Bifurcation and chaos in a bilinear constrained column. Part Ⅰ: Stability analysis and nonlinear unperturbed dynamics

摘要

This paper analyzes the problem of an axially loaded compressed elastic column, fixed at its base and constrained at the top by a bilinear boundary condition. The exact bifurcation analysis results in a typical asymmetric branching diagram, which is shown and discussed. The dynamic analysis is performed by means of Ritz-Kantorovich's method, and single mode dynamics were analyzed. Vertical inertia is taken into account, as well as geometric nonlinearities and the effects of the bilinear boundary condition. For different values of the dimensionless axial load in the range [0,(3/2 π)~2], three typical situations are first recognized and then studied. The three typical resulting phase portraits are shown. Finally, the presence of homoclinic loops, which will be the starting point for the second part of this paper, is evidenced.