Reconstructed drill-bit motion for sonic drillstring dynamics

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Reconstructed drill-bit motion for sonic drillstring dynamics

Kaïs Ammari, Lotfi Beji

To cite this version:

Kaïs Ammari, Lotfi Beji. Reconstructed drill-bit motion for sonic drillstring dynamics. European

Journal of Control, Elsevier, 2019, �10.1016/j.ejcon.2019.04.003�. �hal-02151646�

Reconstructed drill-bit motion for sonic drillstring dynamics

Kaïs Ammari

^a

, Lotfi Beji

^b,∗

aUR Analysis and Control of PDEs, UR13ES64, Department of Mathematics, Faculty of Sciences of Monastir, University of Monastir, Monastir 5019, Tunisia

bIBISC-EA4526 Laboratory, University of Evry, 40 rue du Pelvoux, Evry 91020, France

IntunnelexcavationthatintegratesResonantSonicHeadDrilling(RSHD)machine,oneoftheimportant stabilityproblemtodealwithisrepresentedbythenecessitytoassignaxialvibrationamplitudeinduced bytheRSHDtoaresonantvibrationmode.Ingeneral,acontrollawdependsonsystemvariableswhich arepartiallyavailableformeasurements.Indrillingoperation,thedrillstringvariablesatthetipboundary aren’teasy,ifnotimpossible,tobemeasured.Thesonicdrillstringinfinitedimensiondynamics,derived inthispaper,willplayanimportantroleinPDE(PartialDerivativeEquation)observerdesignforthedrill bitmotion.Thewell-posednessproblemisaddressed,andthedesignedPDE-observerstabilityisdetailed usingtheresolventmethod.Thewavepeakamplitudeisdeterminedfromtheresonantdrillstringmode shape,andthesystem’sstabilityisanalyzedaroundapracticalfrequencymode.

1. Introduction

Ashard rockdrilling inducesseverfrictionphenomenaaround thedrillstringsystemandalsoatthebit,severvibrationvariations areinherentfortheequipmentandmayaffectRatesOfPenetration (ROP).ROParetheprincipalmonitoringfactorlikeperformancepa- rameterofdrilling.Indrillingenvironment,largermachinesarede- signedtodrillharderformations;thesemachinevibrate,rotateand injecta fluidforcoolingandremovalofwastecuttings. Machines equippedwithheadrotaryare usefulinoilwell drilling.However, for tunnel drilling support wherethe depth is limited, machines that integrate axial vibration tool are considered more efficient.

Thevariationoffrequencybytheoperatorandthedrillbitweight to matchthe materialheisgoingthrough, ensurethebestpene- trationrateandmostaccuratesamplingisobtained.

Sonic drilling has been used in industry for many years [17]. Themajorityoftheresearchhasbeenperformedbyprivateindus- trywheretheyhavekepttheknowhowthatthey havedeveloped internalandproprietary[3].Mostofthemodel,thatdescribesthe sonicdrillstringfound intheliterature, didnot take intoaccount the damping of the soil along the length of the drill and they modelpoorlytheinteractionbetweenthesonicdrillstringbitand thecrown[7].Thequantificationofthesonicdrillsystemvariables integratestheunderestimatedfrictionfromthedrillstringandthe crown. The essence of sonic drilling is high-frequency vibration

∗ Corresponding author.

E-mail addresses: kais.ammari@fsm.rnu.tn (K. Ammari), Lotfi.Beji@ibisc.univ- evry.fr (L. Beji).

drillingwhereenergywavesaretransmittedthoughthedrillstring totheendstringandthenreflectedback.Asonicdrillheadworks bysendinghighfrequencyresonantvibrationsdownthedrillstring tothedrillbit,asshowninFig.1,whiletheoperatorcontrolsthese frequenciestosuitthespecificconditionsofthesoil/rockgeology.

Vibrationsmay also be generated within the drill head. The fre- quency is generally between50 and150 Hz (cycles per second) andcan be varied by the operator. Resonancemagnifies the am- plitudeofthedrillbit,whichfluidizes thesoilparticlesatthebit face,allowing forfastandeasypenetration throughmostgeologi- calformations.Aninternalspringsystemisolatesthesevibrational forcesfromtherestofthedrillrig.

Despitetheperformance oftheaxialvibration controllerspro- posedin[6,7]bymeansofadequatesimulationresults,implemen- tationonrealmachinemaybeimpractical.Actually,thedownhole system’svariables measurements, are unrealizableduringdrilling inundergroundtunnelingunlikeintheoilwellrotarydrillingfield.

Observability is a property of a dynamical system which means that we can determine exactly the state of the system from the observerofitsinput andoutput onasufficientlylong timeinter- val.Notethatanobserverforagivensystemisananotherdynam- icalsystem thatproduces an estimate of thecurrentstate of the givensystembasedonpast observations.Theconstruction ofob- serversisimportantincontrolapplicationsforinfinite-dimensional systems.The locationofmeasurementandactuationplays anim- portantrole in thedesign oftheobserverfor PDEs.Ifthe sensor andtheactuatorareplacedattheoppositeboundaries,wecallthis ananti-collocatedsetup,otherwise,whenthesensorandtheactu- atorare locatedat thesameside, itis calledcollocatedsetup. In

Fig. 1. Drilling machine (RSHD) tip and top boundary conditions.

general,twoapproachesareusedtodesignaconvergentboundary observerforadistributedparametersystems.Thefirstapproachis basedonthebacksteppingtechnique:in[19],backstepping-based observersfor a classof linearparabolic integro-differential equa- tionswaspresented.Anobserverdesignforgeneralperiodicquasi- linearparabolic PDEs in one space dimension is used in [5]. For the case of a linear first-order hyperbolic system, an estimation of boundary parameters has been proposed in [11]. The contin- uum backsteppingapproach wasapplied to a hyperbolic PDEs in [8]to derive a collocated observer. The second approach is Lya- punov.Usingthistheory,boundaryobserverisdesignedforamo- torizedEuler–BernoulliBeamin[13],foratowedseismiccablein [14]andforaflexible-linkmanipulatorin[20].

In thepresentpaperthemain contributionhereistopropose acollocatedinfinitedimensionalobserverforaxialvibrationswith sensingrestrictedtothe boundary.Totheauthor’sknowledge,all previous works considering the vibrations control problem were carriedoutinthe fieldofoilwell drilling,wherethevariables in the bottom hole can be measured as there are sensors fixed on thedrill collarswhich is notthe casein thetunneling field. The estimationproblem ofthe unmeasured parameters relatedto vi- brationshasnotbeenfoundintheliterature.Theonlyresultsdeal withestimatingthepressureandtemperatureconditions.

The paper is organizedasfollows. The second section is con- cerned by the system model, we present a distributed parame- termodelofthedrillstringaxial vibrationsandwestate theob- serverproblem.InSection3,wedefinethecollocatedobserverde- sign.Sections5and6address thewell-posedness andthe stabil- ityproblems,respectively.Finally,simulationresultsareshownin Section7,andaconclusionwillendthepaper.

2. Mathematicalmodel

Toreducethecomplexityofthesystemandthusderiveamath- ematicalmodel,it isnecessaryto makesome initial assumptions andsimplifications of the system when the boundary conditions areselected[16].Itisassumedthatthedrillstringisalongpipe having a uniform cross-sectional area A and assumed to have a negligibleeffectofthetorsionalvibrationsduetothepipelimited length(3–18 m) andthe naturalapplied input forces(axial).The forcesthatexcitethedrillstringare assumedtoactonthetopof thesonicdrill.Inoilandgasfield,arotary tableisdeployed,and wherethepipelengthisveryimportant(km).Hence,torsionalvi- brationcannotbeneglected.Thus,thisphenomenahasintensively

studiedintheliterature.One mayreferto Saldivaretal.[18]and Zhao[21]andreferencestherein.

Because dampingalong the length is very low,the sonicdrill operator has to be very careful not to overstress the drill string atresonancewhenthebottomdrilltip isnotengagedindrilling.

Thedampingatthedrillbitisthemostimportantvariableofthe drillingsystem,asitdefinesthedrillingworkthatistakingplace.

The governing differential equations of motion for the sonic drillisderivedfromforcebalance.Wedenotebyu(x,t)thelongi- tudinaldisplacementofarod’ssectionA,thatisadistancexfrom theverticesattimetasshowninFig.2.

ρ

Adx

∂

²u

(

x,t

)

∂

t² +2bAdx

∂

u

(

x,t

)

∂

t +aAdxu

(

x,t

)

+

σ

A−

σ

A+ d

dx

( σ

A

)

dx

=0

where

ρ

isthe pipedensity,EistheYoungmodulus,a andb are respectivelythecouplinganddampingconstantsalong thelength ofthedrillstring, and

σ

isthestressgivenby

σ

=E^∂u_∂^(x,t_x ⁾.So,we get

ρ

Adx

∂

²u

(

x,t

)

∂

t² +2bAdx

∂

u

(

x,t

)

∂

t +aAdxu

(

x,t

)

−EAdx

∂

²u

(

x,t

)

∂

x² =0 (1)

Dividingby

ρ

Adx,weobtain

∂

²u

(

x,t

)

∂

t² +2b

ρ

∂

u

(

x,t

)

∂

t + a

ρ

^u

⁽

^x,t

⁾

⁻

E

ρ

∂

²u

(

x,t

)

∂

x² =0 (2) Wedefinethespeed ofthesoundthroughthesteeldrillc byc=

E ρ.

Eq.(1)became

∂

²u

(

x,t

)

∂

t² +2b

ρ

∂

u

(

x,t

)

∂

t + a

ρ

^u

⁽

^x,t

⁾

^−c

2

∂

²u

(

x,t

)

∂

x² =0 (3) Itremainstodefinetheboundaryconditionsofthedrillstringdy- namicsgivenabove.

In order to calculate the drill string natural frequencies, the boundary conditions at the ends of the string must be known whenthe frequencyfunctionsare derived. Thesonicdrivermass, the input force from the sonic driver, and the air spring all re- side, wherexis equalto zero.The sonicdrivermass andtheair springarealwaysboundaryconditions.Atthedrilltipofthestring, where x is equal to the drillstring length L, a boundary condi- tion causedby couplingof thesonic drilltip tothe material be- ing drilledthrough exists.Allboundary conditionsarelocated on theendsofthedrillstringandbecauseofthis, alltheconditions havetoequaltheapparentforcesattheendconditions.Theforces fortheendsare foundbytakingthedrillstring’selastic constant E multiplied by the crosssectional area ofthe drill stringA and also multipliedby the partial derivative of the localdeflection u withrespect to the location in space xand setting thisequal to theboundarycondition,asshowninFig.2.

Topboundarycondition:

EA

∂

u

(

0,t

)

∂

x =msh

∂

²u

(

0,t

)

∂

t² +csh

∂

u

(

0,t

)

∂

t −U

(

t

)

+kshu

(

0,t

)

(4) wherem_sh isthe massofthe sonichead,k_shand c_sh are respec- tivelythespringandthedampingratesoftheairspringontopof thesonicdrill.

The input force is typically fixed, as it is dependent on the sizeoftheeccentricsandonthesquareoftheangularfrequency.

U(t)=mecrec(2

π ξ

)²sin(2

π ξ

t) where mec, rec and

ξ

are respec- tivelythemassofthetwoeccentrics,eccentricityoftheeccentrics, andthetemporalfrequencyenHz.

Fig. 2. Sonic drill model [10] .

Tipboundarycondition:

EA

∂

u

(

L,t

)

∂

x =−m_bit

∂

²u

(

L,t

)

∂

t² −c_bit

∂

u

(

L,t

)

∂

t −k_bitu

(

L,t

)

(5) where m_bit is themass of thesonic drillbit, k_bit andc_bit arere- spectivelythe springandthe dampingratesofthedrillbitwhile drilling.

Despitetheperformance ofthe axialvibrationcontrollers pro- posedin[6,7]bymeansofadequatesimulationresults,implemen- tationonrealmachinemaybeimpractical.Actually,thedownhole system’s variables measurements, are unrealizable duringdrilling inundergroundtunnelingunlikeintheoilwellrotarydrillingfield.

Theproblemistodesignanobserverforthesystemwithonly boundary measurements in the top of the drillstring (i.e. x=0) available, to estimate the drill bit parameters not accessible by measurements.

3. Observerdesign

We considertheaxialvibrations model(3)–(5)representedby adampedwaveequationandactuatedinthetopextremityofthe drillstring(x=0)bytheboundarycontrolinputU(t).

Theinvestigationoftheobservererrordynamics’s(9)–(11)sta- bility isreally defiant,crucialand innovative inthe underground tunnelingfield.

Inthefollowingdistributedparameterobserverdesign,we as- sume the availability of u(0, t) for measurement, as described above.Considerthefollowingdistributedparameterobserver.

∂

²uˆ

(

x,t

)

∂

t² +2b

ρ

∂

uˆ

(

x,t

)

∂

t + a

ρ

^uˆ

⁽

^x,t

⁾

^−c

2

∂

²uˆ

(

x,t

)

∂

x² =0 (6)

EA

∂

uˆ

(

0,t

)

∂

x =msh

∂

²uˆ

(

0,t

)

∂

t² +csh

∂

uˆ

(

0,t

)

∂

t

−U

(

t

)

+kshuˆ

(

0,t

)

−G

(

uˆ

(

0,t

)

−u

(

0,t

) )

(7)

EA

∂

uˆ

(

L,t

)

∂

x =−m_bit

∂

²uˆ

(

L,t

)

∂

t² −c_bit

∂

uˆ

(

L,t

)

∂

t −k_bituˆ

(

L,t

)

(8) whereuˆistheobservedvalueofuandGistheobservergain pa- rameter.

Letu¯=uˆ−utheerrordynamicandsubtracting(6)–(8)by(3)– (5)givestheobservererrormodel:

∂

²u¯

(

x,t

)

∂

t² +2b

ρ

∂

u¯

(

x,t

)

∂

t +a

ρ

^u¯

⁽

^x,t

⁾

^−c

2

∂

²u¯

(

x,t

)

∂

x² =0 (9) withtheboundaryconditions

EA

∂

u¯

(

0,t

)

∂

x =m_sh

∂

²u¯

(

0,t

)

∂

t² +c_sh

∂

u¯

(

0,t

)

∂

t +G˜u¯

(

0,t

)

(10)

EA

∂

u¯

(

L,t

)

∂

x =−m_bit

∂

²u¯

(

L,t

)

∂

t² −c_bit

∂

u¯

(

L,t

)

∂

t −k_bitu¯

(

L,t

)

(11) whereG˜=k_sh−G>0.

4. Well-posedness

In this section, we will prove the global existence and the uniqueness of the solution of problem (3)–(5). For this purpose wewilluseasemigroupformulationoftheinitial-boundaryvalue problem(6)–(8).

Ifwedenote V:=(u,u_t,u(0),u_t(0),u(L),u_t(L))^T,we definethe energyspace:

H=

{ (

u,

v

_,w1,w2,z1,z2

₎

_∈H¹

₍

_0,L

₎

_×L²

₍

_0,L

₎

×R⁴,w1=u

(

0

)

,z1=u

(

L

) }

.

Clearly,HisaHilbertspacewithrespecttotheinnerproduct

V1,V2

H= a

ρ

L 0

u¹u²dx+c² L

0

u¹_xu²_xdx+ L

0

v

¹

v

²_dx

+c² G˜

EAw¹₁w²₁+c²msh

EA w¹₂w²₂ +c²kbit

EA z₁¹z²₁+c²mbit

EA z¹₂z²₂ (12) forV₁=(u¹,

v

¹_,w¹

1,w¹₂,z¹₁,z¹₂)^T,V₂=(u²,

v

²_,w²

1,w²₂,z²₁,z²₂)^T. Therefore,ifU∈L²_comp(0,+∞),¹ theproblem(3)–(5)isformally equivalent to the following abstract evolution equation in the HilbertspaceH:

_V′

(

t

)

=AV

(

t

)

+BU

(

t

)

,t>0, V

(

0

)

=V⁰:=

(

u⁰,

v

⁰_,w⁰

1,w⁰₂,z⁰₁,z⁰₂

)

^T, (13)

where^′denotesthederivativewithrespecttotimet,B:=

⎛

⎜

⎜

⎜

⎝

0 0 0

1 msh

0 0

⎞

⎟

⎟

⎟

⎠

andtheoperatorAisdefinedby:

A

⎛

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎝

u

v

w1

w2

z1

z2

⎞

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎠

=

⎛

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎝

v

c²uxx−2b

ρ v

₋^a

ρ

^u

w2

EA msh

ux

(

0

)

− csh

msh

w2− G˜ msh

w1

z2

−EA

m_bitux

(

L

)

− c_bit

m_bitz2− k_bit m_bitz1

⎞

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎠

.

ThedomainofAisgivenby

D

(

^A

)

=

{ (

u,

v

_,w1,w2,z1,z2

₎

^T_∈H;u∈H²

(

0,L

)

,

v

_∈H¹

₍

_0,L

₎

_,

w2=

v ₍

₀

₎

_,z2=

v ₍

_L

₎ }

. Wehavethefollowing.

Theorem4.1. TheoperatorAgenerates aC₀ semigroupofcontrac- tions(e^tA)_t≥0onH.

Proof.AccordingtotheLumer–Phillips’theorem,weshouldprove thattheoperatorAism-dissipative.

LetV=(u,

v

_{,w1,w2,z1,z2)}^T_∈D(^A).Bydefinitionofthe oper- atorAandthescalarproductofH,wehave:

AV,V

H

= a

ρ

L 0

v ₍

_x

₎

_u

₍

_x

₎

_dx+c²

L 0

v

_x

₍

_x

₎

_ux

₍

_x

₎

_dx

+ L

0

c²uxx

(

x

)

−2b

ρ v ₍

_x

₎

₋ ^a

ρ

v ₍

_x

₎

_dx+c²

G˜ EAw2w1

+c²m_sh EA

EA msh

ux

(

0

)

− c_sh msh

w2− G˜ msh

w1

w2+c²k_bit EA z1z2

+c²m_bit EA

−EA

m_bitux

(

L

)

− c_bit

m_bitz2− k_bit m_bitz1

z2.

1L ²_comp(0 , + ∞ )=

f ∈ L ²(0 , + ∞ );the support of f is compact .

ByGreen’sformulaweobtain:

^AV,V

H=−2b

ρ

L 0

v

²

₍

_x

₎

_dx−c^2csh

EAw²₂−c²kbit

EAz²₂≤0. (14) ThustheoperatorAisdissipative.

Nowwewanttoshowthatfor

λ

>0,

λ

I−Aissurjective.

For F=(f₁,f₂,f₃,f₄,f₅,f₆)^T∈H, let V= (u,

v

_{,w1,w2,z1,z2)}^T_∈D(^A)solutionof

( λ

I−A

)

V=F, whichis:

λ

u−

v

_{= f1, (15)}

λ v

_−c²_uxx+^2b

ρ v

₊^a

ρ

^{u=f2, (16)}

λ

w1−w2=f3, (17)

λ

w2− EA

m_shux

(

0

)

+ csh

m_shw2+ G˜

m_shw1=f4 (18)

λ

z1−z2= f5 (19)

λ

z2+ EA

m_bitux

(

L

)

+ c_bit

m_bitz2+ k_bit

m_bitz1= f6. (20) TofindV=(u,

v

_{,w1,w2,z1,z2)}^T_∈D(^A)solutionofthesystem (15)–(20),wesupposeuisdeterminedwiththeappropriateregu- larity.Thenfrom(15),(17)and(15),wegetrespectively:

v

₌

λ

u−f1,w2=

λ

u

(

0

)

−f3,z2=

λ

u

(

L

)

−f5. (21) Consequently,knowingu,wemaydeduce

v

_{,w1=u(0),w2,z1=}

u(L),z2 by(21).

Werecall thatsinceV=(u,

v

_{,w1,w2,z1,z2)}^T_∈D(^A) weauto- maticallygetw₂=

v

_(0)andz2=

v

_(L).

FromEqs.(16),(18),(20),and(21),umustsatisfy:

λ

²u−c²uxx+2b

ρ λ

u+ a

ρ

^u=f2+

2b

ρ

^f1+

λ

f1, in

(

0,L

)

(22) withtheboundaryconditions

λ

²u

(

0

)

− EA msh

ux

(

0

)

+

λ

^csh

msh

u

(

0

)

+ G˜ msh

u

(

0

)

= f4+

λ

f3+ c_sh msh

f3, (23)

λ

²u

(

L

)

+ EA

m_bitux

(

L

)

+

λ

^cbit

m_bitu

(

L

)

+ k_bit

m_bitu

(

L

)

= f6+

λ

f5+ c_bit m_bitf5. The variational formulation of problem (22), (23) is to find (u,w₁,z₁)∈H:=

{

(u,w₁,z₁);

ω

∈H¹(0,L),w₁=u(0),z₁=u(L)

}

suchthat:

L 0

λ

²+2b

ρ λ

+a

ρ

u

ω

+ux

ω

_x

dx

+c²m_sh EA

λ

²+

λ

^csh

m_sh+ G˜ m_sh

u

(

0

)

w

(

0

)

+c²m_bit EA

λ

²+

λ

^cbit

mbit

+ k_bit mbit

u

(

L

)

w

(

L

)

= L

0

f2+

λ

+2b

ρ

f1

ω

dx +c²w

(

0

)

f4+

λ

f3+ c_sh m_shf3

+c²w

(

L

)

f6+

λ

+ c_bit mbit

f5

, (24)

forany(

ω

,

ξ

1,

ξ

2)∈H. Since

λ

>0,the lefthand sideof(24)de- fines a coercive bilinear form on H. Thus by applying the Lax–

Milgramtheorem,thereexistsaunique (u,w1,z1)∈Hsolutionof (24).Now,choosing

ω

∈C^∞_c ,(u,w₁,z₁)isasolutionof(22)inthe senseofdistributionandthereforeu∈H²(0,L).ThususingGreen’s formulaandexploitingEq.(22)on(0,L),weobtainfinally:

c²m_sh EA

λ

²+

λ

^csh

m_sh+ G˜ m_sh

w

(

0

)

u

(

0

)

+c²m_bit EA

λ

²+

λ

^cbit

m_bit + k_bit m_bit

w

(

L

)

u

(

L

)

=c²m_sh EA

f4+

λ

+ c_sh msh

f3

w

(

0

)

+c²m_bit EA

f6+

λ

+ c_bit

m_bit

f5

w

(

L

)

.

Sou∈H²(0,L)verifies(18),(20)andwe recoveru,w₁=u(0),z₁= u(L)and

v

_∈H¹_{(0,L) andthusby (21),weobtain w2=}

v

_(0),z2=

v

_{(L), we have found V=(u,}

v

_{,w1,w2,z1,z2)}^T_∈D(^A) solution of (I−A)V=F.

ThiscompletestheproofofTheorem4.1. Wehave,inparticular,that theproblem(6)–(8),whichcanbe rewrittenasfollows:

Z^′

(

t

)

=AZ

(

t

)

,t>0, Z

(

0

)

=Z⁰=

(

u⁰,

v

⁰_,w⁰

1,w⁰₂,z⁰₁,z⁰₂

)

^{T (25)}

admits for all Z⁰∈H a unique solution Z(t)=e^tAZ⁰∈C(^R+;H). Moreover, forZ⁰∈D(^A),the system(25)admitsan unique solu- tion

Z

(

t

)

=

(

u

(

t

)

,ut

(

t

)

,u

(

0

)

,ut

(

0

)

,u

(

L

)

,ut

(

L

))

∈C

(

^R+;D

(

^A

) )

andsatisifesthefollowingenergyidentity:

E

(

t

)

−E

(

0

)

=−2b

ρ

L 0

u²_t

(

x

)

dx−c²c_sh EAu_t²

(

0,t

)

−c²kbit

EAu²_t

(

L,t

)

,

∀

t≥0, (26) where

E

(

t

)

:=1

2

Z

(

t

)

²H,

∀

t≥0. (27)

Thusthewell-posednessofproblem(3)–(5)isensuredby:

Proposition 4.2. LetU∈L²_comp(0,+∞),V⁰∈H, then there exists a uniquemildsolutionV(t)=e^tAV⁰+t

0e^(t−s)ABU(s)ds∈C(^R+;H)of problem(13).

5. Stability

Recallthefollowingfrequencydomaintheoremforexponential stabilityfrom[4,15]ofaC₀-semigroupofcontractionsonaHilbert space:

Theorem5.1 [4,15].LetAbethegeneratorofaC₀-semigroupofcon- tractions S(t) ona Hilbert space X.Then, e^tAis exponentially stable, i.e.,forallt>0,

||

e^tA

||

L(X)≤Ce^−δt,

forsomepositiveconstantsCand

δ

ifandonlyif

ρ (

A

)

⊃

i

γ γ

∈R

≡iR, (28)

and limsup

|γ|→+∞

(

i

γ

I−A

)

⁻¹

L(X)<∞, (29)

where

ρ

(A)denotestheresolventsetoftheoperatorA.

Weare nowina positiontostate thefirst mainresultofthis section:

Theorem5.2. ThereexistC,

δ

>0suchthat

e^tA

L(^H)≤Ce^−δt,

∀

t>0.

Ourfirst concern is to show that i

γ

is not on the spectra of Aforanyrealnumber

γ

,whichclearlyimplies(28).Wehavethe following:

Lemma5.3. ThespectrumofAcontainsno pointon theimaginary axis.

Proof. SincetheresolventofAiscompact,itsspectrum

σ

(^A)only consistsofeigenvaluesofA.Wewillshowthattheequation

AV=i

β

V (30)

withV =(u,

v

_{,w1,w2,z1,z2)}^T_∈D(A)and

β

∈Rhasonlythetriv- ialsolution(i.e.V=(0,0,0,0,0,0)^T).

Bytakingtheinnerproductof(30)withVandusing

ℜ

^AV,V

H=−2b

ρ

L 0

v

²

₍

_x

₎

_dx−c^2csh

EAw²₂−c²k_bit

EAz²₂, (31) since

^AV,V

H=ℜ(i

β

V

²H)=0,thenweobtainthat

v

_=0,w2=

0,z2=0.

Next,wegetthefollowingordinarydifferentialequation:

⎧

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎨

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎩

i

β

u=0,

(

0,L

)

,

−c²uxx+^a_ρu=0,

(

0,L

)

, i

β

u

(

0

)

=0,

−_m^EA

shux

(

0

)

+_m^G˜

shw1=0,

i

β

u

(

L

)

=0,

EA

mbitux

(

L

)

+_m^kbit

bitz1=0.

(32)

• If

β

=0then

0= L

0

−c²uxx+ a

ρ

^u

dx

=c² L

0

|

ux

(

x

) |

²dx+a

ρ

L 0

|

u

(

x

) |

²dx+c² G˜ m_sh

|

u

(

0

) |

²

+c² k_bit m_bit

|

u

(

L

) |

²

⇓

u=0,w1=u

(

0

)

=0,z1=u

(

L

)

=0. WhichimpliesthatV≡0.

• If

β

=0thenu=0,w1=u(0)=0andz1=u(L)=0.So V≡(0, 0,0,0,0,0)^T.

Wededucethatthesystem(32)hasonlythetrivialsolution.

Proofof Theorem5.2. We suppose that condition(29)doesnot hold. This gives rise, thanks to Banach–Steinhaus Theorem² (see [2,Theorem2.2]),to theexistenceofa sequenceofrealnumbers

γ

n→∞ and a sequence of vectors Vn=(un,

v

_{n,wn,pn,zn,qn)}^T_∈

D(^A)with

Vn

H=1 such that

(

i

γ

nI−A

)

Vn

H→0 as n→∞, (33) i.e.,

i

γ

nun−

v

_{n≡fn→0 in H}¹

₍

_0,L

₎

_{, (34)}

2Let E and F be two Banach spaces and ( T i) i∈Ibe a family (not necessarily count- able) of continuous operators from E into F . Assume that sup i∈IT ix _F<∞∀x ∈E.

Then sup _i∈IT iL(E,F)< ∞ .

Fig. 3. Practical and estimated axial displacements at the tip boundary ( x = L ), G = 1 .62 ∗10 ⁶.

Fig. 4. Practical and estimated axial displacements at the tip boundary ( x = L ), G = 1 .5 ∗10 ⁶.

i

γ

n

v

_n−c²

₍

_un

₎

_xx+^2b

ρ v

_n+^a

ρ

^{un≡gn→0 in L}

2

(

0,L

)

, (35)

i

γ

_nwn−pn=an→0 in C, (36)

i

γ

_npn− EA msh

(

un

)

x

(

0

)

+ c_sh msh

pn+ G˜ msh

wn≡bn→0 in C, (37) i

γ

nzn−qn=rn→0 in C, (38)

i

γ

nqn+ EA

m_bit

(

un

)

x

(

L

)

+ c_bit

m_bitqn+ k_bit

m_bitzn≡sn→0 in C. (39) The ultimateoutcomewillbeconvergenceof

Vn

Htozeroas n→∞,whichcontradictsthefactthat

∀

n∈N,

Vn

H=1.

Firstly,since

(

i

γ

nI−A

)

Vn

H≥

|

ℜ

( (

i

γ

nI−A

)

Vn,Vn

H

) |

=−ℜ

^AVn,Vn

H

=2b

ρ

L 0

| v

_n

₍

_x

₎ |

²dx+c²c_sh

EA

|

pn

|

²+c²k_bit EA

|

qn

|

², itfollowsfrom(33)that

v

_{n→0,→0inL}²

₍

_0,L

₎

_{and pn→0,qn→0in}C. (40)

Fig. 5. Drill-bit axial displacements at the boundary x = L under G = 1 . 7 ∗10 ⁶.

Fig. 6. Drill-bit axial velocities at the boundary x =L under G =1 .7 ∗10 ⁶.

Fig. 7. Practical and estimated axial positions at the tip boundary ( x =L ), G =1 .5 ∗ 10 ⁶.

Fig. 8. Practical and estimated axial velocities at the tip boundary ( x =L ), G =1 .5 ∗ 10 ⁶.

Therewith

wn→0,zn→0inC. (41)

Now,letustakethe innerproductof(35)withun.Astraight- forwardcomputationgives

c² L

0

| (

un

)

x

|

²dx+ a

ρ

L 0

|

un

|

²dx

=− L

0

i

γ

nun

v

_ndx+

L 0

gnundx−2b

ρ

L 0

v

_nundx

−c² G˜

m_sh

|

wn

|

²−c²k_bit m_bit

|

pn

|

²−

i

γ

nwnpn+ c_sh m_shpnwn

−

i

γ

nznpn+ cbit

mbit

qnpn

→0. (42)

Inthelightof(40),(41)and(42),weconcludethat

Vn

H→0 whichwasourobjective.

Lastly,thesufficientconditionsofTheorem5.1arefulfilledand theproofofTheorem5.2iscompleted.

Table 1

Numerical values used for simulations [12] .

L 3 m ρ 7850 Kg/m ³

E 2 . 1 ∗10 ¹¹Pa A 8 . 6 ∗10 ⁻³m ²

m sh 453.6 Kg m bit 8 Kg

k sh 84040034.023 N/m c sh 10 N s/m c ²=E/ρ 2 .6752 ∗10 ⁷m ²/s ² m ec 28.4 Kg

r ec 0.06 m k bit 1194.519 N/m

c bit 1 N s/m b 0 N s/m ⁴

a 2334.434 N/m ⁴ ξ 68 Hz

6. Simulationresults

UsingtheobtainedanalyticalformofthesonicdrillPDEmodel and integrating the boundaries, at first, we have computing the differentmodelresonant frequencies. Forthis, we are referred to the work of Lieu and Jovanovic [9] and the MatlabChebfun’tool inorder to solvethe analytical formin a frequencydomain. The detailedfrequency analysiswas addressedin [12] wherethe ob- tained

ξ

=68Hzmatchesthevaluerequestedinpracticeintunnel reinforcingdomain [7].Recall here, inorderto achievethe simu- lation procedure,we used the Fouriertransform to compute the frequencyoperatorofthesystem. Thispermitstoquantifying the system’sperformance in the presence ofa stimulus, and it char- acterizesthesteady-stateresponseofastablesystemtopersistent harmonicforcing.Consequently,themodelintegratingtheobserver istransformedin(x,

ξ

)domain(Fouriertransformw.r.t.timetand temporal frequency

ξ

). In a second step,we study the temporal response using the intbvp andtrapz Matlab functions. Numerical resultsare performedwiththe initialmodel (3)–(5), andtheob- serverscheme(6)–(8).Forthis, thephysicalparameters aregiven inTable1.

TheresultsaresketchedinFigs.3–9fromwhichtheproposed observerschemeisshownstable/practicallystableintermsofthe drill-bitdisplacementsandvelocities.Forexample,showsthesta- bilitybehavioroftheobservedandtherealwavesignals.Thesys- tem’stotal energyis represented in a frequency domain forG= 1.62∗10⁶whereweproveanimportantquantityofenergyatthe resonance

ω

=68Hz(Fig.9(left)).Fig.9(right)showsalessquan- tityofenergyundertheobservergainparameterG=0.5∗10⁶.Un- derthislast value, thesystemlossesthe resonance(figureisnot givenhere).Hence,we mayconclude, theadequate choice ofthe

Fig. 9. Total energy, the maximum is emphasized at the resonance ( ω= 68 Hz), G = 1 . 62 ∗10 ⁶(left) and G = 0 . 5 ∗10 ⁶(right).

gainparameterwillguaranteethattheobserverworksatthedrill- stringresonancemode. Otherparameters relatedto nonhomoge- neous underground layers can be studied under a control based observer.

7. Conclusion

From the idea ofbeing not able to measure the drill bitam- plitude at the bottom, defined here as the top boundary, a PDE observerwasperformedfromthedrillstringdynamics.Theoretical computationswereconductedtowardthewell-posednessproblem andthe stabilitysufficient conditionsof the observer.In general, sonicdrilloperatorswillmonitorhydraulicpressureofmotorsthat driveatresonantrotationofeccentricsandproducethemaximum axialvibrations.Notethattheoperatorhastoadjustthedrillstring behavior and fluid injection till a resonant value. Consequently, to implement axial vibration control and reach an autonomous drillingoperation,andtherebytheimprovementofsafetylevelon construction sitesand also reducing the drilling time, a PDE ob- serverisdeveloped forthetip boundary. Futureresultsrequirea newcontrolinputelaborationinagreementwiththedesignedob- server. Possibility of integrate and evaluate the effect of moving boundaryonthesystemstability,asforexampleduetothepres- enceofthecuttingsinthecrown(Thewaveequationcaseisstud- iedin[1]).

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