Abstract When avibratingstructureisrotatedwithrespecttoinertialspace,thevibratingpatternrotatesatarateproportionalto the inertialrateofrotation.Bryanfirstobservedthiseffectin1890.Theeffect,calledBryan’seffectinthesequel,has numerous navigationalapplicationsandcouldbeusefulinunderstandingthedynamicsofpulsatingstarsandearthquake series inastrophysicsandseismology.Bryan’sfactor(thecoefficientofproportionalitybetweentheinertialandvibrating pattern rotationrates)dependsonthegeometryofthestructureandthevibrationmodenumber.The‘‘gyroscopiceffects’’ of ahollowisotropicsolidspherefilledwithaninviscidacousticmediumareconsideredhere,butthetheoryisreadily adapted toahollowisotropicsolidcylinderfilledwithaninviscidacousticmedium.Alineartheoryisdevelopedassuming, among othermildconditions,thattherotationrateisconstantandmuchsmallerthanthelowesteigenfrequencyofthe vibrating system.Thuscentrifugalforcesareconsideredtobenegligible.Beforecalculatingsolutionsforthedisplacement of aparticleintheisotropic,spherical,distributedbody,Bryan’sfactorisinterpretedusingacomplexfunction.Hereitis demonstrated thatneitherBryan’seffectnorBryan’sfactorisinfluencedbyincludinglight,isotropic,viscousdampingin the mathematicalmodel.Hencedampingisneglectedinthesequel.Twoscenariosarethenidentified.Firstly,wemay assume thattheacousticmediumiscompletelyinvolvedintherotation(thespheroidalmode).Secondly,wemayassume that theacousticmediumremainsstaticwithrespecttotheinertialreferenceframe(thetorsionalmode).Weinvestigatethe spheroidal modeusinganumericalexperimentthatcomparestherotationalangularrateofasphere(filledwithaninviscid acoustic medium)withthoseofitsvibratingpatternsatbothhighandlowvibrationfrequency.