Experiments on chaotic motions of a forced nonlinear oscillator: Strange attractors

Abstract

The forced vibrations of a buckled beam show nonperiodic, chaotic behavior for forced deterministic excitations. Using magnetic forces to buckle the beam, two and three stable equilibrium positions for the postbuckling state of the beam are found. The deflection of the beam under nonlinear magnetic forces behaves statically as a butterfly catastrophe and dynamically as a strange attractor. The forced nonperiodic vibrations about these multiple equilibrium positions are studied experimentally using Poincare plots in the phase plane. The apparent chaotic motions are shown to possess an intricate but well-defined structure in the Poincare plane for moderate damping. The structure of the strange attractor is unravelled experimentally by looking at different Poincare projections around the toroidal product space of the phase plane and phase angle of the forcing function. An experimental criterion on the forcing amplitude and frequency for strange attractor motions is obtained and compared with the Holmes-Melnikov criterion and a heuristic formula developed by the author. © 1980 by ASME.