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DOI: 10.1177/0954406212455126

online 3 August 2012 publishedProceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science

Ahmad Ghasemloonia, D Geoff Rideout and Stephen D Buttload

Coupled transverse vibration modeling of drillstrings subjected to torque and spatially varying axial

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Original Article

Coupled transverse vibration modeling ofdrillstrings subjected to torque andspatially varying axial load

Ahmad Ghasemloonia, D Geoff Rideout and Stephen D Butt

Abstract

Predicting and mitigating unwanted vibration of drillstrings is an important subject for oil drilling companies. Uncontrolled

vibrations cause premature failure of the drillstring and associated components. The drillstring is a long slender structure

that vibrates in three primary coupled modes: torsional, axial and transverse. Among these coupled modes, the transverse

mode is the major cause of drillstring failures and wellbore washout. Modal analysis of drillstrings reveals critical frequen-

cies and helps drillers to avoid running the bit near critical modes. In this article, the coupled orthogonal modes of

transverse vibration of a drillstring in the presence of torque and spatially varying axial force (due to mud hydrostatic

effect, self-weight and hook load) are derived and the mode shapes and natural frequencies are determined through the

expanded Galerkin method. The results are verified by the nonlinear finite element method. Modal mass participation

factor, which represents how strongly a specific mode contributes to the motion in a certain direction, is used to

determine the appropriate number of modes to retain so that computational efficiency can be maximized.

Keywords

Drillstring, transverse vibration, coupled modes, Galerkin’s method, mode participation factor, finite element method,

modal mass participation factor

Date received: 28 November 2011; accepted: 26 June 2012

Introduction

The subject of drillstring vibration is an ongoing chal-lenging for drillers in oil fields. The effects of vibrationon drilling performance, wellbore stability, joint fail-ures, fatigue, etc. have led drilling companies tostrengthen components or try to control and mitigatethese effects to attain higher performance. In order tocontrol or mitigate the vibration, its behavior andcharacteristics should be revealed and modeled ana-lytically,1,2 experimentally in laboratory scale,3 orthrough field verification.4

A drillstring is a slender structure which consists ofdrill pipes at the upper sections and drill collars andstabilizers at the bottom sections. The drill pipes arehollow pipes (assumed 120mm outer diameter, 10mmthickness in this article) that are lighter than the col-lars (normally with an outer diameter of 120–240mmand thickness of 30–80mm). The bit is attached to thebottom of the collars. The lower section is called the‘bottom hole assembly’ or BHA. The lower, heavierBHA is more easily excited to vibrate than the pipesection. This is due to the presence of an axial load,which is varying spatially along the drillstring length.This linearly varying axial load is a result of threeinteracting axial forces: hook load, self-weight of thedrillstring and mud hydrostatic force. Drilling

performance is very sensitive to the axial force atthe bit, i.e. weight on bit (WOB), and WOB is oneof the main parameters adjusted during drilling toimprove penetration rate. Rate of penetration, orROP, is the conventional index for measuring the effi-ciency of the drilling process.

The dominant role of BHA vibrations on the totaldrillstring vibration was verified by Dareing,5 whoshowed that the collars are easily excited in thelower modes. The pipes, in tension, vibrate at higherexcitation frequencies,6 as will be shown in the latersection on finite element modeling. In most cases, therotational speed at the normal operational conditionsis not high enough to excite the higher modes.The other reason for analyzing the BHA is that mea-surement-while-drilling (MWD) tools are mostlylocated near the bit in the BHA and any type ofBHA vibrations interfere with the interpretation of

Advanced Drilling Group, Faculty of Engineering and Applied Science,

Memorial University, St. John’s, NL, Canada

Corresponding author:

Ahmad Ghasemloonia, Advanced Drilling Group, Faculty of Engineering

and Applied Science, Memorial University, St. John’s, NL, Canada.

Email: [email protected]

Proc IMechE Part C:

J Mechanical Engineering Science

0(0) 1–15

! IMechE 2012

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down-the-hole (DTH) status at the surface. Finally,the BHA is composed of collars which are heavy andstiff and any type of unwanted vibration dissipates aportion of energy which is supposed to be delivered tothe bit. Therefore, an increased understanding of theBHA vibrations will give valuable insight into poten-tial vibration, and ways to avoid it, under normaloperating conditions.

The primarymodes of drillstring vibrations are axial,transverse and torsional.7–9 These modes are coupledtogether via terms containing variables like torque.10

Coupled torsional-bending,11 coupled axial-bending12

and coupled axial-torsional13 are three common com-binations of coupled modes. Stick-slip oscillations14 aretorsional, while whirling and bit bounce15 are examplesof lateral and axial vibrations respectively. These vibra-tions can be transient or steady, depending on the dril-ling parameters such as WOB, torque, rotational speedand many other drilling conditions.

Among these primary modes, the transverse modeis said to be responsible for 75% of failures.16 Bendingwaves are not propagated to the surface via the drill-string as are torsional and longitudinal waves, due tothe difference in the wave speed for different types ofmodes. The propagation speed for axial and torsionalmotions is quite high compared to the lateral motion.Therefore, for any given length of the drillstring, axialand torsional waves travel a few wavelengths toreach to the surface, while in contrast, the lateralwave travels many wavelengths to be felt at the sur-face. Furthermore, transverse vibration is more highlydamped than the other modes due to mud effects andwellbore contact.16 Therefore, there could be severebending vibrations deep in the hole, which the surfacemeasuring tools do not indicate. As later finite elem-ent method (FEM) modeling will show, in the lowerfrequencies the collars are vibrating transversely,while the pipes do not vibrate and remain approxi-mately undeflected. This is due to the axial load dis-tribution along the drillstring, the collars of which aremainly under compression while the pipes are undertension. As a result of the tension, the natural fre-quencies of the pipe section increase, while the naturalfrequencies of the collar section are reduced.

Literature review

Due to the importance of the transverse mode, severalstudies have been done to understand the behavior ofthe drillstring in this mode. Jansen17 studied the contactbehavior between the stabilizer and the wellbore at thepoint of contact, using a lumped segment approach. Henoticed that gravity and lateral coupling should betaken into account for more quantitative analysis.Chen et al.18 studied the lateral vibration of a BHA inthe presence of constant weight on bit, but neglectingthe torque. Berlioz et al.19 performed a laboratory testto study lateral vibration of the drillstring and showedthat the influence of axial force is greater than that of the

torque on the natural frequencies of the drillstring.However, they did not consider spatially varyingaxial load and coupled orthogonal lateral modes.Christoforou et al.20 used the Lagrangian method toderive the equations of motion and study the drillstringtrajectory in the lateral mode at the contact point. Theyused a sine wave as the dynamic WOB, without con-sidering the torque as the coupling term for the lateralmodes. Stability of the drillstring in the lateralmodewasstudied by Gulyaev et al.21 They investigated the buck-ling mode of the drillstring as a function of its length forspecial cases which have analytical solution. Theyshowed that the buckling mode will occur at the sectionwith the compressive axial force (collar section). Inanother work by Dareing,22 the sensitivity of drillcollar vibration to the length was studied using simplebeam equations without torque and axial load. Khuliefet al.23 used the FEM to derive the frequency andmodalresponse of a rotating drillstring without torque. Theycompared the results for the full order model and devel-oped a reduced order model for the total drillstring. Thestring borehole interaction was also an interesting sub-ject in the lateral mode analysis and several studies havebeen carried out on this subject.24,25 Laboratory testrigs, field data and FEM models have been used exten-sively for verification of derived natural frequencies ortime response of the BHA.4,19,23

Coupled orthogonal transverse vibration of thedrillstring in presence of torque and linearly varyingaxial force has not been previously addressed. Thegoals of this article are analytical and numerical mod-eling of bending vibration of drill collars and accurateprediction of natural frequencies. Knowledge of suchfrequencies and understanding of the underlying phys-ics will help drilling companies avoid resonance andreduce drillstring failures. In contrast to previous stu-dies, this work includes the effects of steady torque andspatially varying axial load, thereby revealing couplingbetween the orthogonal components of lateral vibra-tion. The FEM method is used to validate analyticalmodel predictions. This article also uses the concept ofmodal mass participation factor to determine therequired number of modes to retain. Retainingunnecessary modes can increase computation burdenwithout significantly increasing model fidelity. Thenatural frequencies and mode shapes calculated usingthe methods of this article can be exported for use inlow-order modal expansion models.26 While notmatching the fidelity of high-order FEM models,low-order models can be more computationally effi-cient, more easily interfaced to other subsystemmodels, and still useful for top-level design studies.

In the following section, analytical equations of thedrillstring lateral vibration, which are coupled viatorque under linearly varying axial load, will be derived.The next section derives the governing equations andapplied the expanded Galerkin method to solve them.Section ‘Analytical solution of the equations’ describesthe FEM application and results. Section ‘Application

2 Proc IMechE Part C: J Mechanical Engineering Science 0(0)




of the FEM method to the transverse vibration of thedrillstring’ discusses proper model order, and the finalsection states conclusions and future work.

Derivation of governing equations

Drillstrings are assumed to be beamlike structures.Due to the high slenderness ratio of the drillstringand low rotational speed, among the conventionalmodels of beam theory (Timoshenko, Rayleigh andEuler–Bernoulli), the Euler–Bernoulli beam theory isused to model the lateral vibration of drillstrings.Since the aspect ratio is greater than 10 for the drill-string, the use of classical thin beam element (Euler–Bernoulli) is valid. Several studies have been carriedout to model different types of vertical beams, e.g.Hijmissen et al.,27 which modeled the vertical beamunder the effect of gravity.

In this article, the direct Newtonian approach is usedto derive the mathematical model. The rotation of thedrillstring (Figure 1) and gyroscopic effect are neglected,since the string could be assumed as a low-speedrotor.6,9,12,18 An element of the beam, which is undertorque and linearly varying axial load, is considered.This element is located on a portion of the drillstring(beam), which is between two stabilizers (Figure 2). Twolateral directions, namely u and v, are considered forextracting the equations of motion. The elements inthe uz plane and for vz plane are shown in Figure 2.The following derivation refers to equations (15) to(21) which can be found in Appendix 2.

The torque ~T is resolved along the selected elementas two separate bending moments in the u and v dir-ections as a result of bending curvature of the drill-string (Figure 3)

~T ¼ ~Tt þ ~Tn ð1Þ

The ~T vector in the normal direction will be

~Tn

�� ¼ ~T:~n¼ð0,0,TÞ� 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffidu

dz

� �2

þdv

dz

� �2

þ1

s du

dz,dv

dz,1

� �

¼Tffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

du

dz

� �2

þdv

dz

� �2

þ1

s

~Tn¼ ~Tn

�� :~n¼ Tffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffidu

dz

� �2

þdv

dz

� �2

þ1

s

�1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

du

dz

� �2

þdv

dz

� �2

þ1

s du

dz,dv

dz,1

� �

¼T

du

dz

� �2

þdv

dz

� �2

þ1

du

dz,dv

dz,1

� �ð2Þ

Hence, the tangential component of the torque willbe

~Tt ¼ ~T� ~Tn ¼ ð0, 0,TÞ

�T

du

dz

� �2

þdv

dz

� �2

þ1

du

dz,dv

dz, 1

� �

¼T

du

dz

� �2

þdv

dz

� �2

þ1

� �du

dz, �

dv

dz,

du

dz

� �2

þdv

dz

� �2 ! !

ð3Þ

~Tt acts as a bending moment in the u and v direc-tions. The corresponding coupled term in the equa-tion of motion will be third order. The correspondingcomponent of Tt as a bending moment in the u direc-

tion is �Tðdv=dzÞ

ððdu=dzÞÞ2þððdv=dzÞÞ2þ1. According to equation (16)

and substituting for Mv (neglecting the second orderdifferential terms)

@

@z�T

dv

dz

� �¼ Su ð4Þ

As it is clear from equations (18) and (19), theterms related to Su will appear in the final equation

Figure 1. Schematic diagram of the drillstring in the wellbore.

Ghasemloonia et al. 3




of motion as � @Su

@z . Therefore, equation (20) is mod-ified by substituting � @Su

@z and assuming the torque isconstant

�@Su

@z¼ �

@

@z

@

@z�T

dv

dz

� �� ¼ T

@3vðz, tÞ

@z3ð5Þ

Therefore, the T term will be added to the equationof motion in the u direction as a third-order derivativeof v. The above procedure can be repeated in the vdirection with the similar results, except a negativesign due to the opposite direction.

Adding these two coupling effects to equations (20)and (21), and assuming the spatially varying axial forceas Faxial ¼ F0 � �Agz (Figure 4), the final form of theequations of motion for the two lateral directions is

E:Iv@4u ðz, tÞ

@z4� �Ag

@u ðz, tÞ

@z

þ ðF0 � �AgzÞ �@2u ðz, tÞ

@z2þ T

@3v ðz, tÞ

@z3

þ �A@2u ðz, tÞ

@t2¼ 0

E:Iv@4v ðz, tÞ

@z4� �Ag

@v ðz, tÞ

@z

þ ðF0 � �AgzÞ �@2v ðz, tÞ

@z2� T

@3u ðz, tÞ

@z3

þ �A@2v ðz, tÞ

@t2¼ 0 ð6Þ

where F0 is the amount of compressive axial force atthe top point of the collar section. The concentrated

load at this point is due to mismatch of projected areasand the resulting resolved hydrostatic axial forces.28

The derivation of the spatially varying axial forcealong the collar section can be found in Appendix 3.

The above set of equations is coupled by order of 3through the torque. Although it is a set of linear equa-tions, the varying axial load means there is not a closedform solution. In the following section, the above equa-tions will be solved to find the natural frequencies of thedrillstring as well as mode participation factors (equa-tion (7)) for the first five modes of lateral vibration.

Analytical solution of the equations

In this section the expanded Galerkin method is usedto derive the characteristics of the lateral vibration ofthe drillstring. Transforming the set of coupled PDEsto a set of linear differential ODEs is the major benefitof this method. Due to the discretized mode shapefunctions in this method, the results can be analyzedseparately in each mode and the predominance of themodes for different sets of initial conditions can beverified. To apply this method to the above problem,uðz, tÞ and vðz, tÞ should be assumed as

uðz, tÞ ¼Xni¼1

�iðzÞ � piðtÞ

vðz, tÞ ¼Xmj¼1

jðzÞ � qjðtÞ

ð7Þ

Figure 2. Drillstring elements between two stabilizers: (a) the uz plane, (b) the vz plane.





where �iðxÞ and jðxÞ are comparison functions, andqiðtÞ and pjðtÞ are mode participation factors.29 The iand j subscripts depend on the desired mode shapesaccording to the frequency range of interest. For this

problem five modes will be retained. Furtherdiscussion on the number of mode shapes thatshould be retained can be found in the section‘Application of the FEM method to the transversevibration of the drillstring’. The comparison func-tions based on assumption of simply supportedBCs3,5,9,11,12 are

�iðzÞ ¼ sini�:z

l

� �

jðzÞ ¼ sinj�:z

l

� � ð8Þ

If the above assumptions are substituted in equa-tion (7) and finally in equation (6), the result is

�A

�sin

�:z

l

� �€p1ðtÞ þ sin

2�:z

l

� �€p2ðtÞ

þ sin3�:z

l

� �€p3ðtÞ þ sin

4�:z

l

� �€p4ðtÞ

þ sin5�:z

l

� �€p5ðtÞ

þ EIv

��4 sin �:z

l

�l4

p1ðtÞ þ16�4 sin 2�:z

l

�l4

p2ðtÞ

þ81�4 sin 3�:z

l

�l4

p3ðtÞ þ256�4 sin 4�:z

l

�l4

p4ðtÞ

þ625�4 sin �:z

l

�l4

p5ðtÞ

þ T

��3 cos �:z

l

�l3

q1ðtÞ �8�3 cos 2�:z

l

�l3

q2ðtÞ

�27�3 cos 3�:z

l

�l3

q3ðtÞ �64�3 cos 4�:z

l

�l3

q4ðtÞ

�125�3 cos 5�:z

l

�l3

q5ðtÞ

� �Ag

�� cos �:z

l

�l

p1ðtÞ þ2� cos �:z

l

�l

p2ðtÞ

þ3� cos �:z

l

�l

p3ðtÞ þ4� cos �:z

l

�l

p4ðtÞ

þ5� cos �:z

l

�l

p5ðtÞ

þ ðF0 � �AgzÞ �

��2 sin �:z

l

�l2

p1ðtÞ

�4�2 sin �:z

l

�l2

p2ðtÞ �9�2 sin �:z

l

�l2

p3ðtÞ

�16�2 sin �:z

l

�l2

p4ðtÞ �25�2 sin �:z

l

�l2

p5ðtÞ

¼ 0

ð9Þ

Figure 3. Torque in the z direction decomposed to tangential

and normal components.

Figure 4. The profile of axial load along drillstring.





For the v direction the result is as follows

�A

�sin

�:z

l

� �€q1ðtÞ þ sin

2�:z

l

� �€q2ðtÞ

þ sin3�:z

l

� �€q3ðtÞ þ sin

4�:z

l

� �€q4ðtÞ

þ sin5�:z

l

� �€q5ðtÞ

þ EIv

��4 sin �:z

l

�l4

q1ðtÞ þ16�4 sin 2�:z

l

�l4

q2ðtÞ

þ81�4 sin 3�:z

l

�l4

q3ðtÞ þ256�4 sin 4�:z

l

�l4

q4ðtÞ

þ625�4 sin �:z

l

�l4

q5ðtÞ

� T

��3 cos �:z

l

�l3

p1ðtÞ �8�3 cos 2�:z

l

�l3

p2ðtÞ

�27�3 cos 3�:z

l

�l3

p3ðtÞ �64�3 cos 4�:z

l

�l3

p4ðtÞ

�125�3 cos 5�:z

l

�l3

p5ðtÞ

� �Ag

�� cos �:z

l

�l

q1ðtÞ þ2� cos �:z

l

�l

q2ðtÞ

þ3� cos �:z

l

�l

q3ðtÞ þ4� cos �:z

l

�l

q4ðtÞ

þ5� cos �:z

l

�l

q5ðtÞ

þ ðF0 � �AgzÞ

�

��2 sin �:z

l

�l2

q1ðtÞ �4�2 sin �:z

l

�l2

q2ðtÞ

�9�2 sin �:z

l

�l2

q3ðtÞ �16�2 sin �:z

l

�l2

q4ðtÞ

�25�2 sin �:z

l

�l2

q5ðtÞ

¼ 0 ð10Þ

The above equations are simplified according tothe fact that the expanded Galerkin method is basedon the orthogonality of modes; i.e.

Zl0

�iðzÞ:�jðzÞdz ¼l

2�ij

Zl0

iðzÞ: jðzÞdz ¼l

2�ij

ð11Þ

Since the comparison functions are assumed to be�iðzÞ ¼ sinði�:zl Þ and jðzÞ ¼ sinðj�:zl Þ, applying integra-tion by parts to equation (11) results in

Zl0

�00i ðzÞ:�00j ðzÞdz ¼

l

2

j�

l

� �4

�ij

Zl0

00i ðzÞ: 00j ðzÞdz ¼

l

2

j�

l

� �4

�ij

ð12Þ

Using the above comparison functions, equation(9) is multiplied by �iðzÞ ¼ sinði��zl Þ for i¼ 1, 2, . . . , 5and then is integrated term by term over ðzÞ 2 ½0, l�,resulting in five coupled time-dependent equations (inthe case of uncoupled equations, this method willresult in five ordinary uncoupled time-varying differ-ential equations). Equation set 10 is multiplied by jðzÞ ¼ sinðj��zl Þ for j¼ 1, 2, . . . , 5 and the results areintegrated term by term over the same domain. Theresult is another set of five ordinary coupled time-varying differential equations. The resulting set of 10equations after simplification of integrations is shownin Appendix 2.

The 10 equations make a system of coupledsecond-order time-varying differential equations.This system was numerically solved using Fehlbergfourth/fifth order Runge–Kutta method with degreefour interpolant, which is an adaptive numericprocedure for solving the initial value problems com-bining fourth-order and fifth-order Runge–Kuttatechniques. The great advantage of this methodis the dynamic step reduction strategy compared tothe fixed-step fourth-order Runge–Kutta method.The initial conditions were derived from the FEMsolver as discussed later in the next section, toensure geometric compatibility. Any compatible setof initial conditions will suffice, as natural frequenciesare not initial condition dependent. Parameter valuesare shown in Table 1.

The results for p1(t) up to p5(t) are shown in Figure5 and for q1(t) up to q5(t) in Figure 6. Figure 7 is anexpanded view of Figure 6 without q1(t). Using theimposed set of initial conditions, the third mode asshown in Figure 5 and the first mode as shown inFigure 6 are the dominant modes.

If a specific location is selected along the beam, inthe vicinity of an anti-node, then the product of�iðzÞ � piðtÞ and jðzÞ � qjðtÞ determines the transversemotion of that location over the time period.The final transverse motion is the summation ofthese transverse motions in each mode.

In order to derive the eigenfrequencies ofeach mode in both directions the values of p(t)and q(t) are stored for the first five modes. Ifthe FFT of each p(t) and q(t) is determined separ-ately, the natural frequency will be revealed.The sampling rate was 1000, and 512 points wereselected for FFT computations. The natural fre-quencies are compared for both directions inFigure 8. There is a small variation between reson-ance frequencies in the u and v directions as a resultof the numerical solution. The maximum differenceis 0.03Hz.

From a practical drilling standpoint, the rotationalspeed should be adjusted so that it does not corres-pond to one of the eigenfrequencies. Effect of torqueand WOB on the natural frequencies in the u and vdirections has been studied as well. The torque wasvaried from 1 to 10 kN�m and the WOB from 30 to





150 kN. The sensitivity of natural frequency tochanges in WOB (Figure 9) is higher than the sensi-tivity to changes in torque (Figure 10). For the changein WOB the maximum change in the naturalfrequency is around 17 r/min, while for the torquethe maximum change is around 4 r/min. Due to thesimilarity of the u and v frequencies, the sensitivity

analysis in Figures 9 and 10 are just for the udirection.

The following section applies the FEM method tothe current problem and verifies the results derived bythe expanded Galerkin method. Linear and nonlinearFEM has also been used to verify linear analyticalresults by Heisig et al.9

Figure 6. The function qðtÞ for the first five modes (the v direction).

Figure 5. The function pðtÞ for the first five modes (the u direction).





Application of the FEM method to thetransverse vibration of the drillstring

The ABAQUS FEM solver package (SIMULIA Inc.,version 6.7.1) was used, with Euler–Bernoulli beamelements chosen to maintain the same conditions asthe mathematical model. The material specifications,

given in Table 1, are the same as for the analyticalmodel. The beam is modeled by a planar wire shapewith a pipe profile to build the hollow drill collar pipe.Solution follows a three-step process with initial, gen-eral static and perturbation steps. The boundary con-ditions are applied at the initial step to constrain themodel. The general static step is defined by the fixed

Figure 7. The function qðtÞ for the second mode to the fifth mode.

Figure 8. The natural frequency in the u and v directions for the first five modes by the analytical model.





time increments with the direct full-Newtonian equa-tion solver method to apply the static loads (body andhydrostatic forces). The resulting deformations of thisstep are propagated to the next step. In the last step

(perturbation step) the ‘Lanczos’ eigensolver was usedto extract the eigenfrequencies. This method is in con-trast to the ‘subspace iteration method’ and falls intothe class of transformation methods (transformation

Figure 9. The effect of WOB on the natural frequency in the u direction.

WOB: weight on bit.

Figure 10. The effect of torque on the natural frequency in the u direction.





of the normalized eigenvectors through the displace-ment). It is widely used when the higher modes are ofinterest.30 Simply supported boundary conditions areused in the lowest node and all the DOFs except two(rotation along the beam axis and the downwardmotion) are constrained. The ‘cubic element’ is used,in which the shear flexibility is not considered, andthis is in agreement with the Euler–Bernoulli beamtheory assumption in the analytical model. Eightyelements were assumed, giving a total collar lengthof 200m, and using the convergence analysis,24 the

results did not change significantly when more elem-ents were added.

As discussed in the introduction section, this factthat the collar section is more easily excited thanthe pipe section and vibrates in lower modes, whilethe pipes do not vibrate significantly, is verified by theFEM model as shown in Figure 11(a) to (f). Figure11(a) and (b) are the second and the fifth modes of thedrillstring, respectively, with the pipe section remain-ing steady. Figure 11(c), (d) and (e) are the modes 10,25 and 50 respectively, with the vibration propagatingup to the pipe section and the amplitude becominglarger in the pipe section. Figure 11(f) is the axialmode of the drillstring, which is a higher frequencymode, than the transverse vibration and the effectivemass for the axial direction is a large value as asso-ciated with higher modes. The mode shapes magni-tudes derived by the FEM are not absolute values,and are intended only to represent the correct shapedistribution. They are shown exaggerated for clarity.

The values of frequencies for the u and v directionsextracted by the FEM are shown in Figure 12 andcompared with the results derived by the expandedGalerkin method. These values are in agreementwith the results of the last section with slightly lowervalues. This is due to selection of a comparison func-tion in the last section that is not exactly the same asthe real displacement function (eigenfunction).

Determination of appropriatemodel order

Another interesting result of the last section is the effect-ive mass of each mode in any direction. The generalizedmass, m�, associated with the mode � is defined as

m� ¼ �N

�MNM �

M

�ð13Þ

whereMNM is the structural matrix and �N� is the eigen-vector for mode �. M and N are degrees of freedom ofthe FEM model. In the case of eigen vector normaliza-tion, m� is defined as unity. After finding the m�, themodal mass participation factor, ��i, is defined as

��i ¼1

m��

N

�MNM YM

i ð14Þ

Table 1. Dimensions and characteristics of the drillstring and collar section.

Drillstring lenght 1000 m Collar density � ¼ 7860 kg=m3

Drill collar length 200 m Collar modules of elasticity E ¼ 210 GPa

Length between 2 adjacent stabilizers 30 m Gravity acceleration g ¼ 9:81 m=s2

Outside diameter of the collar section do ¼ 0:22 m Rotational torque T ¼ 5 kN�m

Inside diameter of the collar section di ¼ 0:08 m Axial load at the top pint

of the collar section

F0 ¼ �30 kN

Area moment of inertia of

the collars in the u direction

Iu ¼ 1:129� 10�4 m4 Area moment of inertia of the

collars in the v direction

Iv ¼ 1:129� 10�4 m4

Figure 11. The deformed shape of the drillstring in different

modes.





The modal mass participation factor indicatesthe participation factor of a certain motion (globaltranslation or rotation) in the eigenvector of themode � in the i direction. YM

i defines the magnitudeof the rigid body response of the degree of freedom Min the model.31 Effective masses are plotted in Figures13 and 14.

As can be seen, for the u direction, modes up to thefifth mode account almost entirely for the motion in

this direction. This fact validates the truncation ofmodes after the fifth mode in the expanded Galerkinmethod. Another result is that the third mode is thedominant mode in the u direction with higher effectivemass and the first mode is the dominant mode in the vdirection, which is in agreement with the results of theanalytical solutions. The same initial conditions wereused in both methods. The effective mass in the axialdirection happens at higher modes, compared to

Figure 12. Comparison of the natural frequencies in the u and v directions for the FEM and the expanded Galerkin method.

FEM: finite element method.

Figure 13. Effective mass in the ‘u’ direction.





lateral modes, which is in agreement with beamtheory, since the first natural frequency of the beamin the longitudinal direction is much higher than inthe transverse direction. The modal mass participa-tion factor in the v direction is very similar to the udirection. The difference is due to the different cou-pling terms in equation (6). However, the mass par-ticipation factors after the fifth mode decreasesdrastically in both directions. The numerical valueof the modal participation factor is much higher inthe axial direction with respect to the lateral motion.The approach of this section, and the ability to trun-cate unnecessary modes, helps to avoid excessivenumerical computational costs.

Conclusions

Coupled lateral vibration of a drillstring was studiedin two orthogonal transverse directions, under theaction of a steady torque and varying axial load.The axial force arises from the interaction of themud hydrostatic force, the drillstring self-weight andhook weight. As a result, a linear force profile wasassumed along the drillstring. The torque along thedrillstring was resolved into tangential and normalcomponents, with the tangential component actingas the bending moment for the lateral modes. Thecoupled equations were derived using the directNewtonian approach. The expanded Galerkinmethod was used to solve the coupled equations andreveal the natural frequencies as well as the mode par-ticipation factors. Then, the nonlinear FEM wasapplied to the problem with the same conditions toverify the results. The modal mass participation factorwas derived for each direction and the effectivenumber of modes for each direction was selected

according to this criterion. Transverse coupled nat-ural frequencies are more sensitive to changes in theWOB than torque. The rotary speed of the drillstringshould be kept far enough from the natural frequen-cies to avoid excessive deflections and contact with thewellbore, both of which can cause premature failureof bottom-hole assembly components.

Funding

This work was conducted at the Advanced ExplorationDrilling Technology Laboratory at Memorial University inSt. John’s, Canada. Financial support was provided byHusky Energy, Suncor Energy, Newfoundland and

Labrador Research and Development Corporation andAtlantic Canada Opportunities Agency under AIF contractnumber 781-2636-192044.

Acknowledgment

The authors would like to thank Dr. Hajnayeb at ChamranUniversity of Ahvaz for his valuable suggestions on theanalytical code.

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Figure 14. Effective mass in the ‘v’ direction.





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Appendix 1

Notation

A cross-sectional area of the collar sectionEIu, EIv flexural rigidity of the beam in the u and

the v directionsF0 axial compressive force at the top point

of the collar sectiong gravity accelerationm� generalized mass associated with the

mode �M, N degrees of freedom of the FEM modelpiðtÞ mode participation factor for the u

directionqjðtÞ mode participation factor for

.v.directionSu, Sv shear force in the u and the v directions~T torque vector in the z direction~Tt, ~Tn torque components in the tangent and

normal planes of the beam elementuðz, tÞ displacement in first lateral directionvðz, tÞ displacement in second lateral direction�N� eigen vector of the mode �YM

i magnitude of rigid body response of the

degree of freedom M in the i direction��i modal mass participation factor� BHA density�iðzÞ comparison function in the u direction jðzÞ comparison function in the v direction





Appendix 2

The element in the uz plane is shown in Figure 2a andin the vz plane in Figure 2b. The equilibrium equa-tions in the ‘uz’ plane are

XMv ¼ 0)Mv þ

@Mv

@zdz�Mv

þ Su þ@Su

@zdz

� ��dz

2þ Su �

dz

2¼ 0 ð15Þ

Neglecting the second and higher order terms of dzwill result in

@Mv

@z¼ Su ð16Þ

The force equilibrium in the ‘uz’ plane is

Su þ@Su

@zdz� Su þ Faxial þ

@Faxial

@zdz

� �

� sin � þ@�

@zdz

� �� Faxial � sinð�Þ

þ �A@2uðz, tÞ

@t2¼ 0 ð17Þ

Assuming small angle �, and discarding second orhigher order terms of dz, the above equation reducesto

@Su

@zdzþ Faxial

@�

@zdzþ

@Faxial

@z�dz

þ �A@2uðz, tÞ

@t2¼ 0 ð18Þ

Using the result of equation (2) in equation (4),considering that Mv ¼ E:Iv

@2uðz, tÞ@z2

�@

@z

@Mv

@z

� �¼@

@zFaxial �

@uðz, tÞ

@z

� �@

@z

@

@zE:Iv

@2uðz, tÞ

@z2

� ��

þ@

@zFaxial �

@uðz, tÞ

@z

� �¼ ��A

@2uðz, tÞ

@t2ð19Þ

Assuming constant E:Iv for the drillstring in thecollar section and a linearly varying axial load Faxial

E:Iv@4uðz, tÞ

dz4þ@

@zFaxial �

@uðz, tÞ

@z

� �þ �A

@2uðz, tÞ

@t2¼ 0

ð20Þ

The equation of motion in the v direction is obtainedin the similar way to the u direction. Therefore

E:Iu@4vðz, tÞ

dz4þ@

@zFaxial �

@vðz, tÞ

@z

� �þ �A

@2vðz, tÞ

@t2¼ 0

ð21Þ

Ten coupled ordinary time differentialequations in the u and v directions

�4

2l3EIp1ðtÞ �

20

9�Agp2ðtÞ �

136

225�Agp4ðtÞ

þ 0:5�Al €p1ðtÞ þ16�2

3l2Tq2ðtÞ

þ128�2

15l2Tq4ðtÞ �

�2

2lF0p1ðtÞ

þ�2

4�Agp1ðtÞ ¼ 0

�4

2l3EIq1ðtÞ �

20

9�Agq2ðtÞ �

136

225�Agq4ðtÞ

þ 0:5�Al €q1ðtÞ �16�2

3l2Tp2ðtÞ

�128�2

15l2Tp4ðtÞ �

�2

2lF0q1ðtÞ

þ�2

4�Agq1ðtÞ ¼ 0

0:5�LA €p2ðtÞ �20

9�Agp1ðtÞ þ �

2�Agp2ðtÞ

�156

25�gAp3ðtÞ �

580

441�gAp5ðtÞ �

4�2

3l2Tq1ðtÞ

þ108�2

5l2Tq3ðtÞ þ

500�2

21l2Tq5ðtÞ

�2�2

lF0p2ðtÞ þ

8�4

l3EIp2ðtÞ ¼ 0

8�4

l3Iq2ðtÞ þ 0:5�LA €q2ðtÞ �

580

441�gAq5ðtÞ

�156

25�gAq3ðtÞ �

500�2

21l2Tp5ðtÞ �

20

9�Agq1ðtÞ

þ �2�Agq2ðtÞ

�108�2

5l2Tp3ðtÞ þ

4�2

3l2Tp1ðtÞ �

2�2

lF0q2ðtÞ ¼ 0

384�2

7l2Tq4ðtÞ þ

81�4

2l3EIp3ðtÞ �

156

25�gAp2ðtÞ

�48�2

5l2Tq2ðtÞ �

600

49�gAp4ðtÞ þ 0:5�LA €p3ðtÞ

þ9�2

4�gAp3ðtÞ �

9�2

2lF0p3ðtÞ ¼ 0

0:5�LA €q3ðtÞ �156

25�gAq2ðtÞ �

600

49�gAq4ðtÞ

þ48�2

5l2Tp2ðtÞ �

384�2

7l2Tp4ðtÞ þ

9�2

4�gAq3ðtÞ

þ81�4

2l3EIq3ðtÞ �

9�2

2lF0q3ðtÞ ¼ 0





0:5 �LA €p4ðtÞ �136

225�gAp1ðtÞ þ 4�2�gAp4ðtÞ

þ1000�2

9l2Tq5ðtÞ �

216�2

7l2Tq3ðtÞ

�1640

81�gAp5ðtÞ þ

128�4

l3EIp4ðtÞ

�8�2

15l2Tq1ðtÞ �

600

49�gAp3ðtÞ

�8�2

lF0p4ðtÞ ¼ 0

0:5 �LA €q4ðtÞ �136

225�gAq1ðtÞ �

1640

81�gAq5ðtÞ

�8�2

lF0q4ðtÞ þ 4�2 �gAq4ðtÞ �

600

49�gAq3ðtÞ

�1000�2

9l2Tp5ðtÞ þ

8�2

15l2Tp1ðtÞ

þ216�2

7l2Tp3ðtÞ þ

128�4

l3EIq4ðtÞ ¼ 0

�580

441�gAp2ðtÞ �

1640

81�gAp4ðtÞ

�25�2

2lF0p5ðtÞ �

80�2

21l2Tq2ðtÞ þ 0:5 �LA €p5ðtÞ

þ625�4

2l3EIp5ðtÞ �

640�2

9l2Tq4ðtÞ

þ25�2

4�gAp5ðtÞ ¼ 0

�580

441�gAq2ðtÞ �

1640

81�gAq4ðtÞ

þ80�2

21l2Tp2ðtÞ þ

25�2

4�gAq5ðtÞ

þ640�2

9l2Tp4ðtÞ þ 0:5 �LA €q5ðtÞ þ

625�4

2l3EIq5ðtÞ

�25�2

2lF0q5ðtÞ ¼ 0 ð22Þ

Appendix 3

The effects of the hook load, WOB, mud hydrostaticforce and self-weight are presented as a spatially vary-ing axial force along the drillstring. The buoyant forcein the drillstring should not be treated with theArchimedes’ rule and the effective tension point ofview should be implemented for more preciseresults.28 Therefore, at the last point of the collar sec-tion, there are two axial upward forces, namely theWOB and the hydrostatic force at the lower crosssection. The varying axial force in the collar section is

Fcollar ¼ �collar Acollar gz�WOB� Fhydrostatic ð23Þ

Substituting for the hydrostatic force at the bottomof the collar section, the collar force will be

Fcollar ¼ �collar Acollar gz�WOB� �mud glð Þ � Acollar

ð24Þ

According to dimensions and material propertiesgiven in Table 1, the varying axial force along thecollar section is

Fcollar ¼ �30 � 103 � 2543:49 � z ð25Þ

The value of the axial force at the top point of thecollar section (neutral point) is �30 kN, as shown inTable 1 and Figure 15.

Figure 15. Force profile along the collar section.




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