Theoretical and Experimental Finite Forced Motions of a Thin Walled, Cantilever Beam: Dynamic Instability and Modal Coupling

The 3-d dynamics of a cantilever beam undergoing large displacements under a sinusoidally varying, concentrated, vertical force at its free end is analyzed. The pde’s of the motion are obtained by using the Hamilton principle and then a reduced 3 degree-of-freedom model is obtained using in a Galerkin discretization, three eigenfunctions of the linearized model. A path following procedure is used to describe the global dynamic behavior in the excitation control parameter plane. The results obtained using the simple 3 d.o.f. analytical model are then compared with those of an experimental steel model of the cantilever. The regions of instability of the unimodal planar solution in which the nonlinear modal coupling excites the torsional component are studied; the planar motion usually looses stability via Hopf bifurcations after which quasi-periodic and/or chaotic motions are found. Inside the regions in which the system shows a complex time-evolution the complexity-level of the response is analyzed using, for the experimental model, a time series analysis tool.