Research Article Mathematical Modeling and Analysis of Drill String … · 2019. 7. 30. · drill string, , , the expressions can be written as = ... uence of design parame-ters on

Research ArticleMathematical Modeling and Analysis of Drill StringLongitudinal Vibration with Lateral Inertia Effect

Jialin Tian12 Chunming Wu1 Lin Yang1 Zhi Yang1 Gang Liu1 and Changfu Yuan1

1School of Mechatronic Engineering Southwest Petroleum University Chengdu 610500 China2School of Mechanical Engineering Southwest Jiaotong University Chengdu 610031 China

Correspondence should be addressed to Jialin Tian tianjialin001gmailcom and Chunming Wu 1769752908qqcom

Received 16 July 2015 Revised 8 January 2016 Accepted 17 January 2016

Academic Editor Sergio De Rosa

Copyright copy 2016 Jialin Tian 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

Comparative analysis whether considering the lateral inertia or not aiming at the longitudinal vibration of the drill string in drillingprogress is proposed In the light of the actual condition the mechanical model of the drill string about vibration is establishedon the basis of the theoretical analysis Longitudinal vibration equation of the drill string is derived from the Rayleigh-Love modeland one-dimensional viscoelastic model According to the Laplace transform method and the relationships among parameters ofthe model the solutions to complex impedance at the bottom of the drill string are obtained and then the comparison results areanalyzed which is the lateral inertia effect on longitudinal vibration characteristicsThe researches show that the smaller the lengthof the drill string the greater the cross-sectional area of the drill string the greater the damping coefficient of bottom hole on thebottom of the drill string and the more evident the effect on the dynamic stiffness of the drill string with lateral inertia effect ThePoisson ratio of the drill string only has some effects on it taking account of the lateral inertia effect and the influence is relativelysmall compared with the former three conditions

1 Introduction

In oil and gas drilling engineering the drill string is a slenderstructure connecting downhole tools and drilling rig Itsdynamic performance is one of the key factors affecting thedrilling efficiency and cost Also the results caused by drillstring vibration features include rock-breaking performanceof drill bit drilling ROP (rate of penetration) wellbore qual-ity and downhole tools fatigues [1ndash4] According to drillingcondition the drill string vibration can be divided into threeforms such as longitudinal lateral and torsional vibration inwhich the longitudinal vibration is the main factor of drillingefficiency and safety The longitudinal vibration makes thedrill bit move up and down which results in the constantchange of ROP the service life of drill bit cutting elementor drilling footage [5ndash7] To itself longitudinal vibration maycause the drill string fatigue fracture [8]

For the solutions of drill string vibration analysismethodswith complex influence factors and boundary conditionsthe related researches include taking the drill string as elas-tomer or obtaining vibration equations with dynamics FEM

(finite element method) or analyzing longitudinal vibrationconsidering the coupling effects of wellbore friction or thehydrodynamic interaction of drilling fluid [9ndash14] Howeverin light of the drilling actual condition and well structureas an elongated rod the lateral inertia has an importantinfluence on the longitudinal vibration of drill string Infact the drill longitudinal string vibration characteristicscannot be fully reflectedwith ignoring the lateral inertia effect[15] Moreover with the search results of reference articlesprevious researchworkwas not found about the lateral inertiaeffect on drill string longitudinal vibration Based on thisthe influence of the lateral inertia effect on the drill stringlongitudinal vibration is discussed in this paper Accordingto well structure and drill string working conditions thelongitudinal vibrationmodels whether considering the lateralinertia or not are established Moreover the solution methodof drill string key parameter impedance at the bottomhole isobtained

With the presented analysis models and solution equa-tions the influences of considering the lateral inertia effector not on drill string longitudinal vibration are discussed

Hindawi Publishing CorporationShock and VibrationVolume 2016 Article ID 6281264 8 pageshttpdxdoiorg10115520166281264

2 Shock and Vibration

rPk

r998400Pk120591Pkminusi

120591Pkminuso

H

hk+1

hk

h1 120575Pk

kPkz

middot middot middot


rPk


120591Pkminuso

H

hk+1

hk

h1 120575Pk

kPkz



Figure 1 Longitudinal vibration model of drill string

Meanwhile by comparing different calculation results somekey parametersrsquo influences on vibration features are pre-sented such as the drill string length inside and outsideradius damping coefficient and Poissonrsquos ratio The researchmethod and model can provide reference for drill stringvibration research At the same time the research resultscan also present theoretical and actual value for optimizationof downhole tools design and safety evaluation of drillingoperations

2 Analysis Method and Mathematical Model

Based on the real condition of the drill string relatedassumptions are proposed to simplify the research for drillstring longitudinal vibration [16] The following assumptionsare made the cross-sectional surface of the drill string isequivalently circular the drill string axis coincides withborehole axis the drilling fluid is regarded as viscoelasticlayer which can be divided into several layers and thedrilling fluids inside and outside drill string have identicalcharacteristics the vibration velocity of drill string is muchlarger than the flow rate of drilling fluid

According to drilling condition the drill string vibrationanalysis model is presented as in Figure 1 wherein ℎ119896 denotesthe location layer of 119896th drill string 119867 is the total lengthof the drill string 119903119875119896 and 119903

1015840

119875119896 respectively denote the outer

and inner radius of 119896th drill string 120591119875119896minus119894 120591119875119896minus119900 respectivelydenote the shear stress of drilling fluid inside and outside119896th drill string 119896119875119896 is the stiffness coefficient and 120575119875119896 meansthe damping coefficient of the 119896th drill string Usually inthe absence of inner and outer mud pressure this vibrationreduces to the classical wave vibration for longitudinal wavesin a uniform rod

21 Model of Drill String without Lateral Inertia Effect Thedeformation produced in a uniform drill string while it

vibrates in a borehole is symmetric about its axis Forestablishing the longitudinal vibration model of drill stringwithout lateral inertia effect regarding the drill string as aone-dimensional sticky elastomer and by using the laws ofmass andmomentum conservation the vibration equation ofthe 119896th drill string segment can be obtained as

119864119875119896119860119896

1205972119908119896

1205971199112minus 2120587120591119875119896minus119900119903119875119896 minus 2120587120591119875119896minus119894119903

1015840

119875119896= 120588119875119896119860119896

1205972119908119896

1205971199052

(V119891 gt VV V119891 gt 0)

119864119875119896119860119896

1205972119908119896

1205971199112+ 2120587120591119875119896minus119900119903119875119896 minus 2120587120591119875119896minus119894119903

1015840

119875119896= 120588119875119896119860119896

1205972119908119896

1205971199052

(V119891 lt VV V119891 gt 0)

119864119875119896119860119896

1205972119908119896

1205971199112+ 2120587120591119875119896minus119900119903119875119896 + 2120587120591119875119896minus119894119903

1015840

119875119896= 120588119875119896119860119896

1205972119908119896

1205971199052

(V119891 gt VV V119891 lt 0)

119864119875119896119860119896

1205972119908119896

1205971199112minus 2120587120591119875119896minus119900119903119875119896 + 2120587120591119875119896minus119894119903

1015840

119875119896= 120588119875119896119860119896

1205972119908119896

1205971199052

(V119891 lt VV V119891 lt 0)

(1)

wherein the right-hand side of this equation denotes themudpressures which is substituted with axial displacements Theleft-hand side of it denotes inertia force and the shear stressof drilling fluid inside and outside the drill string119908119896 denotesthe longitudinal displacement of the 119896th drill string segmentwhich is function of time 119905 and coordinate 119911 119908119896 = 119908119896(119911 119905)VV V119891 represent the vibration velocity of drill string and theflow rate of drilling fluid respectively Moreover 119864119875119896 denotesthe elastic modulus of the 119896th drill string segment 120588119875119896 is

Shock and Vibration 3

the density119860119896 represents the cross-sectional area of 119896th drillstring segment and the expression is given by

119860119896 = 120587 (1199032

119875119896minus 11990310158402

119875119896) (2)

According to the real working condition of the drillingthe relationship between VV and V119891 is V119891 ≪ VV and themovement of the drill string is always to the right So thecalculation formula should be denoted as

119864119875119896119860119896

1205972119908119896


1015840

119875119896

= 120588119875119896119860119896

1205972119908119896

1205971199052

(3)

To the shear stress of drilling fluid inside and outside 119896thdrill string 120591119875119896minus119894 120591119875119896minus119900 the expressions can be written as

120591119875119896minus119894 = 119870119896minus119894119908119896 (119911 119905)

120591119875119896minus119900 = 119870119896minus119900119908119896 (119911 119905)

(4)

where 119870119896minus119894 119870119896minus119900 are the longitudinal shear stiffness ofdrilling fluid at the position of inside and outside 119896th drillstring segment With applied Laplace transform of (1) andconsistent with the assumption conditions the followingequation can be obtained

119864119875119896119860119896

1205972119882119896

1205971199112

minus [1205881198751198961198601198961199042minus 2120587119870119896minus119900119903119875119896 + 2120587119870119896minus119894119903

1015840

119875119896]119882119896 = 0

(5)

in which the Laplace transform with respect to time of 119908119896 =119908119896(119911 119905) is represented by119882119896 = 119882119896(119911 119904) and the transformrelation is

119882119896 (119911 119904) = 119871 [119908119896 (119911 119905)] = int

+infin

0

119908119896 (119911 119905) 119890minus119904119905119889119905 (6)

The definition of symbol 120572 is as follows

120572 = radic120588119875119896119860119896119904

2minus 2120587 (119870119896minus119900119903119875119896 minus 119870119896minus119894119903

1015840

119875119896)

119864119875119896119860119896

(7)

Then the general solution of119882119896(119911 119904) can be gained as thefollowing form

119882119896 (119911 119904) = 119862119896119890120572119911+ 119863119896119890

minus120572119911 (8)

where 119862119896 119863119896 denote undetermined coefficients determinedby boundary conditions respectively

During oil and gas field drilling process the vibration ofdrill bit is caused partly because of the uneven bottomholeAssuming that the force of bottomhole to the drill bit is119891119896(119911 119905) according to the transitivity of force the force on thebottom of the drill string can be also denoted as 119891119896(119911 119905) Sothe bottom and top of the drill string can be described as

119864119875119896119860119896

120597119882119896 (119911 119904)

120597119911

10038161003816100381610038161003816100381610038161003816119911=0= 119891119896 (119911 119904) (9)

119864119875119896

120597119882119896 (119911 119904)

120597119911

10038161003816100381610038161003816100381610038161003816119911=119867= minus (119896119875119896 + 120575119875119896119904)119882119896 (119911 119904)

1003816100381610038161003816119911=119867 (10)

where the Laplace transform with respect to time of 119891119896(119911 119905)is represented by 119891119896(119911 119904) Furthermore with (8) and (9)substituted into (5) the following equations can be obtained

119862119896 minus 119863119896 =119891119896 (119911 119904)

120572119864119875119896119860119896

119862119896 =119864119875119896120572 minus (119896119875119896 + 120575119875119896119904)

119864119875119896120572 + (119896119875119896 + 120575119875119896119904)119890minus2120572119867

119863119896

(11)

Here defining the symbol 120574 its expression is given by

120574 =119864119875119896120572 minus (119896119875119896 + 120575119875119896119904)

119864119875119896120572 + (119896119875119896 + 120575119875119896119904)119890minus2120572119867

(12)

Then taking the results of 119862119896 119863119896 and definition of 120574into (7) the displacement of the drill string 119882119896(119911 119904) can beobtained as follows

119882119896 (119911 119904) =119891119896 (119911 119904)

(120574 minus 1) 120572119864119875119896119860119896

(120574119890120572119911+ 119890minus120572119911) (13)

Moreover assume that

119904 = 119894120596 (14)

Then the Laplace transform is equivalent to the unilateralFourier transform so the response of displacement frequencycan be expressed as119882119896(119911 119894119908)The complex impedance of thedrill string is derived as follows

119870119875119889 =120572119860119896 (120574 minus 1) 119864119875119896

120574 + 1 (15)

The complex impedance of the drill string bottom is equalto the complex stiffness which can be expressed in the pluralform as follows

119870119875119889 = 119870119903 + 119894119862119894 (16)

in which the plural real component 119870119903 denotes the realdynamic stiffness which reflects the drill string ability oflongitudinal deformation resistance In other words whenthe drill string is pressed by the vibration force the valueof the dynamic stiffness has close association with the defor-mation of the drill string On the other hand the imaginarycomponent 119862119894 denotes the dynamic damping which reflectsthe drill string energy dissipation

22 Model of Considering Lateral Inertia Effect Consideringthe influence of lateral inertia effect on drill string vibrationcharacteristics with actual situation the vibration of the 119896th


drill string segment can be described by the theory ofRayleigh-Love rod as follows

119864119875119896119860119896

1205972119908119896

1205971199112minus 2120587 (120591119875119896minus119894119903119875119896 + 120591119875119896minus119900119903

1015840

119875119896)

= 120588119875119896119860119896 (1205972119908119896

1205971199052minus ]2119875119896(1199032

119875119896minus 11990310158402

119875119896)1205974119908119896

12059711991121205971199052)

(V119891 gt VV V119891 gt 0)

119864119875119896119860119896

1205972119908119896

1205971199112+ 2120587 (120591119875119896minus119894119903119875119896 minus 120591119875119896minus119900119903

1015840

119875119896)

= 120588119875119896119860119896 (1205972119908119896

1205971199052minus ]2119875119896(1199032

119875119896minus 11990310158402

119875119896)1205974119908119896

12059711991121205971199052)

(V119891 lt VV V119891 gt 0)

119864119875119896119860119896

1205972119908119896

1205971199112+ 2120587 (120591119875119896minus119894119903119875119896 + 120591119875119896minus119900119903

1015840

119875119896)

= 120588119875119896119860119896 (1205972119908119896

1205971199052minus ]2119875119896(1199032

119875119896minus 11990310158402

119875119896)1205974119908119896

12059711991121205971199052)

(V119891 gt VV V119891 lt 0)

119864119875119896119860119896

1205972119908119896


1015840

119875119896)

= 120588119875119896119860119896 (1205972119908119896

1205971199052minus ]2119875119896(1199032

119875119896minus 11990310158402

119875119896)1205974119908119896

12059711991121205971199052)

(V119891 lt VV V119891 lt 0)

(17)

where ]119875119896 denotes Poissonrsquos ratio of the 119896th drill stringsegment It is also worth noticing that both coupling termsare proportional to Poissonrsquos ratio This is reasonable sincethis ratio describes the radial contraction of a rod subjectedto axial strain According to the drilling field situation theanalysis condition can be obtained as V119891 minus VV ≪ 0 and V119891 gt 0and the above equation with (16) can be simplified to

119864119875119896119860119896

1205972119908119896


1015840

119875119896)

= 120588119875119896119860119896 (1205972119908119896

1205971199052minus ]2119875119896(1199032

119875119896minus 11990310158402

119875119896)1205974119908119896

12059711991121205971199052)

(18)

For obtaining the results of the equation similar to themethod which is used for the solution of model without con-sidering the influence of lateral inertia effect on drill stringvibration considering the initial and continuity conditionsthe vibration equation of the drill string is solved by Laplacetransform and the displacement expression of the drill stringwith considering the influence of lateral inertia effect can beobtained

Here defining symbols 120572 120573 and 120574 their expressions aregiven as

120572 = radic120588119875119896119860119896119904


1015840

119875119896)

119864119875119896119860119896 + 120588119875119896119860119896]2119875119896 (1199032

119875119896minus 11990310158402119875119896) 1199042

120573 = 120588119875119896]2

119875119896(1199032

119875119896minus 11990310158402

119875119896) 1199042

120574 =(119864119875119896 + 120573) 120572 minus (119896119875119896 + 120575119875119896119904)

(119864119875119896 + 120573) 120572 + (119896119875119896 + 120575119875119896119904)119890minus2120572119867

(19)

Then the displacement expression of the drill string canbe obtained as

119882119896 (119911 119904) =119891119896 (119911 119904)

(120574 minus 1) 120572 (119864119875119896119860119896 + 119860119896120573)(120574119890120572119911+ 119890minus120572119911) (20)

Moreover the analytical solution of the complex impe-dance at the bottomhole of drill string can be expressed as

119870119875119889 =120572119860119896 (120574 minus 1) (119864119875119896 + 120573)

120574 + 1 (21)

3 Numerical Example and Analysis ofKey Parameters Influence

For contrastively analyzing the influence of design parame-ters on the characteristics of drill string longitudinal vibra-tion the complex impedance results are discussed by usingthe analysis models established above including the modelswith and without lateral inertia effect In the calculationresults figures the horizontal axis is the vibration frequencyand the longitudinal axis is the dynamic stiffness at thebottom of drill string First of all the material parametersof the drill string and drilling fluid including the densityelastic modulus stiffness coefficient and longitudinal shearstiffness are shown in Table 1

31 The Drill String Length The length of the drill string isalso on behalf of the well depth At the same time to theratio of drill string length and diameter the bigger ratio iscorresponding to the poorer vibration stability of drill stringThe influence of drill string length on its dynamic stiffnessat the bottom is analyzed as in Figure 2 including takingaccount of the lateral inertia effect or not The numericalexample parameters are shown in Table 2 including the drillstring Poissonrsquos ratio length radius and damping coefficientand the drill string length value includes 1800m and 2000m

In Figure 2 the symbol119867 denotes the length of drill stringmodel without considering the lateral inertia effect m Thesymbol ℎ denotes the length of drill string with consideringthe lateral inertia effect mThe results of drill string dynamicstiffness are presented corresponding to frequency of 20Hzto 30Hz including different drill string length and modelsof considering the lateral inertia effect or not The followingconclusions can be obtainedwith the increasing of frequencythe amplitude of drill string dynamic stiffness presentsincreasing trend including the results of different length and


Table 1 Material parameters of the drilling string

Parameters name ResultsDensity of drill string 120588

119875119896(kgm3) 800000

Elastic modulus of drill string 119864119875119896 (MPa) 206 times 103

Stiffness coefficient 119896119875119896 (Nm) 100 times 105

Longitudinal shear stiffness of drilling fluid inside drill string119870119896minus119894 (Mpam) 03Longitudinal shear stiffness of drilling fluid outside drill string119870119896minus119900 (Mpam) 01

3

2

1

0

minus1

minus2

Kr

(Nm

)

times104

H1 = 1800

h1 = 1800

H2 = 2000

h2 = 2000

22 23 24 25 26 27 28 29 3021f (Hz)

Figure 2 Influence of the drill string length on dynamic stiffness

Table 2 Calculation parameters of the length of drill stringinfluence

Name of parameters ResultsPoissonrsquos ratio of the drill string ] 030Length of the drill string1198671 ℎ1 (m) 1800Length of the drill string1198672 ℎ2 (m) 2000Outside radius of the drill string 119903119875119896 (m) 010Inside radius of the drill string 1199031015840

119875119896(m) 008

Damping coefficient 120575119875119896 (Nsm) 10000

analysis models Meanwhile to the lateral inertia effect theinfluence on drill string dynamic stiffness can be almostnegligible when the frequency is low or zero However withthe increasing of frequency such that the frequency is greaterthan 25Hz the influence of lateral inertia effect becomesmore obvious To the results of models considering lateralinertia effect or not in the increasing period of dynamicstiffness the absolute value of considering lateral inertia effectis greater than that of ignoring lateral inertia effect To theresults of drill string different lengths the influence of lateralinertia effect is bigger when the drill string is shorter

32 The Inside and Outside Radius The inside and outsideradius of drill string are determined by the size specificationThe influence of radius change on the dynamic stiffness isanalyzed as in Figure 3 including the models of consideringthe lateral inertia effect or not As before the numericalexample parameters are shown in Table 3 including the drillstring different inside and outside radius value

543210

minus1

minus3minus2

minus4

Kr

(Nm

)

times104

22 23 24 25 26 27 28 29 302120f (Hz)

RPk1 = 01

rPk1 = 01

RPk2 = 011

rPk2 = 011

Figure 3 Influence of outside radius on dynamic stiffness

Table 3 Calculated parameters of the inside and outside radiusinfluence

Name of parameters ResultsPoissonrsquos ratio of the drill string ] 030Length of the drill string H (m) 2000Outside radii of the drill string 1198771198751198961 1199031198751198961 (m) 010Outside radii of the drill string 119877

1198751198962 1199031198751198962

(m) 011Inside radii of the drill string 1198771015840

1198751198961 1199031198751198961 (m) 008

Inside radii of the drill string 11987710158401198751198962

1199031198751198962 (m) 009Damping coefficient 120575119875119896 (Nsm) 10000

In Figures 3 and 4 the symbols 119877119875119896 and 1198771198751198961 denote theradius of drill string model without considering the lateralinertia effect m The symbols 119903119875119896 and 119903

1015840

119875119896denote the radius

of drill string model considering the lateral inertia effectm The results of drill string dynamic stiffness are presentedcorresponding to frequency of 20Hz to 30Hz as shown inFigures 3 and 4 including different inside and outside radiusandmodels with lateral inertia effect or not As before the fol-lowing conclusions can be got with the frequency increasingthe dynamic stiffness amplitude presents increasing trendMeanwhile when outside radius increases or inside radiusreduces equivalent to the area increasing the increasingtrend of the dynamic stiffness becomes more notable Alsoto the results of models with lateral inertia effect or not inthe increasing period of dynamic stiffness the value of modelwith lateral inertia effect is greater than that without lateralinertia effect Meanwhile with bigger difference between thedrill string outside and inside radius the lateral inertia effecthas a more important influence on the dynamic stiffness ofdrill string


3

2

1

0

minus1

minus2

Kr

(Nm

)

times104

22 23 24 25 26 27 28 29 302120f (Hz)

R998400Pk1 = 008

r998400Pk1 = 008

R998400Pk2 = 009

r998400Pk2 = 009

Figure 4 Influence of inside radius on dynamic stiffness

Table 4 Calculation parameters of the damping coefficient influ-ence

Name of parameters ResultsPoissonrsquos ratio of the drill string ] 030Length of the drill string119867 (m) 2000Outside radius of the drill string 119903119875119896 (m) 010Inside radius of the drill string 1199031015840

119875119896(m) 008

Damping coefficients 12057510158401198751198961

1205751198751198961 (Nsm) 8000Damping coefficients 1205751015840

1198751198962 1205751198751198962 (Nsm) 10000

33 The Damping Coefficient The damping coefficient ofthe bottomhole to drill string is influential to the drillstring longitudinal vibration and the lateral inertia effect isdirectly related to the damping coefficient According to thedrilling field conditions the calculation parameters analyzingdamping coefficient are shown in Table 4 taking the dampingcoefficient as 8000 and 10000Nsm

In Figure 5 the symbol 120575119875119896 denotes the damping coeffi-cient ofmodelwithout lateral inertia effectNsmThe symbol1205751015840

119875119896denotes the damping coefficient of model considering

lateral inertia effect Nsm Figure 5 shows the influence of thedamping coefficient on the drill string dynamic stiffness cor-responding to frequency of 30Hz to 40Hz It can be observedthat the increasing trend of the dynamic stiffness becomesmore notable when the damping coefficient decreased andthe cyclic frequency increased including models with lateralinertia effect or not In a single cycle when it is closer tothe crest or trough the difference of drill string dynamicstiffness is bigger When considering the lateral inertia effectthe absolute value of dynamic stiffness in increasing periodis bigger than that of the model without lateral inertia effectHowever the bigger damping coefficient does notmeanmoreobvious influence in the drill string dynamic stiffness whichindicates that the damping of bottomhole to drill string canweaken the influence of lateral inertia effect on the dynamicstiffness

34 Poissonrsquos Ratio Poissonrsquos ratio is the lateral deformationcoefficient of material which reflects the lateral deformationelastic constants of material In the static force condition

120575Pk1 = 8000

120575998400Pk1 = 8000

120575Pk2 = 10000

120575998400Pk2 = 10000

4

2

0

minus2

minus4

Kr

(Nm

)

32 33 34 35 36 37 38 39 403130f (Hz)

times104

Figure 5 Influence of the damping coefficient on dynamic stiffness

3

4

2

1

0

minus1

minus3

minus2

Kr

(Nm

)

32 33 34 35 36 37 38 39 403130f (Hz)

times104

1 = 01

9984001 = 01

2 = 03

9984002 = 03

Figure 6 Influence of Poissonrsquos ratio on dynamic stiffness

Table 5 Calculated parameters of Poissonrsquos ratio influence

Name of parameters ResultsPoissonrsquos ratios of the drill string ]1015840

1 ]1 010

Poissonrsquos ratios of the drill string ]10158402 ]2 030

Length of the drill string119867 (m) 2000Outside radius of the drill string 119903119875119896 (m) 010Inside radius of the drill string 1199031015840

119875119896(m) 008


Poissonrsquos ratio change of drill string is very small and it isgenerally taken as ] = 025 However in the dynamic loadcase the dynamic Poissonrsquos ratio is commonly taken as ] =01sim03 According to the analysis above the calculatedparameters analyzing Poissonrsquos ratio are shown in Table 5taking Poissonrsquos ratio as 010 and 030

In Figure 6 the symbol ] denotes Poissonrsquos ratio of modelwithout lateral inertia effect On the other hand the symbol]1015840 denotes Poissonrsquos ratio of model considering lateral inertiaeffect Figure 6 shows the influence of Poissonrsquos ratio on thedrill string dynamic stiffness in the frequency of 30Hz to40Hz It can be seen that the increasing trend of the drillstring dynamic stiffness becomes more obvious when Pois-sonrsquos ratio and cyclic frequency increase both of the modelswith lateral inertia effect or not


Table 6 The free vibration natural frequency in different conditions

Boundary conditions Without lateral inertia Lateral inertiaThe first-order natural frequency (rads) 003 004The second-order natural frequency (rads) 005 006The third-order natural frequency (rads) 007 009

However different from other parameters results thechange of the drill string Poissonrsquos ratio has influence on theresults of model considering lateral inertia effect while ithas no influence on the results of model without consideringlateral inertia effect which indicates that the lateral inertiaeffect includes the effect of Poissonrsquos ratio Meanwhile whenconsidering the lateral inertia effect the absolute value ofdynamic stiffness is increasing with the drill string Poissonrsquosratio in the increasing period

35 Natural Frequency Assume that the length of the drillstring is 3500m and the boundary conditions are prescribedby the above as well as the structure parameters of the drillstring The longitudinal vibration model of the drill stringis analyzed with different conditions The first three-ordernatural frequencies of the drill string longitudinal vibrationare shown in Table 6 whether considering the lateral inertiaeffect or not

The calculation results illustrate that the natural fre-quency increases with the increase of frequency order Andwhen there is no inertia it is less than the situation withlateral inertia effect When considering the inertia effect thevibration is coupled with the longitudinal vibration whichwill cause the vibration difference According to adjusting thedrilling parameters the drill string resonance can be avoidedand make drilling optimization possible

4 Conclusion

According to the drilling field conditions the analysis modelof drill string longitudinal vibration is established with thelateral inertia effect and the Laplace transform is used forthe complex impedance solution of drill string bottom Byusing the established equations and analyzing the resultsof numerical examples the influence of design parameterson drill string dynamic stiffness is discussed including theresults of models considering the lateral inertia effect or notWithin the frequency range of drilling field conditions theseconclusions can be drawn

(1) The cyclical transformation amplitude of drill stringdynamic stiffness whether considering the radialinertia effect or not presents increasing with fre-quency and the length of the drill string Thereforefor resisting drill string longitudinal deformation andexcessive vibration the appropriate length of drillstring should be considered in the well structuredesign

(2) For the drill string dynamic stiffness increasing withthe cross-sectional area combining the relationshipof sectional area and inside and outside radius

the influence of sectional area on drill string dynamicstiffness can provide reference for the design of BHA(Bottomhole Assembly)

(3) With the damping coefficient increasing the ampli-tude of drill string dynamic stiffness is reducingwhich would lead to the vibration force increasingThe results are coincidingwith the drilling actual con-dition Moreover the consequences can provide basicdata for the design of downhole friction reductiontools

(4) To the influence of Poissonrsquos ratio on drill stringdynamic stiffness the results are some differencecomparing with other parameters Based on themodel without considering the lateral inertia effectthe change of Poissonrsquos ratio has no influence onthe results of drill string dynamic stiffness Howeverto the model including the lateral inertia effect theinfluence of Poissonrsquos ratio on drill string dynamicstiffness cannot be ignored According to the drillingfield condition the results are important for drillingdynamics analysis especially in deep and ultradeepdrilling or some new wells which is the basis forimproving the rock-breaking efficiency and ROP

Conflict of Interests

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

Acknowledgments

This work is supported by Open Fund (OGE201403-05) ofKey Laboratory of Oil amp Gas Equipment Ministry of Edu-cation (Southwest Petroleum University) National NaturalScience Foundation of China (no 51074202 and no 11102173)Major Cultivation Foundation of Sichuan Education Depart-ment (12ZZ003 no 667) and Graduate Student InnovationFund of Southwest Petroleum University (CX2015SY06)

References

[1] J J Bailey and I Finnie ldquoAn analytical study of drill-stringvibrationrdquo Journal of Engineering for Industry vol 82 no 2 pp122ndash127 1960

[2] S A Al-Hiddabi B Samanta and A Seibi ldquoNon-linear controlof torsional and bending vibrations of oilwell drillstringsrdquoJournal of Sound andVibration vol 265 no 2 pp 401ndash415 2003

[3] Q Xue R Wang F Sun and Z Huang ldquoChaotic vibrationanalysis of the bottom rotating drill stringrdquo Shock andVibrationvol 2014 Article ID 429164 8 pages 2014


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

[5] W Wang Q Hu H Liu andW Zhang ldquoAnalysis on longitudi-nal vibration of drill string and the corresponding applicationrdquoJournal of Vibration and Shock vol 6 pp 229ndash233 2011

[6] J Li T Yan X Sun and S Peng ldquoFinite element analysis ondrilling string axial vibration in a crooked holerdquo in Proceedingsof the International Conference on Pipelines and TrenchlessTechnology (ICPTT rsquo12) vol 10 pp 1328ndash1334 Wuhan ChinaOctober 2012

[7] T Yan XWang S Li X Bi andCHan ldquoFinite element analysisof longitudinal and lateral coupled vibration of drillstringrdquo OilField Equipment vol 3 p 12 2012

[8] X Yan J Sun M Zhang et al ldquoAnalysis of longitudinalvibration of drill strings using equivalent networkrdquo Oil FieldEquipment vol 4 p 5 2010

[9] S M Sahebkar M R Ghazavi S E Khadem and M HGhayesh ldquoNonlinear vibration analysis of an axially movingdrillstring system with time dependent axial load and axialvelocity in inclined wellrdquoMechanism and Machine Theory vol46 no 5 pp 743ndash760 2011

[10] W B Wu K H Wang D H Wu and B N Ma ldquoStudy ofdynamic longitudinal impedance of tapered pile consideringlateral inertial effectrdquo Chinese Journal of Rock Mechanics andEngineering vol 30 no 2 pp 3618ndash3625 2011

[11] T Ding andC Li ldquoAnalysis of coupling axial vibrations betweendrilling fluids and drillstringrdquo Chinese Journal of MechanicalEngineering vol 43 no 9 pp 215ndash219 2007

[12] V I Gulyayev and O I Borshch ldquoFree vibrations of drill stringsin hyper deep vertical bore-wellsrdquo Journal of Petroleum Scienceand Engineering vol 78 no 3-4 pp 759ndash764 2011

[13] C Bu X Li L Sun and B Xia ldquoArithmetic solution for the axialvibration of drill string coupling with a down-the-hole hammerin rock drillingrdquo Journal of Vibration and Control 2014

[14] Y A Khulief F A Al-Sulaiman and S Bashmal ldquoVibrationanalysis of drillstrings with self-excited stick-slip oscillationsrdquoJournal of Sound and Vibration vol 299 no 3 pp 540ndash5582007

[15] W-L Wang Q-A Hu H-W Liu and W Zhang ldquoAnalysison longitudinal vibration of drill string and the correspondingapplicationrdquo Journal of Vibration and Shock vol 30 no 6 pp229ndash233 2011

[16] A A Jafari R Kazemi and M F Mahyari ldquoThe effects ofdrilling mud and weight bit on stability and vibration of a drillstringrdquo Journal of Vibration and Acoustics Transactions of theASME vol 134 no 1 Article ID 011014 2012

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



rPk


120591Pkminuso

H

hk+1

hk

h1 120575Pk

kPkz



rPk


120591Pkminuso

H

hk+1

hk

h1 120575Pk

kPkz



Figure 1 Longitudinal vibration model of drill string

Meanwhile by comparing different calculation results somekey parametersrsquo influences on vibration features are pre-sented such as the drill string length inside and outsideradius damping coefficient and Poissonrsquos ratio The researchmethod and model can provide reference for drill stringvibration research At the same time the research resultscan also present theoretical and actual value for optimizationof downhole tools design and safety evaluation of drillingoperations

2 Analysis Method and Mathematical Model

Based on the real condition of the drill string relatedassumptions are proposed to simplify the research for drillstring longitudinal vibration [16] The following assumptionsare made the cross-sectional surface of the drill string isequivalently circular the drill string axis coincides withborehole axis the drilling fluid is regarded as viscoelasticlayer which can be divided into several layers and thedrilling fluids inside and outside drill string have identicalcharacteristics the vibration velocity of drill string is muchlarger than the flow rate of drilling fluid

According to drilling condition the drill string vibrationanalysis model is presented as in Figure 1 wherein ℎ119896 denotesthe location layer of 119896th drill string 119867 is the total lengthof the drill string 119903119875119896 and 119903

1015840

119875119896 respectively denote the outer

and inner radius of 119896th drill string 120591119875119896minus119894 120591119875119896minus119900 respectivelydenote the shear stress of drilling fluid inside and outside119896th drill string 119896119875119896 is the stiffness coefficient and 120575119875119896 meansthe damping coefficient of the 119896th drill string Usually inthe absence of inner and outer mud pressure this vibrationreduces to the classical wave vibration for longitudinal wavesin a uniform rod

21 Model of Drill String without Lateral Inertia Effect Thedeformation produced in a uniform drill string while it

vibrates in a borehole is symmetric about its axis Forestablishing the longitudinal vibration model of drill stringwithout lateral inertia effect regarding the drill string as aone-dimensional sticky elastomer and by using the laws ofmass andmomentum conservation the vibration equation ofthe 119896th drill string segment can be obtained as

119864119875119896119860119896

1205972119908119896

1205971199112minus 2120587120591119875119896minus119900119903119875119896 minus 2120587120591119875119896minus119894119903

1015840

119875119896= 120588119875119896119860119896

1205972119908119896

1205971199052

(V119891 gt VV V119891 gt 0)

119864119875119896119860119896

1205972119908119896


1015840

119875119896= 120588119875119896119860119896

1205972119908119896

1205971199052

(V119891 lt VV V119891 gt 0)

119864119875119896119860119896

1205972119908119896

1205971199112+ 2120587120591119875119896minus119900119903119875119896 + 2120587120591119875119896minus119894119903

1015840

119875119896= 120588119875119896119860119896

1205972119908119896

1205971199052

(V119891 gt VV V119891 lt 0)

119864119875119896119860119896

1205972119908119896

1205971199112minus 2120587120591119875119896minus119900119903119875119896 + 2120587120591119875119896minus119894119903

1015840

119875119896= 120588119875119896119860119896

1205972119908119896

1205971199052

(V119891 lt VV V119891 lt 0)

(1)

wherein the right-hand side of this equation denotes themudpressures which is substituted with axial displacements Theleft-hand side of it denotes inertia force and the shear stressof drilling fluid inside and outside the drill string119908119896 denotesthe longitudinal displacement of the 119896th drill string segmentwhich is function of time 119905 and coordinate 119911 119908119896 = 119908119896(119911 119905)VV V119891 represent the vibration velocity of drill string and theflow rate of drilling fluid respectively Moreover 119864119875119896 denotesthe elastic modulus of the 119896th drill string segment 120588119875119896 is



119860119896 = 120587 (1199032

119875119896minus 11990310158402

119875119896) (2)


119864119875119896119860119896

1205972119908119896


1015840

119875119896

= 120588119875119896119860119896

1205972119908119896

1205971199052

(3)


120591119875119896minus119894 = 119870119896minus119894119908119896 (119911 119905)

120591119875119896minus119900 = 119870119896minus119900119908119896 (119911 119905)

(4)


119864119875119896119860119896

1205972119882119896

1205971199112


1015840

119875119896]119882119896 = 0

(5)


119882119896 (119911 119904) = 119871 [119908119896 (119911 119905)] = int

+infin

0

119908119896 (119911 119905) 119890minus119904119905119889119905 (6)


120572 = radic120588119875119896119860119896119904


1015840

119875119896)

119864119875119896119860119896

(7)


119882119896 (119911 119904) = 119862119896119890120572119911+ 119863119896119890

minus120572119911 (8)



119864119875119896119860119896

120597119882119896 (119911 119904)

120597119911

10038161003816100381610038161003816100381610038161003816119911=0= 119891119896 (119911 119904) (9)

119864119875119896

120597119882119896 (119911 119904)

120597119911

10038161003816100381610038161003816100381610038161003816119911=119867= minus (119896119875119896 + 120575119875119896119904)119882119896 (119911 119904)

1003816100381610038161003816119911=119867 (10)


119862119896 minus 119863119896 =119891119896 (119911 119904)

120572119864119875119896119860119896

119862119896 =119864119875119896120572 minus (119896119875119896 + 120575119875119896119904)

119864119875119896120572 + (119896119875119896 + 120575119875119896119904)119890minus2120572119867

119863119896

(11)


120574 =119864119875119896120572 minus (119896119875119896 + 120575119875119896119904)

119864119875119896120572 + (119896119875119896 + 120575119875119896119904)119890minus2120572119867

(12)


119882119896 (119911 119904) =119891119896 (119911 119904)

(120574 minus 1) 120572119864119875119896119860119896

(120574119890120572119911+ 119890minus120572119911) (13)


119904 = 119894120596 (14)


119870119875119889 =120572119860119896 (120574 minus 1) 119864119875119896

120574 + 1 (15)


119870119875119889 = 119870119903 + 119894119862119894 (16)





119864119875119896119860119896

1205972119908119896


1015840

119875119896)

= 120588119875119896119860119896 (1205972119908119896

1205971199052minus ]2119875119896(1199032

119875119896minus 11990310158402

119875119896)1205974119908119896

12059711991121205971199052)

(V119891 gt VV V119891 gt 0)

119864119875119896119860119896

1205972119908119896


1015840

119875119896)

= 120588119875119896119860119896 (1205972119908119896

1205971199052minus ]2119875119896(1199032

119875119896minus 11990310158402

119875119896)1205974119908119896

12059711991121205971199052)

(V119891 lt VV V119891 gt 0)

119864119875119896119860119896

1205972119908119896

1205971199112+ 2120587 (120591119875119896minus119894119903119875119896 + 120591119875119896minus119900119903

1015840

119875119896)

= 120588119875119896119860119896 (1205972119908119896

1205971199052minus ]2119875119896(1199032

119875119896minus 11990310158402

119875119896)1205974119908119896

12059711991121205971199052)

(V119891 gt VV V119891 lt 0)

119864119875119896119860119896

1205972119908119896


1015840

119875119896)

= 120588119875119896119860119896 (1205972119908119896

1205971199052minus ]2119875119896(1199032

119875119896minus 11990310158402

119875119896)1205974119908119896

12059711991121205971199052)

(V119891 lt VV V119891 lt 0)

(17)


119864119875119896119860119896

1205972119908119896


1015840

119875119896)

= 120588119875119896119860119896 (1205972119908119896

1205971199052minus ]2119875119896(1199032

119875119896minus 11990310158402

119875119896)1205974119908119896

12059711991121205971199052)

(18)



120572 = radic120588119875119896119860119896119904


1015840

119875119896)

119864119875119896119860119896 + 120588119875119896119860119896]2119875119896 (1199032

119875119896minus 11990310158402119875119896) 1199042

120573 = 120588119875119896]2

119875119896(1199032

119875119896minus 11990310158402

119875119896) 1199042

120574 =(119864119875119896 + 120573) 120572 minus (119896119875119896 + 120575119875119896119904)

(119864119875119896 + 120573) 120572 + (119896119875119896 + 120575119875119896119904)119890minus2120572119867

(19)


119882119896 (119911 119904) =119891119896 (119911 119904)

(120574 minus 1) 120572 (119864119875119896119860119896 + 119860119896120573)(120574119890120572119911+ 119890minus120572119911) (20)


119870119875119889 =120572119860119896 (120574 minus 1) (119864119875119896 + 120573)

120574 + 1 (21)








119875119896(kgm3) 800000




3

2

1

0

minus1

minus2

Kr

(Nm

)

times104

H1 = 1800

h1 = 1800

H2 = 2000

h2 = 2000

22 23 24 25 26 27 28 29 3021f (Hz)




119875119896(m) 008




543210

minus1

minus3minus2

minus4

Kr

(Nm

)

times104

22 23 24 25 26 27 28 29 302120f (Hz)

RPk1 = 01

rPk1 = 01

RPk2 = 011

rPk2 = 011




1198751198962 1199031198751198962


1198751198961 1199031198751198961 (m) 008




1015840




3

2

1

0

minus1

minus2

Kr

(Nm

)

times104

22 23 24 25 26 27 28 29 302120f (Hz)

R998400Pk1 = 008

r998400Pk1 = 008

R998400Pk2 = 009

r998400Pk2 = 009




119875119896(m) 008



1198751198962 1205751198751198962 (Nsm) 10000






120575Pk1 = 8000

120575998400Pk1 = 8000

120575Pk2 = 10000

120575998400Pk2 = 10000

4

2

0

minus2

minus4

Kr

(Nm

)

32 33 34 35 36 37 38 39 403130f (Hz)

times104


3

4

2

1

0

minus1

minus3

minus2

Kr

(Nm

)

32 33 34 35 36 37 38 39 403130f (Hz)

times104

1 = 01

9984001 = 01

2 = 03

9984002 = 03




1 ]1 010



119875119896(m) 008










4 Conclusion









Acknowledgments


References




















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119860119896 = 120587 (1199032

119875119896minus 11990310158402

119875119896) (2)


119864119875119896119860119896

1205972119908119896


1015840

119875119896

= 120588119875119896119860119896

1205972119908119896

1205971199052

(3)


120591119875119896minus119894 = 119870119896minus119894119908119896 (119911 119905)

120591119875119896minus119900 = 119870119896minus119900119908119896 (119911 119905)

(4)


119864119875119896119860119896

1205972119882119896

1205971199112


1015840

119875119896]119882119896 = 0

(5)


119882119896 (119911 119904) = 119871 [119908119896 (119911 119905)] = int

+infin

0

119908119896 (119911 119905) 119890minus119904119905119889119905 (6)


120572 = radic120588119875119896119860119896119904


1015840

119875119896)

119864119875119896119860119896

(7)


119882119896 (119911 119904) = 119862119896119890120572119911+ 119863119896119890

minus120572119911 (8)



119864119875119896119860119896

120597119882119896 (119911 119904)

120597119911

10038161003816100381610038161003816100381610038161003816119911=0= 119891119896 (119911 119904) (9)

119864119875119896

120597119882119896 (119911 119904)

120597119911

10038161003816100381610038161003816100381610038161003816119911=119867= minus (119896119875119896 + 120575119875119896119904)119882119896 (119911 119904)

1003816100381610038161003816119911=119867 (10)


119862119896 minus 119863119896 =119891119896 (119911 119904)

120572119864119875119896119860119896

119862119896 =119864119875119896120572 minus (119896119875119896 + 120575119875119896119904)

119864119875119896120572 + (119896119875119896 + 120575119875119896119904)119890minus2120572119867

119863119896

(11)


120574 =119864119875119896120572 minus (119896119875119896 + 120575119875119896119904)

119864119875119896120572 + (119896119875119896 + 120575119875119896119904)119890minus2120572119867

(12)


119882119896 (119911 119904) =119891119896 (119911 119904)

(120574 minus 1) 120572119864119875119896119860119896

(120574119890120572119911+ 119890minus120572119911) (13)


119904 = 119894120596 (14)


119870119875119889 =120572119860119896 (120574 minus 1) 119864119875119896

120574 + 1 (15)


119870119875119889 = 119870119903 + 119894119862119894 (16)





119864119875119896119860119896

1205972119908119896


1015840

119875119896)

= 120588119875119896119860119896 (1205972119908119896

1205971199052minus ]2119875119896(1199032

119875119896minus 11990310158402

119875119896)1205974119908119896

12059711991121205971199052)

(V119891 gt VV V119891 gt 0)

119864119875119896119860119896

1205972119908119896


1015840

119875119896)

= 120588119875119896119860119896 (1205972119908119896

1205971199052minus ]2119875119896(1199032

119875119896minus 11990310158402

119875119896)1205974119908119896

12059711991121205971199052)

(V119891 lt VV V119891 gt 0)

119864119875119896119860119896

1205972119908119896

1205971199112+ 2120587 (120591119875119896minus119894119903119875119896 + 120591119875119896minus119900119903

1015840

119875119896)

= 120588119875119896119860119896 (1205972119908119896

1205971199052minus ]2119875119896(1199032

119875119896minus 11990310158402

119875119896)1205974119908119896

12059711991121205971199052)

(V119891 gt VV V119891 lt 0)

119864119875119896119860119896

1205972119908119896


1015840

119875119896)

= 120588119875119896119860119896 (1205972119908119896

1205971199052minus ]2119875119896(1199032

119875119896minus 11990310158402

119875119896)1205974119908119896

12059711991121205971199052)

(V119891 lt VV V119891 lt 0)

(17)


119864119875119896119860119896

1205972119908119896


1015840

119875119896)

= 120588119875119896119860119896 (1205972119908119896

1205971199052minus ]2119875119896(1199032

119875119896minus 11990310158402

119875119896)1205974119908119896

12059711991121205971199052)

(18)



120572 = radic120588119875119896119860119896119904


1015840

119875119896)

119864119875119896119860119896 + 120588119875119896119860119896]2119875119896 (1199032

119875119896minus 11990310158402119875119896) 1199042

120573 = 120588119875119896]2

119875119896(1199032

119875119896minus 11990310158402

119875119896) 1199042

120574 =(119864119875119896 + 120573) 120572 minus (119896119875119896 + 120575119875119896119904)

(119864119875119896 + 120573) 120572 + (119896119875119896 + 120575119875119896119904)119890minus2120572119867

(19)


119882119896 (119911 119904) =119891119896 (119911 119904)

(120574 minus 1) 120572 (119864119875119896119860119896 + 119860119896120573)(120574119890120572119911+ 119890minus120572119911) (20)


119870119875119889 =120572119860119896 (120574 minus 1) (119864119875119896 + 120573)

120574 + 1 (21)








119875119896(kgm3) 800000




3

2

1

0

minus1

minus2

Kr

(Nm

)

times104

H1 = 1800

h1 = 1800

H2 = 2000

h2 = 2000

22 23 24 25 26 27 28 29 3021f (Hz)




119875119896(m) 008




543210

minus1

minus3minus2

minus4

Kr

(Nm

)

times104

22 23 24 25 26 27 28 29 302120f (Hz)

RPk1 = 01

rPk1 = 01

RPk2 = 011

rPk2 = 011




1198751198962 1199031198751198962


1198751198961 1199031198751198961 (m) 008




1015840




3

2

1

0

minus1

minus2

Kr

(Nm

)

times104

22 23 24 25 26 27 28 29 302120f (Hz)

R998400Pk1 = 008

r998400Pk1 = 008

R998400Pk2 = 009

r998400Pk2 = 009




119875119896(m) 008



1198751198962 1205751198751198962 (Nsm) 10000






120575Pk1 = 8000

120575998400Pk1 = 8000

120575Pk2 = 10000

120575998400Pk2 = 10000

4

2

0

minus2

minus4

Kr

(Nm

)

32 33 34 35 36 37 38 39 403130f (Hz)

times104


3

4

2

1

0

minus1

minus3

minus2

Kr

(Nm

)

32 33 34 35 36 37 38 39 403130f (Hz)

times104

1 = 01

9984001 = 01

2 = 03

9984002 = 03




1 ]1 010



119875119896(m) 008










4 Conclusion









Acknowledgments


References




















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119864119875119896119860119896

1205972119908119896


1015840

119875119896)

= 120588119875119896119860119896 (1205972119908119896

1205971199052minus ]2119875119896(1199032

119875119896minus 11990310158402

119875119896)1205974119908119896

12059711991121205971199052)

(V119891 gt VV V119891 gt 0)

119864119875119896119860119896

1205972119908119896


1015840

119875119896)

= 120588119875119896119860119896 (1205972119908119896

1205971199052minus ]2119875119896(1199032

119875119896minus 11990310158402

119875119896)1205974119908119896

12059711991121205971199052)

(V119891 lt VV V119891 gt 0)

119864119875119896119860119896

1205972119908119896

1205971199112+ 2120587 (120591119875119896minus119894119903119875119896 + 120591119875119896minus119900119903

1015840

119875119896)

= 120588119875119896119860119896 (1205972119908119896

1205971199052minus ]2119875119896(1199032

119875119896minus 11990310158402

119875119896)1205974119908119896

12059711991121205971199052)

(V119891 gt VV V119891 lt 0)

119864119875119896119860119896

1205972119908119896


1015840

119875119896)

= 120588119875119896119860119896 (1205972119908119896

1205971199052minus ]2119875119896(1199032

119875119896minus 11990310158402

119875119896)1205974119908119896

12059711991121205971199052)

(V119891 lt VV V119891 lt 0)

(17)


119864119875119896119860119896

1205972119908119896


1015840

119875119896)

= 120588119875119896119860119896 (1205972119908119896

1205971199052minus ]2119875119896(1199032

119875119896minus 11990310158402

119875119896)1205974119908119896

12059711991121205971199052)

(18)



120572 = radic120588119875119896119860119896119904


1015840

119875119896)

119864119875119896119860119896 + 120588119875119896119860119896]2119875119896 (1199032

119875119896minus 11990310158402119875119896) 1199042

120573 = 120588119875119896]2

119875119896(1199032

119875119896minus 11990310158402

119875119896) 1199042

120574 =(119864119875119896 + 120573) 120572 minus (119896119875119896 + 120575119875119896119904)

(119864119875119896 + 120573) 120572 + (119896119875119896 + 120575119875119896119904)119890minus2120572119867

(19)


119882119896 (119911 119904) =119891119896 (119911 119904)

(120574 minus 1) 120572 (119864119875119896119860119896 + 119860119896120573)(120574119890120572119911+ 119890minus120572119911) (20)


119870119875119889 =120572119860119896 (120574 minus 1) (119864119875119896 + 120573)

120574 + 1 (21)








119875119896(kgm3) 800000




3

2

1

0

minus1

minus2

Kr

(Nm

)

times104

H1 = 1800

h1 = 1800

H2 = 2000

h2 = 2000

22 23 24 25 26 27 28 29 3021f (Hz)




119875119896(m) 008




543210

minus1

minus3minus2

minus4

Kr

(Nm

)

times104

22 23 24 25 26 27 28 29 302120f (Hz)

RPk1 = 01

rPk1 = 01

RPk2 = 011

rPk2 = 011




1198751198962 1199031198751198962


1198751198961 1199031198751198961 (m) 008




1015840




3

2

1

0

minus1

minus2

Kr

(Nm

)

times104

22 23 24 25 26 27 28 29 302120f (Hz)

R998400Pk1 = 008

r998400Pk1 = 008

R998400Pk2 = 009

r998400Pk2 = 009




119875119896(m) 008



1198751198962 1205751198751198962 (Nsm) 10000






120575Pk1 = 8000

120575998400Pk1 = 8000

120575Pk2 = 10000

120575998400Pk2 = 10000

4

2

0

minus2

minus4

Kr

(Nm

)

32 33 34 35 36 37 38 39 403130f (Hz)

times104


3

4

2

1

0

minus1

minus3

minus2

Kr

(Nm

)

32 33 34 35 36 37 38 39 403130f (Hz)

times104

1 = 01

9984001 = 01

2 = 03

9984002 = 03




1 ]1 010



119875119896(m) 008










4 Conclusion









Acknowledgments


References




















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












119875119896(kgm3) 800000




3

2

1

0

minus1

minus2

Kr

(Nm

)

times104

H1 = 1800

h1 = 1800

H2 = 2000

h2 = 2000

22 23 24 25 26 27 28 29 3021f (Hz)




119875119896(m) 008




543210

minus1

minus3minus2

minus4

Kr

(Nm

)

times104

22 23 24 25 26 27 28 29 302120f (Hz)

RPk1 = 01

rPk1 = 01

RPk2 = 011

rPk2 = 011




1198751198962 1199031198751198962


1198751198961 1199031198751198961 (m) 008




1015840




3

2

1

0

minus1

minus2

Kr

(Nm

)

times104

22 23 24 25 26 27 28 29 302120f (Hz)

R998400Pk1 = 008

r998400Pk1 = 008

R998400Pk2 = 009

r998400Pk2 = 009




119875119896(m) 008



1198751198962 1205751198751198962 (Nsm) 10000






120575Pk1 = 8000

120575998400Pk1 = 8000

120575Pk2 = 10000

120575998400Pk2 = 10000

4

2

0

minus2

minus4

Kr

(Nm

)

32 33 34 35 36 37 38 39 403130f (Hz)

times104


3

4

2

1

0

minus1

minus3

minus2

Kr

(Nm

)

32 33 34 35 36 37 38 39 403130f (Hz)

times104

1 = 01

9984001 = 01

2 = 03

9984002 = 03




1 ]1 010



119875119896(m) 008










4 Conclusion









Acknowledgments


References




















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










3

2

1

0

minus1

minus2

Kr

(Nm

)

times104

22 23 24 25 26 27 28 29 302120f (Hz)

R998400Pk1 = 008

r998400Pk1 = 008

R998400Pk2 = 009

r998400Pk2 = 009




119875119896(m) 008



1198751198962 1205751198751198962 (Nsm) 10000






120575Pk1 = 8000

120575998400Pk1 = 8000

120575Pk2 = 10000

120575998400Pk2 = 10000

4

2

0

minus2

minus4

Kr

(Nm

)

32 33 34 35 36 37 38 39 403130f (Hz)

times104


3

4

2

1

0

minus1

minus3

minus2

Kr

(Nm

)

32 33 34 35 36 37 38 39 403130f (Hz)

times104

1 = 01

9984001 = 01

2 = 03

9984002 = 03




1 ]1 010



119875119896(m) 008










4 Conclusion









Acknowledgments


References




















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation















4 Conclusion









Acknowledgments


References




















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation

























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









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