Burak Özhan B.Pakdemirli M.2024-07-222024-07-22201010958568http://akademikarsiv.cbu.edu.tr:4000/handle/123456789/18371A general vibrational model of a continuous system with arbitrary linear and cubic operators is considered. Approximate analytical solutions are found using the method of multiple scales. The primary resonances of the external excitation and three-to-one internal resonances between two arbitrary natural frequencies are treated. The amplitude and phase modulation equations are derived. The steady-state solutions and their stability are discussed. The solution algorithm is applied to two specific problems: (1) axially moving Euler-Bernoulli beam, and (2) axially moving viscoelastic beam. © 2010 Elsevier Ltd. All rights reserved.EnglishResonanceApproximate analytical solutionsContinuous systemCubic nonlinearitiesEuler Bernoulli beamsExternal excitationForced vibrationGeneral solutionsInternal resonanceMethod of multiple scalePrimary resonanceSolution algorithmsSpecific problemsSteady state solutionVibrational modelsViscoelastic beamsMathematical operatorsA general solution procedure for the forced vibrations of a system with cubic nonlinearities: Three-to-one internal resonances with external excitationArticle10.1016/j.jsv.2010.01.010