For the beam with simple and flexible supports, a nonlinear dynamic model is established by applying the flexible multi-body dynamic theory. The model can describe both the global rotation and the relative deformation of the beam. The modal and axial-compressive stability of the system are investigated by using analytical and numerical method, and the effect of the movable support stiffness are obtained. The results show that there is great influence on the lower-order frequencies, vibration shape and the buckling mode of the system while the movable support stiffness is smaller than the beam. In this case, they behave to the global rotation characteristic and the lower-order vibration shape in the floating coordinate system is also different to the classical beam, which is affected by the global rotation. However, when the movable support is very stiff, the influence on the lowerorder frequencies, vibration shape and the buckling mode of the system are extremely slight and the uncertainty of the movable support stiffness only lightly affects the higher-order frequencies and vibration shape of the system. The results are important to the constraint boundary design of the beam and the application of the flexible multibody dynamic theory.

A nonlinear dynamic model of thin rectangular plate rotating around its symmetrical axis with two opposite simply-supported edges and two opposite free edges is established using general Hamilton's Variational Principle. Three lower modes and the critical bifurcation values of the plate are analyzed approximately by employing assumed modes method. The results show that the overall motions can result in dynamic softening in the flexible multi-body system. Furthermore, the same method is also used to investigate the post-buckling behaviour of the plate. The symmetrical stable post-buckling solutions which are developed from the trivial solution through first bifurcation, the asymmetrical stable post-buckling solutions which are developed from the symmetrical post-buckling solutions through the second bifurcation and the antisymmetry unstable post-buckling solutions which are developed from the trivial solution through its second bifurcation are obtained.