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An analyticalstudyforfreevibrationofnaturallycurvedandtwistedbeamswith uniformcross-sectionalshapesiscarriedoutusingspatialcurvedbeamtheorybasedon theWashizu’sstaticmodel.Inthegoverningequationsofmotionofthebeams,all displacementfunctionsandthegeneralizedwarpingcoordinatearedefinedatthe centroidaxisandalsotheeffectsofrotaryinertia,transversesheardeformationsand torsion-relatedwarpingareincludedintheproposedmodel.Explicitanalytical expressionsarederivedforthevibratingmodeshapesofacurved,bending-torsional- shearingcoupledbeamunderclamped–clampedboundaryconditionwiththehelpof symboliccomputingpackage Mathematica, andaprocessofsearchingisusedto determinethenaturalfrequencies.ComparisonsofthepresentresultswiththeFEM resultsusingbeamelementsinANSYScodeshowgoodaccuracyincomputationand validityofthemodel.Further,thepresentmodelisusedforcylindricalhelicalsprings with circularcross-sectionfixedatbothends,andtheresultsindicatethatthenatural frequenciesagreewellwiththetheoreticalandexperimentalresultsavailable.
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