On The Nonliner Nonplane Vibration Of The Cantilever Beam

The nonlinear vibration of beam is a typical problem of nonlinear problem. It has coupled vibration,parametric excited,internal resonance, harmonic resonance, stability and bifurcation. As the reason above, we can research the various nonlinear phenomena and has a deep understanding of the nonlinear vibration characterized by doing the research on the nonplane moving beam. At the same time, many huge structures, like tall buildings, large span bridges, vibrate nonplane when suffered heavy wind force, and it will be meaningful to do the research of beam's nonlinear vibration.The first part of this paper is an introduction of nonplane free vibration of the simple cantilever beam. By using the generalized Hamilton theory and three continuous Euler corners are induced, we get the controlling equations of the beam's moving including six generalize coordinates. Next, by considering the practical conditions in the real projects, we introduced the inextensional condition of center axes and deduced the controlling equations of the beam's moving including four generalize coordinates.Lastly, the boundary conditions of cantilever beam and the predigestion of without moment of inertia are concerned ,and also the Tayor's expressions,we deduced the differential equations of cantilever beam in two flexural directions by take the space flexural and torsion and shear deflection into consideration.In the second part, we can get the first normal mode of the above two flexural vibrations by using nonlinear normal mode, Galerkin's method and harmonic balance method.Because of the stability of the normal mode, we can think the vibration of one way as the parameter excitation in the other way,then the equation can transform to the Mathieu equation; by resolving the Mathieu equation's stable area, analyze the stability of flexural vibration, we find the normal mode of flexural vibration is unstable when the rectangle section's length equals its wide. If the ration of the length and wide equals to 1.2, the normal mode is stable when the swing is small and it will become unstable when swing being larger.