Interaction between slow and fast oscillations in an infinite degree-of-freedom linear system coupled to a nonlinear subsystem: Theory and experiment

The interaction dynamics of a cantilever linear beam coupled to a nonlinear pendulum, a prototype for linear/nonlinear coupled structures of infinite degrees-of-freedom, has been studied analytically and experimentally. The spatio-temporal characteristics of the dynamics is analyzed by using tools from geometric singular perturbation theory and proper orthogonal decompositions. Over a wide range of coupling between the linear beam and the nonlinear pendulum, the coupled dynamics is dominated by three proper orthogonal (PO) modes. The first two dominant PO modes stem from those characterizing the reduced slow free dynamics of the stiff/soft (weakly coupled) system. The third mode appears in all interactions and stems from the reduced fast free dynamics. The interaction creates periodic and quasi-periodic motions that reduce dramatically the forced resonant dynamics in the linear substructure These regular Motions are characterized by four PO modes. The irregular interaction dynamics consists of low-dimensional and high-dimensional chaotic motions characterized by three PO modes and six to seven PO modes, respectively. Experimental tests are also carried out and there is satisfactory agreement with theoretical predictions.