Geometrically Non-Linear Frequency Response of Axially Functionally Graded Beams Resting on Elastic Foundation Under Harmonic Excitation

This article presents geometrically nonlinear forced vibration analysis of an axially functionally graded(AFG)non-uniformbeamrestingonanelasticfoundation.Themathematicalformulation isdisplacementbasedandderivationofgoverningequationsisaccomplishedfollowingHamilton’s principle.Thefoundationhasbeenmathematicallyincorporatedintotheanalysisasasetoflinear springs.According to thebasicassumptionof thepresentmethodforceequilibriumcondition is satisfied at a maximum excitation amplitude value. Thus, the dynamic problem is equivalently representedasastaticone,whichissolvedbyfollowinganumericalimplementationoftheBroyden method.ItisamethodthatutilizestheJacobianmatrixandsubsequentcorrectionoftheinitialJacobian tosolveasystemofnonlinearequations.Thelargeamplitudedynamicbehaviourofthesystemin termsofnon-dimensionalfrequencyresponsecurvesisvalidatedagainstestablishedresultsandnew resultsarefurnishedforaparabolictaperedAFGbeamonalinearelasticfoundation. KEywoRdS Axially Inhomogeneous Beam, Broyden Method, Elastic Foundation, Forced Vibration, Large Amplitude

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