DYNAMIC STABILITY OF A SLENDER BEAM WITH INTERNAL RESONANCE UNDER A LARGE LINEAR MOTION
内共振条件下直线运动梁的动力稳定性

Dynamic modeling of a flexible beam undergoing a large linear motion is presentedin this paper at first. The equations of motion for the beam are derived by using Kane's equationand then simplified through the Rayleigh-Ritz method. Different from the linear modeling methodwhere the generalized inertia forces and the generalized active forces are linearized in the model-ing process, the present model takes the coupled cubic non-linearities of geometrical and inertialtypes into consideration. In the case of a simply supported slender beam under certain averageacceleration of base, the second natural frequency of the beam may approximate to the tripled firstone so that the condition of 3:1 internal resonance of the beam holds true. The method of multi-ple scales is used to solve directly the nonlinear differential equations and to derive the nonlinearmodulation equation for either the principal parametric resonance or the combination parametricresonance with 3:1 internal resonance between the first two modes of the beam. The dynamicstability of the trivial state of the system is investigated by using Cartesian transformation indetail. The equations of approximate transition curves in the plane of dimensionless frequency andexcitation parameter that separate stable from unstable solution are derived. For the case of prin-cipal parametric resonance of the first mode, in addition to the principal instability region, thereexist several new narrow instability regions because of the presence of internal resonance. Thesenarrow regions move from the left of the principal instability region to the right of it when thefrequency detuning parameter of internal resonance increases from the negative to the positive. Incontrast with the case of principal parametric resonance of the first mode, single mode equilibriumsolution is possible for the principal parametric resonance of the second mode. Furthermore, thestability of the trivial solution for the third mode or the higher ones has not been affected by theinternal resonance between the first two modes. Finally,the modulation equations are reduced toa two-dimensional system and the type of a Hopf bifurcation is determined in the vicinity of thebifurcation via the center manifold theorem and a limit cvcle is found.