Acomplianttowerismodeledasapartiallydry,partiallytapered,dampedTimoshenkobeamwiththesuperstructuremodeledasaneccentrictipmass,andanon-classicaldampedboundaryatthebase.Thefoundationismodeledasacombinationofalinearspringandatorsionalspring,alongwithparallellinearandtorsionaldampers(Kelvin-Voigtmodel).Thesuperstructureaddstothekineticenergyofthesystemwithoutaffectingthepotentialenergy,therebyreducingthenaturalfrequencies.Theweightofthesuperstructureactsasanaxialcompressiveloadonthebeam,reducingitsnaturalfrequenciesfurther.Theemptyspacefactorduetothetruss-typestructureofthetowerisincluded.Theeffectofsheardeformationandrotaryinertiaareincludedinthevibrationanalysis;withthenon-uniformbeammode-shapesbeingaweightedsumoftheuniformbeammode-shapessatisfyingtheendcondition.TheweightsareevaluatedbytheRayleigh-Ritz(RR)method,andverifiedusingfiniteelementmethod(FEM).Theweightofthesuperstructureactsasanaxialcompressiveloadonthebeam.Kelvin-Voigtmodelofstructuraldampingisincluded.Apartofthestructurebeingunderwater,thevirtualaddedinertiaisincludedtocalculatethewetnaturalfrequencies.Aparametricstudyisdoneforvariousmagnitudesoftipmassandvariouslevelsofsubmergence.ThecomputationalefficiencyandaccuracyoftheRayleigh-Ritzmethod,ascomparedtotheFEA,hasbeendemonstrated.Theadvantageofusingclosed-formtrialfunctionsisclearlyseenintheefficacyofcalculatingthevariousenergycomponentsintheRRmethod.
