Forced Oscillations of a Single Degree of Freedom System with Fractional Dissipation
振动基础
SCIENTIFICPUBLICATIONSOFTHESTATEUNIVERSITYOFNOVIPAZARSER.A:APPL.MATH.INFORM.ANDMECH.vol.3,1(2011),1-10.
ForcedOscillationsofaSingleDegreeofFreedomSystemwith
FractionalDissipation
´Doli´T.M.Atanackovi´c,D.C.canin,S.Pilipovi´c
Abstract:Westudymotionofasingledegreeoffreedommechanicalsystemconsistingofa
visco-elasticrodof nitelengthwithconcentratedmassatthefreeend.Ifthedeformationoftherodisapproximatedsothatrodisconsideredinastateofquasi-staticdeformationor,equivalently,ifitisassumedthattherodislight(densityisequaltozero)weshowthattheseveralknownoscillationequationscanbederived.
Keywords:Fractionalderivative,distributed-orderfractionalderivative.
1Introduction
Fractionalorderviscoelasticityhasbeentreatedrecentlyinmanypublications[12],[13],[11],[2],[8],[9].Inthisworkweconsideraviscoelasticrodof nitelength, xedatoneendandwithamassattachedattheotherend.Themassisrestrictedtomovealongastraightlinecoincidingwiththerodsaxis(see
Figure)
Weconsidermotionofthemassfromthepositiony=0inwhichtherodisundeformd,
i.e.,ithasitsnaturallengthl.Thefunctionhrepresentingtheouterforcesisassumedtobeknownfunctionoftimetanddisplacementofthepointatwhichthemassisconected.TheproblemofdeterminingthemotionofanoscillatorpresentedinFig.1wastreatedearlierin[4].Thedeformationoftherodisdescribedbythefollowingsystemofequations
ManuscriptreceivedNovember232010;acceptedFebruary32011.T.M.Atanackovi´ciswiththeFacultyofTechnicalSciences,InstituteofMechanics,UniversityofNovi
´Doli´Sad,NoviSad,Serbia;D.C.caniniswiththeFacultyofTechnicalSciences,UniversityofPristina-KosovskaMitrovica,Serbia;S.Pilipovi´ciswiththeFacultyofScience,DepartmentofMathematics,University
ofNoviSad,NoviSad,Serbia.
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