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CalculationofresponsesofanonlinearfractionalderivativemodelofimpulsemotionforviscoelasticmaterialsusingRunge-Kutta-Nystrmmethod

Nguyen Van Khang1, Nobuyuki Shimizu2, Mataka fukunaga3, Duong Van Lac1, Bui Thi Thuy4

1Hanoi University of Science and Technology, Vietnam 2Iwaki Meisei University, Japan, 3Nihon University, Japan

4Hanoi University of Mining and Geology, VietnamEmail: khang.nguyenvan2@hust.edu.vn

Abstract. Generally,aforcecanbedescribedasafunctionofdisplacement in themechanicalmodel.FromFukunaga,ShimizuandNasuno [16], a nonlinear fractionalderivativemodelwith respect todisplacement isproposed todescribethe force for aviscoelasticmaterialbasedonthemeasureddataof impulsivemotion.UsingtheRunge-Kutta-Nystrmmethod,thispaperpresents a numerical method to calculate nonlinear responses of fractional derivative model of impulse motion forviscoelasticmaterials.

Key work: fractional derivative, impulse motion, viscoelastic material, Runge-Kutta-Nystrm

1. Introduction

The concepts of fractional derivatives appeared many years ago and were introduced by famousmathematicianslikeRiemann,Liouville,Grunewald,Letnikov,Caputo[1-3].Theconceptoffractionaloperatorsin engineering applications is now increasingly attractive in the formulation of constitutive laws for someviscoelasticmaterials.

Itiswellknownthatmanyviscoelasticmaterialsshowthefractionalpower-lawdependenceonfrequencyoverthewiderange.Suchresponsescanbetransformedintotherelationbetweenthestress andthestrain ,whichincludesthefractionalderivative[4-8]:

0 ( )pD t (1)

Where pD isthefractionaldifferentialoperatordefinedby,for 0 1p ,[1-3]:

1

( ) ( ) ( )(1 )

tp p

a

dD x t t x d

p dt

(2)

Where ( )s isthegammafunction.ThisdefinitionisknownastheRiemann-Liouvillefractionaldifferentiation.

When the deformation of viscoelastic material is large, the material shows nonlinear responses to theappliedforce.Anumberofmodelsareproposedinordertoexplainthenonlinearresponses.Onepossiblemodelis that the right hand-side of Eq. (1) is added by a nonlinear elastic term [7, 8]. The models with multiplefractional derivative termsare proposed by Lee and Rodgers [9] and Rossikin and Shitikova [10]. The time-dependentorderoffractionalderivativesisemployedbyIngmanetal[11,12].However,acomplextreatmentisnecessaryinthesemodelsinordertofitexperimentaldata.

From the experimental analysis by Fukunaga, Shimizu and Nasuno [13-17], a nonlinear fractionalderivativemodelisproposedtodescribetherelationbetweentheappliedforcef andthedisplacementxduringthecompressionasfollows:

0 pf t c x D xb x (3)Where 0 isaconstant,c(x)andb(x)arethenonlinearfunctions.

2. A new numerical algorithm for fractional dynamic system

2.1. Equation of a nonlinear fractional derivative model of impulse motion for viscoelastic materials We consider the motion of a rigid body of mass m0 that collides at time 0t with the viscoelastic

material fixed on a wall or a plate (figure 1). The material is assumed to obey Eq. (3). If the two bodies

516 Nguyen Van Khang, Nobuyuki Shimizu, Mataka fukunaga, Duong Van Lac, Bui Thi Thuy

identicallymove togetherafterthecollision, therelevantnonlinear fractionaldifferentialequation (NFE)afterthecollision 0t isgivenby:

0 , 0, , (0 1).pmx t c x t D x t b x t kx t f t t T p (4)

Figure 1.Thehead-oncollisionofthetwobodies.

Where x t is the displacement of the contact surface between the rigidbody and the viscoelastic material,

D d dt isthedifferentialoperator,m1themassoftheviscoelasticmaterialand f t theappliedforceand

0 1m m m .TheinitialconditionsforEq.(4)aregivenby:

00 0 00 1

0 , (0) 2m

x x x x ghm m

v (5)

Considernowthemotiondifferentialequationinvolvingfractionalderivativeoforderp

0 , 0, , (0 1)pmx t c x t D x t b x t kx t f t t T p (6)Withtheinitialconditions 0 0 0(0) , (0)x x x x v (7)

2.2. Numerical scheme of Runge-Kutta-Nystrm method

Denote: z t x t b x t (8)Wehavethefirstorderandthesecondorderderivativeof z t

xz t x t b x t x t b x t (9)

2 2x x xz t x t b x t x t b x t x t b x t x t b x t (10)Where x t Dx t representsthevelocityand x t representstheacceleration.TheLiouvilleRiemannsfractionalderivativeisdefinedas[1-3]

10

1,

tp u

u

zdD z t D D z t d

u dt t

(11)

where 1 , 0 1u p u .

InordertomakeuseofLiouvilleRiemannsformulatodeduceournumericalschemeandtopresentthe

problemsmentionedabove,weapplythecompositionruleto pD z t [1-3],thatis

1

0

0 0 1,

1 1

tp u u u p

p

z z zD z t D D z t t D z t t d

u p p t

(12)

WeintegratebypartsthesecondtermofEq.(12)

Beforecollision Aftercollision

k , (), (),

m1

m0

h

k , (), (),

m1

m0

Calculation of responses of a nonlinear fractional derivative model of impulse motion 517 for viscoelastic materials using Runge-Kutta-Nystrm method

11

0 0

10 .

1

t tpp

p

zd z t z t d

pt

(13)

Afterintegratingbyparts,theintegration

0

t

p

zd

t

isderivedtoadefiniteintegral

1

0 0

( )t t

p

tI t z t d y d

(14)

Bysubstitutingequations(13)and(14)intoEq.(12),weobtain

101

01 1

p

p p z t I tD z t z tp p

(15)

Then,bysubstitutingEq.(15)intoEq.(4)weobtainthefollowingequation

1

0

01 10

1 1

p

p z t I tx t f t kx t c x t z tm p p

(16)

Weapproximatetheintegral 1

0 0

( )i i

i

t tp

i i tI t z t d y d

foreveryinstance it bytrapezoid

numericalintegrationasfollows:

0 1 1 0 00 ... , , , , 0in i i i j t it t t T h t t t t ih t jh y t (17)

Figure 2.Approximatingtheintegralbytrapezoidnumericalintegration

1 2

1 1 1

0 0

, 12 2 2i i i i i

i i

i t j t j t j t j t i

j j

h h hI t y y y y y t i

(18)

1

/2 /2 1 /2

0

, 02 2 4i i i

i

i t h j t h j t h i

j

h hhI t y y y i

(19)

Thus,integral I t isindependentof iz t attime it .Hence,Eq.(16)becomes

,i i ix g t x (20)

Where and i i i ix x t x x t withsubscriptidenotethedisplacementandaccelerationattime it respectively,

it

y

00 1 2 1... ...j i i i

it j

y

1( )2 i

t i

hy t

iI t

2i hI t

2i

hty

00 1 1... ... 2j i i i

h

2i

h jty

2( )

4 ih it

hy t

518 Nguyen Van Khang, Nobuyuki Shimizu, Mataka fukunaga, Duong Van Lac, Bui Thi Thuy

1

0

01 1, 0

1 1

pi ip

i i i i i i

z t I tg t x f t k x c x z t

m p p

(21)

WehaveRKNnumericalalgorithmforEq.(20)as[18]:

1 1 2 3

1 1 2 3 4

1 1

3

12 2

3

1

i i i

i i

i i i

hx x hx k k k

x x k k k k

x g t h x i n

, ,

(22)

Where

1

2 1

3 2

4 3

, ;2

, ;2 2 2 4

;

, .2

i i

i i i

i i i

hk g t x

h h h hk g t x x k

k k

hk g t h x hx hk

(23)

Wefinallyobtainnumericalsolution ix ofthedifferentialequationofmotionaccordingtoEq.(6).Wesuppose

thatformulas 0 , 0x x and 0x aregiven.

3. Numerical example

Wehave triedout thealgorithmforr someexamples.For the firstexample,thedifferential equationofmotionhasthefollowingform:

4 0,52 1 3 1x t x t D x t x t (24)Wehavechosen:

401, 1, 1, 2, ( ) 1 3 , 0.5, 1f b x m c x x p k ,

Andtheinitialconditions:

0 0, 0 3.x x Fig.3 shows theresultsbyusingRunge-Kutta-Nystrmmethod.Thesymbol attheordinate isdefinedby

2D x t .Inthesecondexample,thedifferentialequationofmotionofthesysteminthiscasetakesthefollowingform

2 0,390,17 616 1043 0.1 342 ( )

x tD x t D x t

x t

(25)

Thenumericalcomputationsarecarriedoutusingthefollowingparameters

1

1

0 1 0 1

1( ) 1, ( ) , ( ) 0, 1.71, 0.005

1

0.17, 0.39, 1043, 616, 342

c x b x f t c h hc x

m m m p k c

Theinitialconditionsarechosenasbelow:

0 0, 0 0.64x x .SomecalculatedresultsofEq.(25)areshowninFig.4.Thecalculatedresultsareingoodagreementwiththeexperimentalonesin[16,17].

Calculation of responses of a nonlinear fractional derivative model of impulse motion 519 for viscoelastic materials using Runge-Kutta-Nystrm method

Figure 3.TheimpulseresponseofEq.(24)

Figure 4.TheimpulseresponseofEq.(25).

520 Nguyen Van Khang, Nobuyuki Shimizu, Mataka fukunaga, Duong Van Lac, Bui Thi Thuy

4. Conclusion

BasedontheideaoftheRunge-Kutta-Nystrmmethod,theapproximationformulasfordynamicsystemsaredevelopedinthispaper.Then,usingtheLiouville-RiemansdefinitionoffractionalderivativesandtheRKNalgorithm,wecalculatednonlinearvibrationsof fractionalderivativemodelofimpulsemotionforviscoelasticmaterials.Accordingtothisalgorithm,acomputerprogramisdevelopedusingMATLABsoftware.

We also compared the results calculated by the Newmark method [13] and Runge-Kutta-Nystrmmethod. The computation time with Runge-Kutta-Nystrm method is greatly reduced in comparison withNewmarkmethod.

TheRKNalgorithmpresentedhereforfractionalordersystemsiseffectiveandsuccessful.

5. Acknowledgment

ThispaperwascompletedwiththefinancialsupportbytheVietnamNationalFoundationforScienceandTechnologyDevelopment(NAFOSTED).

6. Reference

[1] K.B.Oldham,J.Spanier,The Fractional Calculus,AcademicPress,Boston,NewYork(1974).

[2] I.Podluny,Fractional Differential Equations,AcademicPress,Boston,NewYork(1999).

[3] K.Miller,B.Ross,An Introduction to the Fractional Calculus and Fractional Differential Equations,JohnWileg&Sons,NewYork1993.

[4] W.Smit,H.deVries,Rheologicalmodelscontainingfractionalderivatives,Rheological Acta,9,525-534,(1970).

[5] R.L.Bagley,P.J.Torvik,Atheoreticalbasisfortheapplicationoffractionalcalculustoviscoelasticity,Journal of Rheology,27(3),201-210,(1983).

[6] R.L. Bagley, P.J. Torvik, Fractional calculus in the transient analysis of viscoelastically damped structures, AIAAJournal,23(6),918-925,(1985).

[7] S.G. Bardenhagen,M.G. Stout, G.T. Gray, Three-dimensional finite deformation viscoelastic constitutive model forpolymericmaterials,Mech. Materials,25,235-253,(1997).

[8] R.Deng,P.Davis,A.K.Bajaj,Applicationoffractionalmodelingthequasi-staticresponseofpolynomialform,ASME 2003 IDETC,Chicago,III,Sept.2-6,2003,pagesDETC2003/VIB-48397,DistributedonCDROM,(2004).

[9] E.H.Lee,T.G.Rodgers,Solutionofviscoelasticstressusingmeasuredcrepporrelaxationfunctions,ASME J. Applied Mechanics,30,127-133,(1963).

[10] Yu. A. Rossikin, M.V. Shitikova, Applications of fractional calculus to dynamic problems of linear and nonlinearhereditarymechanicsofsolids,Appl. Mech. Rev.,50,15-57,(1997).

[11]D.Ingman,J.Suzdalnitsky,Responseofviscoelasticplatetoimpact,Journal of Vibration andAcoustics,130:011010,

[12]D.Ingman,J.Suzdalnitsky,M.Zeifman,Constitutivedynaic-ordermodelfornonlinearcontactphenomena,ASME J. Applied Mechanics,67,383-390,(2000).

[13] W. Zhang, N. Shimizu, Numerical Algorithm for Dynamic Problems Involving Fractional Operator, International Journal of JSME,SeriesC,Vol.41(1998),No.3,pp.364-370.

[14] W.Zhang,N.Shimizu,DampingPropertiesoftheViscoelasticMaterialDescribedbyFractionalKelvin-VoigtModel,International Journal of JSME,SeriesC,Vol.42(1999),No.1,pp.1-9.

[15] N.Shimizu,W.Zhang,Fractional CalculusApproach toDynamicProblemsofViscoelasticMaterials, International Journal of JSME,SeriesC,Vol.42(1999),No.4,pp.825-837.

[16] M. Fukunaga, N. Shimizu, H. Nasuno, A nonlinear fractional derivative model of impulse motion for viscoelasticmaterials,Physica ScriptaT136(2009)014010(6pp).

[17] M.Fukunaga,N.Shimizu,Nonlinearfractionalderivativemodelsofviscoelastic impactdynamicsbasedonEntropyelasticity andgeneralizedMaxwell Law,Journal ofComputational and Nonlinear Dynamics,Vol.6(2011),021005(6pp).

[18] L.Collatz,Numerische Behandlung von Differentialgleichungen,Springer-Verlag,Berlin1951.