Dynamic analysis of a simply supported beam resting on a nonlinear elastic foundation under compressive axial load using nonlinear normal modes techniques under three-to-one internal resonance condition

摘要: In this paper, the Nonlinear Normal Modes (NNMs) analysis for case of three-to-one (3:1) internal resonance a slender simply supported beam in presence compressive axial load resting on nonlinear elastic foundation is studied. Using Euler–Bernoulli model, governing PDE beam’s transverse vibration and also its associated boundary conditions are extracted. These motion equation condition relations solved simultaneously using four different approximate-analytical solution techniques, namely themethod ofMultiple Time Scales, ofNormal Forms, ofShaw Pierre, ofKing Vakakis. The obtained results at stage methods which all time–space domain compared it concluded that result similar answer amplitude part vibration. At next step, normal modes obtained. Furthermore, effect force dynamic such Finally, under three-to-one-internal NNMs steady-state stability performed. Then changing values parameters response considered. Moreover, 3-D plots frequency curves presented.