

1

384

1205872ℎ31198642

11987121198773

minus

9

512

1205874119877ℎ1198641

1198714

minus

3

32

ℎ31198642

1198775

minus

9

32

ℎ1198642

1198773

minus

1

32

1205872ℎ1198641

1198712119877

minus

1

64

1205872ℎ1205811198641

1198712119877

)

1198866=

13ℎ1205811198641

15120588119871

1198867=

1

120588

(

1

4

1205872ℎ31205811198641

11987121198773

+

13

16

1205872ℎ1205811198641

1198712119877

minus

1

4

ℎ1198641

1198773

+

1

16

1205874119877ℎ1198641

1198714

+

3

4

ℎ1198641

1198713

+

1

24

1205872ℎ31198642

11987121198773

+

9

2

1205874119877ℎ1198641

1198714

)

1198868=

1

120588

(

1

2

1205872ℎ1198642

1198712119877

+

3

2

1205874ℎ31198641

1198714119877

+

1205872ℎ1205811198641

1198712119877

+

3

4

ℎ1198641

1198713

12 Shock and Vibrationx2

minus10

minus05

00

05

10

minus05 00 05 10minus10

x1

(a) Phase diagram of the axial modal

minus10

minus05

00

05

10

x1

1020 1040 1060 10801000t

(b) Oscillogram of the axial modal

minus2

minus1

0

1

2

x4

15 20 25 3010x3

(c) Phase diagram of the torsional modal

10

15

20

25

30x3

1020 1040 1060 10801000t

(d) Oscillogram of the torsional modal

1

0 0

11

2

3

minus1minus1

x3

x2 x 1

(e) Three-dimensional phase diagram

minus15

minus10

minus05

00

05

10

15

x2

0500 10 15minus10 minus05minus15

x1

(f) Poincare section

Figure 9 Period responses of the system (1199022= 05025)

Shock and Vibration 13x2

minus05 00 05 10minus10

x1

minus10

minus05

00

05

10

(a) Phase diagram of the axial modal

1020 1040 1060 10801000t

minus10

minus05

00

05

10

x1

(b) Oscillogram of the axial modal

minus20

minus15

minus10

minus05

00

05

10

15

20

x4

15 20 25 3010x3

(c) Phase diagram of the torsional modal

10

15

20

25

30x3

1020 1040 1060 10801000t

(d) Oscillogram of the torsional modal

1

0 0

11

2

3

minus1 minus1

x3

x2 x 1

(e) Three-dimensional phase diagram

minus10 minus05minus15 05 10 1500x1

minus15

minus10

minus05

00

05

10

15

x2

(f) Poincare section

Figure 10 Period responses of the system (1199022= 0635)

14 Shock and Vibration

minus2

minus1

0

1

2

x2

10 2 3minus1minus2

x1

(a) Phase diagram of the axial modal

1050 1100 1150 12001000t

minus2

minus1

0

1

2

3

x1

(b) Oscillogram of the axial modal

2 4minus2 0minus4

x3

minus4

minus2

0

2

4

x4

(c) Phase diagram of the torsional modal

minus4

minus2

0

2

4x3

1050 1100 1150 12001000t

(d) Oscillogram of the torsional modal

2

00

2

4

0

5

minus2

x3

minus5

x2

minus2

x 1

(e) Three-dimensional phase diagram

minus15

minus10

minus05

00

05

10

15

x2

0500 10 15minus10 minus05minus15

x1

(f) Poincare section

Figure 11 Period responses of the system (1199022= 07725)

Shock and Vibration 15x2

minus05 00 05 10minus10

x1

minus10

minus05

00

05

10

(a) Phase diagram of the axial modal

1020 1040 1060 10801000t

minus10

minus05

00

05

10

x1

(b) Oscillogram of the axial modal

15 20 25 3010x3

minus2

minus1

0

1

2

x4

(c) Phase diagram of the torsional modal

10

15

20

25

30x3

1020 1040 1060 10801000t

(d) Oscillogram of the torsional modal

1

0 0

11

2

3

minus1minus1

x3

x2

x 1

(e) Three-dimensional phase diagram

minus10 minus05minus15 05 10 1500x1

minus15

minus10

minus05

00

05

10

15

x2

(f) Poincare section

Figure 12 Period-doubled responses of the system (1199022= 079)

16 Shock and Vibration

+

1

24

1205872ℎ31198642

11987121198773

+

9

2

1205874119877ℎ1198641

1198714

)

1198869=

1198621

2120587119877radic120588119864

11988610

=

1205872ℎ31198641120581

411987121205881198772

minus

ℎ1198641

41205881198772

minus

1205874ℎ1198771198641

161198714120588

minus

1205872ℎ1198641120581

161198712120588119877

11988611

= minus

91205874ℎ1198771198641

5121198714120588

minus

3ℎ31198641

321205881198775

minus

1205872ℎ1198642

321198712120588119877

minus

1205872ℎ31198642

38411987121205881198773

minus

9ℎ1198641

321205881198773

minus

1205872ℎ1198641120581

641198712120588119877

11988612

=

1198622

2120587119877radic120588119864

11988613

= minus

41205872119877ℎ1198642

1198712120588

minus

ℎ1198641

119877120588

minus

1205872ℎ31198642

31198712120588119877

11988614

=

1205872ℎ31198641120581

411987121205881198773

minus

ℎ1198641

41205881198773

minus

1205874ℎ1198771198641

161198714120588

minus

131205872ℎ1198641120581

161198712120588119877

11988615

= minus

1205872ℎ31198642

2411987121205881198773

minus

1205872ℎ1198642

21198712120588119877

minus

91205874ℎ1198641120581

21198714120588

minus

3ℎ1198641

41205881198773

minus

1205872ℎ1198641120581

1198712120588119877

minus

31205874ℎ31198641

21198714120588119877

(A1)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to acknowledge financial support byNatural Science Foundation of China Project 11372071 andPostdoctoral Fund of China Project 2013M541339

References

[1] P D Spanos A M Chevallier N Politis and B Payne ldquoOilwell drilling a vibration perspectiverdquo The Shock and VibrationDigest vol 35 pp 81ndash99 2013

[2] W D Aldred andM C Sheppard ldquoDrillstring vibrations a newgeneration mechanism and control strategiesrdquo in Proceedingsof the SPE Annual Technical Conference and Exhibition SPE-24582-MS pp 353ndash363 Washington DC USA October 1992

[3] T V Aarrestand H A Tonnesen and A Kyllingstad ldquoDrill-string vibrations theory and experiments on full-scale researchdrilling rigrdquo inProceedings of the IADCSPEDrilling ConferenceSPE Paper No 14760 pp 311ndash321 Dallas Tex USA 1986

[4] J D Jansen ldquoNon-linear rotor dynamics as applied to oilwelldrillstring vibrationsrdquo Journal of Sound and Vibration vol 147no 1 pp 115ndash135 1991

[5] A S Yigit and A P Christoforou ldquoCoupled axial and transversevibrations of oilwell drillstringsrdquo Journal of Sound and Vibra-tion vol 195 no 4 pp 617ndash627 1996

[6] A S Yigit and A P Christoforou ldquoCoupled torsional and bend-ing vibration of drill-strings subject to impact with frictionrdquoJournal of Sound and Vibration vol 215 no 1 pp 167ndash181 1998

[7] K Z Khan Longitudinal and torsional vibration of drill-strings [MS thesis] Massachusetts Institute of TechnologyCambridge Mass USA 1986

[8] R J Shyu Bending of rotating drill-strings [PhD thesis] Mas-sachusetts Institute of Technology 1989

[9] A P Christoforou and A S Yigit ldquoFully coupled vibrations ofactively controlled drillstringsrdquo Journal of Sound and Vibrationvol 267 no 5 pp 1029ndash1045 2003

[10] M A Trindade C Wolter and R Sampaio ldquoKarhunen-Loevedecomposition of coupled axialbending vibrations of beamssubject to impactsrdquo Journal of Sound and Vibration vol 279 no3ndash5 pp 1015ndash1036 2005

[11] R Sampaio M T Piovan and G Venero Lozano ldquoCoupledaxialtorsional vibrations of drill-strings bymeans of non-linearmodelrdquo Mechanics Research Communications vol 34 no 5-6pp 497ndash502 2007

[12] H Hakimi and S Moradi ldquoDrillstring vibration analysis usingdifferential quadrature methodrdquo Journal of Petroleum Scienceand Engineering vol 70 no 3-4 pp 235ndash242 2010

[13] F S Ren S Chen and Z G Yao ldquoDynamics analysis andvibration suppression of a flexible rotation beamrdquo AppliedMechanics and Materials vol 214 pp 165ndash172 2012

[14] F-S Ren and Z-G Yao ldquoStudy on nonlinear dynamics andbifurcations in rotating compressive-drill stringrdquo EngineeringMechanics vol 30 no 10 pp 251ndash256 2013

[15] Z-F Li Y-G Zhang X-T Hou W-D Liu and G-Q XuldquoAnalysis of longitudinal and torsion vibration of drill-stringsrdquoEngineering Mechanics vol 21 no 6 pp 203ndash210 2004 (Chi-nese)

[16] G J Sheu and S M Yang ldquoDynamic analysis of a spinningRayleigh beamrdquo International Journal of Mechanical Sciencesvol 47 no 2 pp 157ndash169 2005

[17] X Zhou X Qichong D Hu et al ldquoThe nonlinear dynam-ics analysis of bottom drillstring of air drillingrdquo Journal ofChongqing University of Science and Technology vol 4 pp 119ndash121 2013 (Chinese)

[18] Y-J Jia P Jiang X-H Zhu and Y-F Zhang ldquoNonlinearbuckling simulation of drillstring system during pilot holedrilling in horizontal directional drillingrdquo Journal of SystemSimulation vol 25 no 4 pp 821ndash825 2013 (Chinese)

[19] W-S Xiao Z-Y Liu H-Y Wang X-F Wang L Fu and Y-C Yin ldquoTensional vibration analysis of drill-string by FSIrdquo OilField Equipment vol 42 pp 23ndash26 2013 (Chinese)

[20] D Shao Z Guan X Wen and Y Shi ldquoExperiment on lateralvibration characteristics of horizontal rotary drilling stringrdquoJournal of China University of Petroleum vol 37 pp 100ndash1052013 (Chinese)

[21] W Zhang X Zhu Z Zhou and W Ma ldquoEffect of rotationalspeeds mutation on the dynamic characteristics of the drill-string system in a vertical wellrdquo Journal of PetrochemicalUniversities vol 26 no 2 pp 47ndash51 2013 (Chinese)

[22] Y Tang ldquoNonlinear vibrations of axially accelerating viscoelas-tic Timoshenko beamsrdquo Chinese Journal of Theoretical andApplied Mechanics vol 45 no 6 pp 965ndash973 2013 (Chinese)

Shock and Vibration 17

[23] P B Goncalves and Z J G N Del Prado ldquoNonlinear oscilla-tions and stability of parametrically excited cylindrical shellsrdquoMeccanica vol 37 no 6 pp 569ndash597 2002

[24] M Strozzi and F Pellicano ldquoNonlinear vibrations of function-ally graded cylindrical shellsrdquo Thin-Walled Structures vol 67pp 63ndash77 2013

[25] M Amabili Nonlinear Vibrations and Stability of Shells andPlates Cambridge University Press Cambridge CambridgeUK 2008

[26] A W Leissa Vibrations of Shells Government Printing OfficeWashington DC USA 1973

[27] N Yamaki Elastic Stability of Circular Cylindrical Shells North-Holland Amsterdam The Netherlands 1984

[28] J N ReddyMechanics of LaminatedComposite Plates and ShellsCRC Press Boca Raton Fla USA 2004

[29] A H Nayfeh and D T Mook Nonlinear Oscillations Wiley-VCH Press Weinheim Germany 2004

[30] P D Spanos A K Sengupta R A Cunningham and P RPaslay ldquoModeling of roller cone bit lift-off dynamics in rotarydrillingrdquo Journal of Energy Resources Technology vol 117 no 3pp 197ndash207 1995

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

12 Shock and Vibrationx2

minus10

minus05

00

05

10

minus05 00 05 10minus10

x1

(a) Phase diagram of the axial modal

minus10

minus05

00

05

10

x1

1020 1040 1060 10801000t

(b) Oscillogram of the axial modal

minus2

minus1

0

1

2

x4

15 20 25 3010x3

(c) Phase diagram of the torsional modal

10

15

20

25

30x3

1020 1040 1060 10801000t

(d) Oscillogram of the torsional modal

1

0 0

11

2

3

minus1minus1

x3

x2 x 1

(e) Three-dimensional phase diagram

minus15

minus10

minus05

00

05

10

15

x2

0500 10 15minus10 minus05minus15

x1

(f) Poincare section

Figure 9 Period responses of the system (1199022= 05025)

Shock and Vibration 13x2

minus05 00 05 10minus10

x1

minus10

minus05

00

05

10

(a) Phase diagram of the axial modal

1020 1040 1060 10801000t

minus10

minus05

00

05

10

x1

(b) Oscillogram of the axial modal

minus20

minus15

minus10

minus05

00

05

10

15

20

x4

15 20 25 3010x3

(c) Phase diagram of the torsional modal

10

15

20

25

30x3

1020 1040 1060 10801000t

(d) Oscillogram of the torsional modal

1

0 0

11

2

3

minus1 minus1

x3

x2 x 1

(e) Three-dimensional phase diagram

minus10 minus05minus15 05 10 1500x1

minus15

minus10

minus05

00

05

10

15

x2

(f) Poincare section

Figure 10 Period responses of the system (1199022= 0635)

14 Shock and Vibration

minus2

minus1

0

1

2

x2

10 2 3minus1minus2

x1

(a) Phase diagram of the axial modal

1050 1100 1150 12001000t

minus2

minus1

0

1

2

3

x1

(b) Oscillogram of the axial modal

2 4minus2 0minus4

x3

minus4

minus2

0

2

4

x4

(c) Phase diagram of the torsional modal

minus4

minus2

0

2

4x3

1050 1100 1150 12001000t

(d) Oscillogram of the torsional modal

2

00

2

4

0

5

minus2

x3

minus5

x2

minus2

x 1

(e) Three-dimensional phase diagram

minus15

minus10

minus05

00

05

10

15

x2

0500 10 15minus10 minus05minus15

x1

(f) Poincare section

Figure 11 Period responses of the system (1199022= 07725)

Shock and Vibration 15x2

minus05 00 05 10minus10

x1

minus10

minus05

00

05

10

(a) Phase diagram of the axial modal

1020 1040 1060 10801000t

minus10

minus05

00

05

10

x1

(b) Oscillogram of the axial modal

15 20 25 3010x3

minus2

minus1

0

1

2

x4

(c) Phase diagram of the torsional modal

10

15

20

25

30x3

1020 1040 1060 10801000t

(d) Oscillogram of the torsional modal

1

0 0

11

2

3

minus1minus1

x3

x2

x 1

(e) Three-dimensional phase diagram

minus10 minus05minus15 05 10 1500x1

minus15

minus10

minus05

00

05

10

15

x2

(f) Poincare section

Figure 12 Period-doubled responses of the system (1199022= 079)

16 Shock and Vibration

+

1

24

1205872ℎ31198642

11987121198773

+

9

2

1205874119877ℎ1198641

1198714

)

1198869=

1198621

2120587119877radic120588119864

11988610

=

1205872ℎ31198641120581

411987121205881198772

minus

ℎ1198641

41205881198772

minus

1205874ℎ1198771198641

161198714120588

minus

1205872ℎ1198641120581

161198712120588119877

11988611

= minus

91205874ℎ1198771198641

5121198714120588

minus

3ℎ31198641

321205881198775

minus

1205872ℎ1198642

321198712120588119877

minus

1205872ℎ31198642

38411987121205881198773

minus

9ℎ1198641

321205881198773

minus

1205872ℎ1198641120581

641198712120588119877

11988612

=

1198622

2120587119877radic120588119864

11988613

= minus

41205872119877ℎ1198642

1198712120588

minus

ℎ1198641

119877120588

minus

1205872ℎ31198642

31198712120588119877

11988614

=

1205872ℎ31198641120581

411987121205881198773

minus

ℎ1198641

41205881198773

minus

1205874ℎ1198771198641

161198714120588

minus

131205872ℎ1198641120581

161198712120588119877

11988615

= minus

1205872ℎ31198642

2411987121205881198773

minus

1205872ℎ1198642

21198712120588119877

minus

91205874ℎ1198641120581

21198714120588

minus

3ℎ1198641

41205881198773

minus

1205872ℎ1198641120581

1198712120588119877

minus

31205874ℎ31198641

21198714120588119877

(A1)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to acknowledge financial support byNatural Science Foundation of China Project 11372071 andPostdoctoral Fund of China Project 2013M541339

References

[1] P D Spanos A M Chevallier N Politis and B Payne ldquoOilwell drilling a vibration perspectiverdquo The Shock and VibrationDigest vol 35 pp 81ndash99 2013

[2] W D Aldred andM C Sheppard ldquoDrillstring vibrations a newgeneration mechanism and control strategiesrdquo in Proceedingsof the SPE Annual Technical Conference and Exhibition SPE-24582-MS pp 353ndash363 Washington DC USA October 1992

[3] T V Aarrestand H A Tonnesen and A Kyllingstad ldquoDrill-string vibrations theory and experiments on full-scale researchdrilling rigrdquo inProceedings of the IADCSPEDrilling ConferenceSPE Paper No 14760 pp 311ndash321 Dallas Tex USA 1986

[4] J D Jansen ldquoNon-linear rotor dynamics as applied to oilwelldrillstring vibrationsrdquo Journal of Sound and Vibration vol 147no 1 pp 115ndash135 1991

[5] A S Yigit and A P Christoforou ldquoCoupled axial and transversevibrations of oilwell drillstringsrdquo Journal of Sound and Vibra-tion vol 195 no 4 pp 617ndash627 1996

[6] A S Yigit and A P Christoforou ldquoCoupled torsional and bend-ing vibration of drill-strings subject to impact with frictionrdquoJournal of Sound and Vibration vol 215 no 1 pp 167ndash181 1998

[7] K Z Khan Longitudinal and torsional vibration of drill-strings [MS thesis] Massachusetts Institute of TechnologyCambridge Mass USA 1986

[8] R J Shyu Bending of rotating drill-strings [PhD thesis] Mas-sachusetts Institute of Technology 1989

[9] A P Christoforou and A S Yigit ldquoFully coupled vibrations ofactively controlled drillstringsrdquo Journal of Sound and Vibrationvol 267 no 5 pp 1029ndash1045 2003

[10] M A Trindade C Wolter and R Sampaio ldquoKarhunen-Loevedecomposition of coupled axialbending vibrations of beamssubject to impactsrdquo Journal of Sound and Vibration vol 279 no3ndash5 pp 1015ndash1036 2005

[11] R Sampaio M T Piovan and G Venero Lozano ldquoCoupledaxialtorsional vibrations of drill-strings bymeans of non-linearmodelrdquo Mechanics Research Communications vol 34 no 5-6pp 497ndash502 2007

[12] H Hakimi and S Moradi ldquoDrillstring vibration analysis usingdifferential quadrature methodrdquo Journal of Petroleum Scienceand Engineering vol 70 no 3-4 pp 235ndash242 2010

[13] F S Ren S Chen and Z G Yao ldquoDynamics analysis andvibration suppression of a flexible rotation beamrdquo AppliedMechanics and Materials vol 214 pp 165ndash172 2012

[14] F-S Ren and Z-G Yao ldquoStudy on nonlinear dynamics andbifurcations in rotating compressive-drill stringrdquo EngineeringMechanics vol 30 no 10 pp 251ndash256 2013

[15] Z-F Li Y-G Zhang X-T Hou W-D Liu and G-Q XuldquoAnalysis of longitudinal and torsion vibration of drill-stringsrdquoEngineering Mechanics vol 21 no 6 pp 203ndash210 2004 (Chi-nese)

[16] G J Sheu and S M Yang ldquoDynamic analysis of a spinningRayleigh beamrdquo International Journal of Mechanical Sciencesvol 47 no 2 pp 157ndash169 2005

[17] X Zhou X Qichong D Hu et al ldquoThe nonlinear dynam-ics analysis of bottom drillstring of air drillingrdquo Journal ofChongqing University of Science and Technology vol 4 pp 119ndash121 2013 (Chinese)

[18] Y-J Jia P Jiang X-H Zhu and Y-F Zhang ldquoNonlinearbuckling simulation of drillstring system during pilot holedrilling in horizontal directional drillingrdquo Journal of SystemSimulation vol 25 no 4 pp 821ndash825 2013 (Chinese)

[19] W-S Xiao Z-Y Liu H-Y Wang X-F Wang L Fu and Y-C Yin ldquoTensional vibration analysis of drill-string by FSIrdquo OilField Equipment vol 42 pp 23ndash26 2013 (Chinese)

[20] D Shao Z Guan X Wen and Y Shi ldquoExperiment on lateralvibration characteristics of horizontal rotary drilling stringrdquoJournal of China University of Petroleum vol 37 pp 100ndash1052013 (Chinese)

[21] W Zhang X Zhu Z Zhou and W Ma ldquoEffect of rotationalspeeds mutation on the dynamic characteristics of the drill-string system in a vertical wellrdquo Journal of PetrochemicalUniversities vol 26 no 2 pp 47ndash51 2013 (Chinese)

[22] Y Tang ldquoNonlinear vibrations of axially accelerating viscoelas-tic Timoshenko beamsrdquo Chinese Journal of Theoretical andApplied Mechanics vol 45 no 6 pp 965ndash973 2013 (Chinese)

Shock and Vibration 17

[23] P B Goncalves and Z J G N Del Prado ldquoNonlinear oscilla-tions and stability of parametrically excited cylindrical shellsrdquoMeccanica vol 37 no 6 pp 569ndash597 2002

[24] M Strozzi and F Pellicano ldquoNonlinear vibrations of function-ally graded cylindrical shellsrdquo Thin-Walled Structures vol 67pp 63ndash77 2013

[25] M Amabili Nonlinear Vibrations and Stability of Shells andPlates Cambridge University Press Cambridge CambridgeUK 2008

[26] A W Leissa Vibrations of Shells Government Printing OfficeWashington DC USA 1973

[27] N Yamaki Elastic Stability of Circular Cylindrical Shells North-Holland Amsterdam The Netherlands 1984

[28] J N ReddyMechanics of LaminatedComposite Plates and ShellsCRC Press Boca Raton Fla USA 2004

[29] A H Nayfeh and D T Mook Nonlinear Oscillations Wiley-VCH Press Weinheim Germany 2004

[30] P D Spanos A K Sengupta R A Cunningham and P RPaslay ldquoModeling of roller cone bit lift-off dynamics in rotarydrillingrdquo Journal of Energy Resources Technology vol 117 no 3pp 197ndash207 1995

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 13x2

minus05 00 05 10minus10

x1

minus10

minus05

00

05

10

(a) Phase diagram of the axial modal

1020 1040 1060 10801000t

minus10

minus05

00

05

10

x1

(b) Oscillogram of the axial modal

minus20

minus15

minus10

minus05

00

05

10

15

20

x4

15 20 25 3010x3

(c) Phase diagram of the torsional modal

10

15

20

25

30x3

1020 1040 1060 10801000t

(d) Oscillogram of the torsional modal

1

0 0

11

2

3

minus1 minus1

x3

x2 x 1

(e) Three-dimensional phase diagram

minus10 minus05minus15 05 10 1500x1

minus15

minus10

minus05

00

05

10

15

x2

(f) Poincare section

Figure 10 Period responses of the system (1199022= 0635)

14 Shock and Vibration

minus2

minus1

0

1

2

x2

10 2 3minus1minus2

x1

(a) Phase diagram of the axial modal

1050 1100 1150 12001000t

minus2

minus1

0

1

2

3

x1

(b) Oscillogram of the axial modal

2 4minus2 0minus4

x3

minus4

minus2

0

2

4

x4

(c) Phase diagram of the torsional modal

minus4

minus2

0

2

4x3

1050 1100 1150 12001000t

(d) Oscillogram of the torsional modal

2

00

2

4

0

5

minus2

x3

minus5

x2

minus2

x 1

(e) Three-dimensional phase diagram

minus15

minus10

minus05

00

05

10

15

x2

0500 10 15minus10 minus05minus15

x1

(f) Poincare section

Figure 11 Period responses of the system (1199022= 07725)

Shock and Vibration 15x2

minus05 00 05 10minus10

x1

minus10

minus05

00

05

10

(a) Phase diagram of the axial modal

1020 1040 1060 10801000t

minus10

minus05

00

05

10

x1

(b) Oscillogram of the axial modal

15 20 25 3010x3

minus2

minus1

0

1

2

x4

(c) Phase diagram of the torsional modal

10

15

20

25

30x3

1020 1040 1060 10801000t

(d) Oscillogram of the torsional modal

1

0 0

11

2

3

minus1minus1

x3

x2

x 1

(e) Three-dimensional phase diagram

minus10 minus05minus15 05 10 1500x1

minus15

minus10

minus05

00

05

10

15

x2

(f) Poincare section

Figure 12 Period-doubled responses of the system (1199022= 079)

16 Shock and Vibration

+

1

24

1205872ℎ31198642

11987121198773

+

9

2

1205874119877ℎ1198641

1198714

)

1198869=

1198621

2120587119877radic120588119864

11988610

=

1205872ℎ31198641120581

411987121205881198772

minus

ℎ1198641

41205881198772

minus

1205874ℎ1198771198641

161198714120588

minus

1205872ℎ1198641120581

161198712120588119877

11988611

= minus

91205874ℎ1198771198641

5121198714120588

minus

3ℎ31198641

321205881198775

minus

1205872ℎ1198642

321198712120588119877

minus

1205872ℎ31198642

38411987121205881198773

minus

9ℎ1198641

321205881198773

minus

1205872ℎ1198641120581

641198712120588119877

11988612

=

1198622

2120587119877radic120588119864

11988613

= minus

41205872119877ℎ1198642

1198712120588

minus

ℎ1198641

119877120588

minus

1205872ℎ31198642

31198712120588119877

11988614

=

1205872ℎ31198641120581

411987121205881198773

minus

ℎ1198641

41205881198773

minus

1205874ℎ1198771198641

161198714120588

minus

131205872ℎ1198641120581

161198712120588119877

11988615

= minus

1205872ℎ31198642

2411987121205881198773

minus

1205872ℎ1198642

21198712120588119877

minus

91205874ℎ1198641120581

21198714120588

minus

3ℎ1198641

41205881198773

minus

1205872ℎ1198641120581

1198712120588119877

minus

31205874ℎ31198641

21198714120588119877

(A1)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to acknowledge financial support byNatural Science Foundation of China Project 11372071 andPostdoctoral Fund of China Project 2013M541339

References

[1] P D Spanos A M Chevallier N Politis and B Payne ldquoOilwell drilling a vibration perspectiverdquo The Shock and VibrationDigest vol 35 pp 81ndash99 2013

[2] W D Aldred andM C Sheppard ldquoDrillstring vibrations a newgeneration mechanism and control strategiesrdquo in Proceedingsof the SPE Annual Technical Conference and Exhibition SPE-24582-MS pp 353ndash363 Washington DC USA October 1992

[3] T V Aarrestand H A Tonnesen and A Kyllingstad ldquoDrill-string vibrations theory and experiments on full-scale researchdrilling rigrdquo inProceedings of the IADCSPEDrilling ConferenceSPE Paper No 14760 pp 311ndash321 Dallas Tex USA 1986

[4] J D Jansen ldquoNon-linear rotor dynamics as applied to oilwelldrillstring vibrationsrdquo Journal of Sound and Vibration vol 147no 1 pp 115ndash135 1991

[5] A S Yigit and A P Christoforou ldquoCoupled axial and transversevibrations of oilwell drillstringsrdquo Journal of Sound and Vibra-tion vol 195 no 4 pp 617ndash627 1996

[6] A S Yigit and A P Christoforou ldquoCoupled torsional and bend-ing vibration of drill-strings subject to impact with frictionrdquoJournal of Sound and Vibration vol 215 no 1 pp 167ndash181 1998

[7] K Z Khan Longitudinal and torsional vibration of drill-strings [MS thesis] Massachusetts Institute of TechnologyCambridge Mass USA 1986

[8] R J Shyu Bending of rotating drill-strings [PhD thesis] Mas-sachusetts Institute of Technology 1989

[9] A P Christoforou and A S Yigit ldquoFully coupled vibrations ofactively controlled drillstringsrdquo Journal of Sound and Vibrationvol 267 no 5 pp 1029ndash1045 2003

[10] M A Trindade C Wolter and R Sampaio ldquoKarhunen-Loevedecomposition of coupled axialbending vibrations of beamssubject to impactsrdquo Journal of Sound and Vibration vol 279 no3ndash5 pp 1015ndash1036 2005

[11] R Sampaio M T Piovan and G Venero Lozano ldquoCoupledaxialtorsional vibrations of drill-strings bymeans of non-linearmodelrdquo Mechanics Research Communications vol 34 no 5-6pp 497ndash502 2007

[12] H Hakimi and S Moradi ldquoDrillstring vibration analysis usingdifferential quadrature methodrdquo Journal of Petroleum Scienceand Engineering vol 70 no 3-4 pp 235ndash242 2010

[13] F S Ren S Chen and Z G Yao ldquoDynamics analysis andvibration suppression of a flexible rotation beamrdquo AppliedMechanics and Materials vol 214 pp 165ndash172 2012

[14] F-S Ren and Z-G Yao ldquoStudy on nonlinear dynamics andbifurcations in rotating compressive-drill stringrdquo EngineeringMechanics vol 30 no 10 pp 251ndash256 2013

[15] Z-F Li Y-G Zhang X-T Hou W-D Liu and G-Q XuldquoAnalysis of longitudinal and torsion vibration of drill-stringsrdquoEngineering Mechanics vol 21 no 6 pp 203ndash210 2004 (Chi-nese)

[16] G J Sheu and S M Yang ldquoDynamic analysis of a spinningRayleigh beamrdquo International Journal of Mechanical Sciencesvol 47 no 2 pp 157ndash169 2005

[17] X Zhou X Qichong D Hu et al ldquoThe nonlinear dynam-ics analysis of bottom drillstring of air drillingrdquo Journal ofChongqing University of Science and Technology vol 4 pp 119ndash121 2013 (Chinese)

[18] Y-J Jia P Jiang X-H Zhu and Y-F Zhang ldquoNonlinearbuckling simulation of drillstring system during pilot holedrilling in horizontal directional drillingrdquo Journal of SystemSimulation vol 25 no 4 pp 821ndash825 2013 (Chinese)

[19] W-S Xiao Z-Y Liu H-Y Wang X-F Wang L Fu and Y-C Yin ldquoTensional vibration analysis of drill-string by FSIrdquo OilField Equipment vol 42 pp 23ndash26 2013 (Chinese)

[20] D Shao Z Guan X Wen and Y Shi ldquoExperiment on lateralvibration characteristics of horizontal rotary drilling stringrdquoJournal of China University of Petroleum vol 37 pp 100ndash1052013 (Chinese)

[21] W Zhang X Zhu Z Zhou and W Ma ldquoEffect of rotationalspeeds mutation on the dynamic characteristics of the drill-string system in a vertical wellrdquo Journal of PetrochemicalUniversities vol 26 no 2 pp 47ndash51 2013 (Chinese)

[22] Y Tang ldquoNonlinear vibrations of axially accelerating viscoelas-tic Timoshenko beamsrdquo Chinese Journal of Theoretical andApplied Mechanics vol 45 no 6 pp 965ndash973 2013 (Chinese)

Shock and Vibration 17

[23] P B Goncalves and Z J G N Del Prado ldquoNonlinear oscilla-tions and stability of parametrically excited cylindrical shellsrdquoMeccanica vol 37 no 6 pp 569ndash597 2002

[24] M Strozzi and F Pellicano ldquoNonlinear vibrations of function-ally graded cylindrical shellsrdquo Thin-Walled Structures vol 67pp 63ndash77 2013

[25] M Amabili Nonlinear Vibrations and Stability of Shells andPlates Cambridge University Press Cambridge CambridgeUK 2008

[26] A W Leissa Vibrations of Shells Government Printing OfficeWashington DC USA 1973

[27] N Yamaki Elastic Stability of Circular Cylindrical Shells North-Holland Amsterdam The Netherlands 1984

[28] J N ReddyMechanics of LaminatedComposite Plates and ShellsCRC Press Boca Raton Fla USA 2004

[29] A H Nayfeh and D T Mook Nonlinear Oscillations Wiley-VCH Press Weinheim Germany 2004

[30] P D Spanos A K Sengupta R A Cunningham and P RPaslay ldquoModeling of roller cone bit lift-off dynamics in rotarydrillingrdquo Journal of Energy Resources Technology vol 117 no 3pp 197ndash207 1995

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

14 Shock and Vibration

minus2

minus1

0

1

2

x2

10 2 3minus1minus2

x1

(a) Phase diagram of the axial modal

1050 1100 1150 12001000t

minus2

minus1

0

1

2

3

x1

(b) Oscillogram of the axial modal

2 4minus2 0minus4

x3

minus4

minus2

0

2

4

x4

(c) Phase diagram of the torsional modal

minus4

minus2

0

2

4x3

1050 1100 1150 12001000t

(d) Oscillogram of the torsional modal

2

00

2

4

0

5

minus2

x3

minus5

x2

minus2

x 1

(e) Three-dimensional phase diagram

minus15

minus10

minus05

00

05

10

15

x2

0500 10 15minus10 minus05minus15

x1

(f) Poincare section

Figure 11 Period responses of the system (1199022= 07725)

Shock and Vibration 15x2

minus05 00 05 10minus10

x1

minus10

minus05

00

05

10

(a) Phase diagram of the axial modal

1020 1040 1060 10801000t

minus10

minus05

00

05

10

x1

(b) Oscillogram of the axial modal

15 20 25 3010x3

minus2

minus1

0

1

2

x4

(c) Phase diagram of the torsional modal

10

15

20

25

30x3

1020 1040 1060 10801000t

(d) Oscillogram of the torsional modal

1

0 0

11

2

3

minus1minus1

x3

x2

x 1

(e) Three-dimensional phase diagram

minus10 minus05minus15 05 10 1500x1

minus15

minus10

minus05

00

05

10

15

x2

(f) Poincare section

Figure 12 Period-doubled responses of the system (1199022= 079)

16 Shock and Vibration

+

1

24

1205872ℎ31198642

11987121198773

+

9

2

1205874119877ℎ1198641

1198714

)

1198869=

1198621

2120587119877radic120588119864

11988610

=

1205872ℎ31198641120581

411987121205881198772

minus

ℎ1198641

41205881198772

minus

1205874ℎ1198771198641

161198714120588

minus

1205872ℎ1198641120581

161198712120588119877

11988611

= minus

91205874ℎ1198771198641

5121198714120588

minus

3ℎ31198641

321205881198775

minus

1205872ℎ1198642

321198712120588119877

minus

1205872ℎ31198642

38411987121205881198773

minus

9ℎ1198641

321205881198773

minus

1205872ℎ1198641120581

641198712120588119877

11988612

=

1198622

2120587119877radic120588119864

11988613

= minus

41205872119877ℎ1198642

1198712120588

minus

ℎ1198641

119877120588

minus

1205872ℎ31198642

31198712120588119877

11988614

=

1205872ℎ31198641120581

411987121205881198773

minus

ℎ1198641

41205881198773

minus

1205874ℎ1198771198641

161198714120588

minus

131205872ℎ1198641120581

161198712120588119877

11988615

= minus

1205872ℎ31198642

2411987121205881198773

minus

1205872ℎ1198642

21198712120588119877

minus

91205874ℎ1198641120581

21198714120588

minus

3ℎ1198641

41205881198773

minus

1205872ℎ1198641120581

1198712120588119877

minus

31205874ℎ31198641

21198714120588119877

(A1)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to acknowledge financial support byNatural Science Foundation of China Project 11372071 andPostdoctoral Fund of China Project 2013M541339

References

[1] P D Spanos A M Chevallier N Politis and B Payne ldquoOilwell drilling a vibration perspectiverdquo The Shock and VibrationDigest vol 35 pp 81ndash99 2013

[2] W D Aldred andM C Sheppard ldquoDrillstring vibrations a newgeneration mechanism and control strategiesrdquo in Proceedingsof the SPE Annual Technical Conference and Exhibition SPE-24582-MS pp 353ndash363 Washington DC USA October 1992

[3] T V Aarrestand H A Tonnesen and A Kyllingstad ldquoDrill-string vibrations theory and experiments on full-scale researchdrilling rigrdquo inProceedings of the IADCSPEDrilling ConferenceSPE Paper No 14760 pp 311ndash321 Dallas Tex USA 1986

[4] J D Jansen ldquoNon-linear rotor dynamics as applied to oilwelldrillstring vibrationsrdquo Journal of Sound and Vibration vol 147no 1 pp 115ndash135 1991

[5] A S Yigit and A P Christoforou ldquoCoupled axial and transversevibrations of oilwell drillstringsrdquo Journal of Sound and Vibra-tion vol 195 no 4 pp 617ndash627 1996

[6] A S Yigit and A P Christoforou ldquoCoupled torsional and bend-ing vibration of drill-strings subject to impact with frictionrdquoJournal of Sound and Vibration vol 215 no 1 pp 167ndash181 1998

[7] K Z Khan Longitudinal and torsional vibration of drill-strings [MS thesis] Massachusetts Institute of TechnologyCambridge Mass USA 1986

[8] R J Shyu Bending of rotating drill-strings [PhD thesis] Mas-sachusetts Institute of Technology 1989

[9] A P Christoforou and A S Yigit ldquoFully coupled vibrations ofactively controlled drillstringsrdquo Journal of Sound and Vibrationvol 267 no 5 pp 1029ndash1045 2003

[10] M A Trindade C Wolter and R Sampaio ldquoKarhunen-Loevedecomposition of coupled axialbending vibrations of beamssubject to impactsrdquo Journal of Sound and Vibration vol 279 no3ndash5 pp 1015ndash1036 2005

[11] R Sampaio M T Piovan and G Venero Lozano ldquoCoupledaxialtorsional vibrations of drill-strings bymeans of non-linearmodelrdquo Mechanics Research Communications vol 34 no 5-6pp 497ndash502 2007

[12] H Hakimi and S Moradi ldquoDrillstring vibration analysis usingdifferential quadrature methodrdquo Journal of Petroleum Scienceand Engineering vol 70 no 3-4 pp 235ndash242 2010

[13] F S Ren S Chen and Z G Yao ldquoDynamics analysis andvibration suppression of a flexible rotation beamrdquo AppliedMechanics and Materials vol 214 pp 165ndash172 2012

[14] F-S Ren and Z-G Yao ldquoStudy on nonlinear dynamics andbifurcations in rotating compressive-drill stringrdquo EngineeringMechanics vol 30 no 10 pp 251ndash256 2013

[15] Z-F Li Y-G Zhang X-T Hou W-D Liu and G-Q XuldquoAnalysis of longitudinal and torsion vibration of drill-stringsrdquoEngineering Mechanics vol 21 no 6 pp 203ndash210 2004 (Chi-nese)

[16] G J Sheu and S M Yang ldquoDynamic analysis of a spinningRayleigh beamrdquo International Journal of Mechanical Sciencesvol 47 no 2 pp 157ndash169 2005

[17] X Zhou X Qichong D Hu et al ldquoThe nonlinear dynam-ics analysis of bottom drillstring of air drillingrdquo Journal ofChongqing University of Science and Technology vol 4 pp 119ndash121 2013 (Chinese)

[18] Y-J Jia P Jiang X-H Zhu and Y-F Zhang ldquoNonlinearbuckling simulation of drillstring system during pilot holedrilling in horizontal directional drillingrdquo Journal of SystemSimulation vol 25 no 4 pp 821ndash825 2013 (Chinese)

[19] W-S Xiao Z-Y Liu H-Y Wang X-F Wang L Fu and Y-C Yin ldquoTensional vibration analysis of drill-string by FSIrdquo OilField Equipment vol 42 pp 23ndash26 2013 (Chinese)

[20] D Shao Z Guan X Wen and Y Shi ldquoExperiment on lateralvibration characteristics of horizontal rotary drilling stringrdquoJournal of China University of Petroleum vol 37 pp 100ndash1052013 (Chinese)

[21] W Zhang X Zhu Z Zhou and W Ma ldquoEffect of rotationalspeeds mutation on the dynamic characteristics of the drill-string system in a vertical wellrdquo Journal of PetrochemicalUniversities vol 26 no 2 pp 47ndash51 2013 (Chinese)

[22] Y Tang ldquoNonlinear vibrations of axially accelerating viscoelas-tic Timoshenko beamsrdquo Chinese Journal of Theoretical andApplied Mechanics vol 45 no 6 pp 965ndash973 2013 (Chinese)

Shock and Vibration 17

[23] P B Goncalves and Z J G N Del Prado ldquoNonlinear oscilla-tions and stability of parametrically excited cylindrical shellsrdquoMeccanica vol 37 no 6 pp 569ndash597 2002

[24] M Strozzi and F Pellicano ldquoNonlinear vibrations of function-ally graded cylindrical shellsrdquo Thin-Walled Structures vol 67pp 63ndash77 2013

[25] M Amabili Nonlinear Vibrations and Stability of Shells andPlates Cambridge University Press Cambridge CambridgeUK 2008

[26] A W Leissa Vibrations of Shells Government Printing OfficeWashington DC USA 1973

[27] N Yamaki Elastic Stability of Circular Cylindrical Shells North-Holland Amsterdam The Netherlands 1984

[28] J N ReddyMechanics of LaminatedComposite Plates and ShellsCRC Press Boca Raton Fla USA 2004

[29] A H Nayfeh and D T Mook Nonlinear Oscillations Wiley-VCH Press Weinheim Germany 2004

[30] P D Spanos A K Sengupta R A Cunningham and P RPaslay ldquoModeling of roller cone bit lift-off dynamics in rotarydrillingrdquo Journal of Energy Resources Technology vol 117 no 3pp 197ndash207 1995

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 15x2

minus05 00 05 10minus10

x1

minus10

minus05

00

05

10

(a) Phase diagram of the axial modal

1020 1040 1060 10801000t

minus10

minus05

00

05

10

x1

(b) Oscillogram of the axial modal

15 20 25 3010x3

minus2

minus1

0

1

2

x4

(c) Phase diagram of the torsional modal

10

15

20

25

30x3

1020 1040 1060 10801000t

(d) Oscillogram of the torsional modal

1

0 0

11

2

3

minus1minus1

x3

x2

x 1

(e) Three-dimensional phase diagram

minus10 minus05minus15 05 10 1500x1

minus15

minus10

minus05

00

05

10

15

x2

(f) Poincare section

Figure 12 Period-doubled responses of the system (1199022= 079)

16 Shock and Vibration

+

1

24

1205872ℎ31198642

11987121198773

+

9

2

1205874119877ℎ1198641

1198714

)

1198869=

1198621

2120587119877radic120588119864

11988610

=

1205872ℎ31198641120581

411987121205881198772

minus

ℎ1198641

41205881198772

minus

1205874ℎ1198771198641

161198714120588

minus

1205872ℎ1198641120581

161198712120588119877

11988611

= minus

91205874ℎ1198771198641

5121198714120588

minus

3ℎ31198641

321205881198775

minus

1205872ℎ1198642

321198712120588119877

minus

1205872ℎ31198642

38411987121205881198773

minus

9ℎ1198641

321205881198773

minus

1205872ℎ1198641120581

641198712120588119877

11988612

=

1198622

2120587119877radic120588119864

11988613

= minus

41205872119877ℎ1198642

1198712120588

minus

ℎ1198641

119877120588

minus

1205872ℎ31198642

31198712120588119877

11988614

=

1205872ℎ31198641120581

411987121205881198773

minus

ℎ1198641

41205881198773

minus

1205874ℎ1198771198641

161198714120588

minus

131205872ℎ1198641120581

161198712120588119877

11988615

= minus

1205872ℎ31198642

2411987121205881198773

minus

1205872ℎ1198642

21198712120588119877

minus

91205874ℎ1198641120581

21198714120588

minus

3ℎ1198641

41205881198773

minus

1205872ℎ1198641120581

1198712120588119877

minus

31205874ℎ31198641

21198714120588119877

(A1)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to acknowledge financial support byNatural Science Foundation of China Project 11372071 andPostdoctoral Fund of China Project 2013M541339

References

[1] P D Spanos A M Chevallier N Politis and B Payne ldquoOilwell drilling a vibration perspectiverdquo The Shock and VibrationDigest vol 35 pp 81ndash99 2013

[2] W D Aldred andM C Sheppard ldquoDrillstring vibrations a newgeneration mechanism and control strategiesrdquo in Proceedingsof the SPE Annual Technical Conference and Exhibition SPE-24582-MS pp 353ndash363 Washington DC USA October 1992

[3] T V Aarrestand H A Tonnesen and A Kyllingstad ldquoDrill-string vibrations theory and experiments on full-scale researchdrilling rigrdquo inProceedings of the IADCSPEDrilling ConferenceSPE Paper No 14760 pp 311ndash321 Dallas Tex USA 1986

[4] J D Jansen ldquoNon-linear rotor dynamics as applied to oilwelldrillstring vibrationsrdquo Journal of Sound and Vibration vol 147no 1 pp 115ndash135 1991

[5] A S Yigit and A P Christoforou ldquoCoupled axial and transversevibrations of oilwell drillstringsrdquo Journal of Sound and Vibra-tion vol 195 no 4 pp 617ndash627 1996

[6] A S Yigit and A P Christoforou ldquoCoupled torsional and bend-ing vibration of drill-strings subject to impact with frictionrdquoJournal of Sound and Vibration vol 215 no 1 pp 167ndash181 1998

[7] K Z Khan Longitudinal and torsional vibration of drill-strings [MS thesis] Massachusetts Institute of TechnologyCambridge Mass USA 1986

[8] R J Shyu Bending of rotating drill-strings [PhD thesis] Mas-sachusetts Institute of Technology 1989

[9] A P Christoforou and A S Yigit ldquoFully coupled vibrations ofactively controlled drillstringsrdquo Journal of Sound and Vibrationvol 267 no 5 pp 1029ndash1045 2003

[10] M A Trindade C Wolter and R Sampaio ldquoKarhunen-Loevedecomposition of coupled axialbending vibrations of beamssubject to impactsrdquo Journal of Sound and Vibration vol 279 no3ndash5 pp 1015ndash1036 2005

[11] R Sampaio M T Piovan and G Venero Lozano ldquoCoupledaxialtorsional vibrations of drill-strings bymeans of non-linearmodelrdquo Mechanics Research Communications vol 34 no 5-6pp 497ndash502 2007

[12] H Hakimi and S Moradi ldquoDrillstring vibration analysis usingdifferential quadrature methodrdquo Journal of Petroleum Scienceand Engineering vol 70 no 3-4 pp 235ndash242 2010

[13] F S Ren S Chen and Z G Yao ldquoDynamics analysis andvibration suppression of a flexible rotation beamrdquo AppliedMechanics and Materials vol 214 pp 165ndash172 2012

[14] F-S Ren and Z-G Yao ldquoStudy on nonlinear dynamics andbifurcations in rotating compressive-drill stringrdquo EngineeringMechanics vol 30 no 10 pp 251ndash256 2013

[15] Z-F Li Y-G Zhang X-T Hou W-D Liu and G-Q XuldquoAnalysis of longitudinal and torsion vibration of drill-stringsrdquoEngineering Mechanics vol 21 no 6 pp 203ndash210 2004 (Chi-nese)

[16] G J Sheu and S M Yang ldquoDynamic analysis of a spinningRayleigh beamrdquo International Journal of Mechanical Sciencesvol 47 no 2 pp 157ndash169 2005

[17] X Zhou X Qichong D Hu et al ldquoThe nonlinear dynam-ics analysis of bottom drillstring of air drillingrdquo Journal ofChongqing University of Science and Technology vol 4 pp 119ndash121 2013 (Chinese)

[18] Y-J Jia P Jiang X-H Zhu and Y-F Zhang ldquoNonlinearbuckling simulation of drillstring system during pilot holedrilling in horizontal directional drillingrdquo Journal of SystemSimulation vol 25 no 4 pp 821ndash825 2013 (Chinese)

[19] W-S Xiao Z-Y Liu H-Y Wang X-F Wang L Fu and Y-C Yin ldquoTensional vibration analysis of drill-string by FSIrdquo OilField Equipment vol 42 pp 23ndash26 2013 (Chinese)

[20] D Shao Z Guan X Wen and Y Shi ldquoExperiment on lateralvibration characteristics of horizontal rotary drilling stringrdquoJournal of China University of Petroleum vol 37 pp 100ndash1052013 (Chinese)

[21] W Zhang X Zhu Z Zhou and W Ma ldquoEffect of rotationalspeeds mutation on the dynamic characteristics of the drill-string system in a vertical wellrdquo Journal of PetrochemicalUniversities vol 26 no 2 pp 47ndash51 2013 (Chinese)

[22] Y Tang ldquoNonlinear vibrations of axially accelerating viscoelas-tic Timoshenko beamsrdquo Chinese Journal of Theoretical andApplied Mechanics vol 45 no 6 pp 965ndash973 2013 (Chinese)

Shock and Vibration 17

[23] P B Goncalves and Z J G N Del Prado ldquoNonlinear oscilla-tions and stability of parametrically excited cylindrical shellsrdquoMeccanica vol 37 no 6 pp 569ndash597 2002

[24] M Strozzi and F Pellicano ldquoNonlinear vibrations of function-ally graded cylindrical shellsrdquo Thin-Walled Structures vol 67pp 63ndash77 2013

[25] M Amabili Nonlinear Vibrations and Stability of Shells andPlates Cambridge University Press Cambridge CambridgeUK 2008

[26] A W Leissa Vibrations of Shells Government Printing OfficeWashington DC USA 1973

[27] N Yamaki Elastic Stability of Circular Cylindrical Shells North-Holland Amsterdam The Netherlands 1984

[28] J N ReddyMechanics of LaminatedComposite Plates and ShellsCRC Press Boca Raton Fla USA 2004

[29] A H Nayfeh and D T Mook Nonlinear Oscillations Wiley-VCH Press Weinheim Germany 2004

[30] P D Spanos A K Sengupta R A Cunningham and P RPaslay ldquoModeling of roller cone bit lift-off dynamics in rotarydrillingrdquo Journal of Energy Resources Technology vol 117 no 3pp 197ndash207 1995

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

16 Shock and Vibration

+

1

24

1205872ℎ31198642

11987121198773

+

9

2

1205874119877ℎ1198641

1198714

)

1198869=

1198621

2120587119877radic120588119864

11988610

=

1205872ℎ31198641120581

411987121205881198772

minus

ℎ1198641

41205881198772

minus

1205874ℎ1198771198641

161198714120588

minus

1205872ℎ1198641120581

161198712120588119877

11988611

= minus

91205874ℎ1198771198641

5121198714120588

minus

3ℎ31198641

321205881198775

minus

1205872ℎ1198642

321198712120588119877

minus

1205872ℎ31198642

38411987121205881198773

minus

9ℎ1198641

321205881198773

minus

1205872ℎ1198641120581

641198712120588119877

11988612

=

1198622

2120587119877radic120588119864

11988613

= minus

41205872119877ℎ1198642

1198712120588

minus

ℎ1198641

119877120588

minus

1205872ℎ31198642

31198712120588119877

11988614

=

1205872ℎ31198641120581

411987121205881198773

minus

ℎ1198641

41205881198773

minus

1205874ℎ1198771198641

161198714120588

minus

131205872ℎ1198641120581

161198712120588119877

11988615

= minus

1205872ℎ31198642

2411987121205881198773

minus

1205872ℎ1198642

21198712120588119877

minus

91205874ℎ1198641120581

21198714120588

minus

3ℎ1198641

41205881198773

minus

1205872ℎ1198641120581

1198712120588119877

minus

31205874ℎ31198641

21198714120588119877

(A1)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to acknowledge financial support byNatural Science Foundation of China Project 11372071 andPostdoctoral Fund of China Project 2013M541339

References

[1] P D Spanos A M Chevallier N Politis and B Payne ldquoOilwell drilling a vibration perspectiverdquo The Shock and VibrationDigest vol 35 pp 81ndash99 2013

[2] W D Aldred andM C Sheppard ldquoDrillstring vibrations a newgeneration mechanism and control strategiesrdquo in Proceedingsof the SPE Annual Technical Conference and Exhibition SPE-24582-MS pp 353ndash363 Washington DC USA October 1992

[3] T V Aarrestand H A Tonnesen and A Kyllingstad ldquoDrill-string vibrations theory and experiments on full-scale researchdrilling rigrdquo inProceedings of the IADCSPEDrilling ConferenceSPE Paper No 14760 pp 311ndash321 Dallas Tex USA 1986

[4] J D Jansen ldquoNon-linear rotor dynamics as applied to oilwelldrillstring vibrationsrdquo Journal of Sound and Vibration vol 147no 1 pp 115ndash135 1991

[5] A S Yigit and A P Christoforou ldquoCoupled axial and transversevibrations of oilwell drillstringsrdquo Journal of Sound and Vibra-tion vol 195 no 4 pp 617ndash627 1996

[6] A S Yigit and A P Christoforou ldquoCoupled torsional and bend-ing vibration of drill-strings subject to impact with frictionrdquoJournal of Sound and Vibration vol 215 no 1 pp 167ndash181 1998

[7] K Z Khan Longitudinal and torsional vibration of drill-strings [MS thesis] Massachusetts Institute of TechnologyCambridge Mass USA 1986

[8] R J Shyu Bending of rotating drill-strings [PhD thesis] Mas-sachusetts Institute of Technology 1989

[9] A P Christoforou and A S Yigit ldquoFully coupled vibrations ofactively controlled drillstringsrdquo Journal of Sound and Vibrationvol 267 no 5 pp 1029ndash1045 2003

[10] M A Trindade C Wolter and R Sampaio ldquoKarhunen-Loevedecomposition of coupled axialbending vibrations of beamssubject to impactsrdquo Journal of Sound and Vibration vol 279 no3ndash5 pp 1015ndash1036 2005

[11] R Sampaio M T Piovan and G Venero Lozano ldquoCoupledaxialtorsional vibrations of drill-strings bymeans of non-linearmodelrdquo Mechanics Research Communications vol 34 no 5-6pp 497ndash502 2007

[12] H Hakimi and S Moradi ldquoDrillstring vibration analysis usingdifferential quadrature methodrdquo Journal of Petroleum Scienceand Engineering vol 70 no 3-4 pp 235ndash242 2010

[13] F S Ren S Chen and Z G Yao ldquoDynamics analysis andvibration suppression of a flexible rotation beamrdquo AppliedMechanics and Materials vol 214 pp 165ndash172 2012

[14] F-S Ren and Z-G Yao ldquoStudy on nonlinear dynamics andbifurcations in rotating compressive-drill stringrdquo EngineeringMechanics vol 30 no 10 pp 251ndash256 2013

[15] Z-F Li Y-G Zhang X-T Hou W-D Liu and G-Q XuldquoAnalysis of longitudinal and torsion vibration of drill-stringsrdquoEngineering Mechanics vol 21 no 6 pp 203ndash210 2004 (Chi-nese)

[16] G J Sheu and S M Yang ldquoDynamic analysis of a spinningRayleigh beamrdquo International Journal of Mechanical Sciencesvol 47 no 2 pp 157ndash169 2005

[17] X Zhou X Qichong D Hu et al ldquoThe nonlinear dynam-ics analysis of bottom drillstring of air drillingrdquo Journal ofChongqing University of Science and Technology vol 4 pp 119ndash121 2013 (Chinese)

[18] Y-J Jia P Jiang X-H Zhu and Y-F Zhang ldquoNonlinearbuckling simulation of drillstring system during pilot holedrilling in horizontal directional drillingrdquo Journal of SystemSimulation vol 25 no 4 pp 821ndash825 2013 (Chinese)

[19] W-S Xiao Z-Y Liu H-Y Wang X-F Wang L Fu and Y-C Yin ldquoTensional vibration analysis of drill-string by FSIrdquo OilField Equipment vol 42 pp 23ndash26 2013 (Chinese)

[20] D Shao Z Guan X Wen and Y Shi ldquoExperiment on lateralvibration characteristics of horizontal rotary drilling stringrdquoJournal of China University of Petroleum vol 37 pp 100ndash1052013 (Chinese)

[21] W Zhang X Zhu Z Zhou and W Ma ldquoEffect of rotationalspeeds mutation on the dynamic characteristics of the drill-string system in a vertical wellrdquo Journal of PetrochemicalUniversities vol 26 no 2 pp 47ndash51 2013 (Chinese)

[22] Y Tang ldquoNonlinear vibrations of axially accelerating viscoelas-tic Timoshenko beamsrdquo Chinese Journal of Theoretical andApplied Mechanics vol 45 no 6 pp 965ndash973 2013 (Chinese)

Shock and Vibration 17

[23] P B Goncalves and Z J G N Del Prado ldquoNonlinear oscilla-tions and stability of parametrically excited cylindrical shellsrdquoMeccanica vol 37 no 6 pp 569ndash597 2002

[24] M Strozzi and F Pellicano ldquoNonlinear vibrations of function-ally graded cylindrical shellsrdquo Thin-Walled Structures vol 67pp 63ndash77 2013

[25] M Amabili Nonlinear Vibrations and Stability of Shells andPlates Cambridge University Press Cambridge CambridgeUK 2008

[26] A W Leissa Vibrations of Shells Government Printing OfficeWashington DC USA 1973

[27] N Yamaki Elastic Stability of Circular Cylindrical Shells North-Holland Amsterdam The Netherlands 1984

[28] J N ReddyMechanics of LaminatedComposite Plates and ShellsCRC Press Boca Raton Fla USA 2004

[29] A H Nayfeh and D T Mook Nonlinear Oscillations Wiley-VCH Press Weinheim Germany 2004

[30] P D Spanos A K Sengupta R A Cunningham and P RPaslay ldquoModeling of roller cone bit lift-off dynamics in rotarydrillingrdquo Journal of Energy Resources Technology vol 117 no 3pp 197ndash207 1995

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 17

[23] P B Goncalves and Z J G N Del Prado ldquoNonlinear oscilla-tions and stability of parametrically excited cylindrical shellsrdquoMeccanica vol 37 no 6 pp 569ndash597 2002

[24] M Strozzi and F Pellicano ldquoNonlinear vibrations of function-ally graded cylindrical shellsrdquo Thin-Walled Structures vol 67pp 63ndash77 2013

[25] M Amabili Nonlinear Vibrations and Stability of Shells andPlates Cambridge University Press Cambridge CambridgeUK 2008

[26] A W Leissa Vibrations of Shells Government Printing OfficeWashington DC USA 1973

[27] N Yamaki Elastic Stability of Circular Cylindrical Shells North-Holland Amsterdam The Netherlands 1984

[28] J N ReddyMechanics of LaminatedComposite Plates and ShellsCRC Press Boca Raton Fla USA 2004

[29] A H Nayfeh and D T Mook Nonlinear Oscillations Wiley-VCH Press Weinheim Germany 2004

[30] P D Spanos A K Sengupta R A Cunningham and P RPaslay ldquoModeling of roller cone bit lift-off dynamics in rotarydrillingrdquo Journal of Energy Resources Technology vol 117 no 3pp 197ndash207 1995

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of