Abstract :
Modalanalysisofdampedsystemsoftencannotproceedwithcommonreal-eigenvaluetechniques.Thesystemofequilibriumequationsleadstoamatrixwithelementsbeingquadraticfunctionsofaparameterλ.Thevaluesofthatwhichmakethematrixsingulararethelatentroots,whilethesolutionsoftheassociatedhomogenousequationarethelatentvectors.Theyarethe(generallycomplex)characteristicfrequenciesandthemodeshapesofthesystem,respectively.Althoughthetheoryiswelldeveloped,thenumericalapplicationisopentorefinementsyet.Areductiontobetter-knownreal-domainsubtasksdeservesattention.WithatheoremofPopperandGáspár,an×nλ-matrixproblemcanbecutintwo:inton-sizeasymmetricrealmatriceshavingaseigenvaluesthenlowerandnupperlatentroots,rankedbyabsolutevalue.Thisapproachmaybeofuseforsystemswithhighnumberofdegreesoffreedomwhiledampedbyarelativelyfewconcentrateddevices.Itmightfitalsoanearthquakeanalysis,wherethelowerportionofeigenvaluesiscustomarilywhatcounts.Thedampersappearinthesplittingalgorithmasrestricted-sizemodifications,readyforusebytheSherman–Morrison–Woodburyidentity.Thetaskisre-tracedthiswaytoamoreusualreal-asymmetriceigenproblem.Arequirementofconvergenceisthattheloweranduppern-setoflatentvaluesmustbedistinct.Withodd-numberdegreesoffreedomandneitherover-damped,i.e.alllatentrootsbeingcomplex,thisconditionissurelyviolated.Forsuchcases,asupplementalalgorithmisproposed.
