Local bifurcations and codimension-3 degenerate bifurcations of a quintic nonlinear beam under parametric excitation. (English) Zbl 1143.74334

Summary: This paper presents the analysis of the local and codimension-3 degenerate bifurcations in a simply supported flexible beam with quintic nonlinear terms subjected to a harmonic axial excitation for the first time. The quintic nonlinear equation of motion with parametric excitation is derived using the Hamilton’s principle. The parametrically excited system is transformed to the averaged equations using the method of multiple scales. Numerical method is used to compute the bifurcation response curves based on the averaged equations. The investigations are made on the effects of quintic nonlinear terms and parametric excitation on the local bifurcations. The stability of trivial solution is analyzed. With the aid of normal form theory, the explicit expressions are obtained for normal form associated with a double zero eigenvalues and \(Z_2\)-symmetry of the averaged equations. Based on normal form, the analysis of codimension-3 degenerate bifurcations is performed for a simply supported quintic nonlinear beam with the focus on homoclinic and heteroclinic bifurcations. It is found from the analysis of homoclinic and heteroclinic bifurcations that multiple limit cycles may simultaneously exist for quintic nonlinearity. In particular, the number of limit cycles can be precisely determined analytically. New jumping phenomena are discovered in amplitude modulated oscillations.


74H60 Dynamical bifurcation of solutions to dynamical problems in solid mechanics
34C23 Bifurcation theory for ordinary differential equations
37G15 Bifurcations of limit cycles and periodic orbits in dynamical systems
37N15 Dynamical systems in solid mechanics
70K50 Bifurcations and instability for nonlinear problems in mechanics
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
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