Acompositebeamissymmetricifboththematerialpropertyandsupportaresymmetricwithrespecttothemiddlepoint.Inordertostudythefreevibrationperformanceofthesymmetriccompositebeamswithdifferentcomplexnonsmooth/discontinuousinterfaces,wedevelopanR(x)-orthonormaltheory,whereR(x)isanintegrableflexuralrigidityfunction.TheR(x)-orthonormalbasesinthelinearspaceofboundaryfunctionsareconstructed,ofwhichthesecond-orderderivativesoftheboundaryfunctionsareaskedtobeorthonormalwithrespecttotheweightfunctionR(x).WhenthevibrationmodesofthesymmetriccompositebeamareexpressedintermsoftheR(x)-orthonormalbaseswecanderiveaneigenvalueproblemendowedwithaspecialstructureofthecoefficientmatrixA:=[aij],aij=0ifi+jisodd.Basedonthespecialstructurewecanprovetwonewtheorems,whichindicatethatthecharacteristicequationofAcanbedecomposedintotheproductofthecharacteristicequationsoftwosub-matriceswithdimensionshalflower.Hence,wecansequentiallysolvethenaturalfrequenciesinclosed-formowingtothespecialtyofA.Weusethispowerfulnewtheorytoanalyzethefreevibrationperformanceandthevibrationmodesofsymmetriccompositebeamswiththreedifferentinterfaces.
