Local and global dynamical behavior in nonlinear mechanical models: Theory and experiments.

摘要

This dissertation explores local and global dynamic bifurcations within the context of experimental mechanical models. Making use of the "ball-rolling-on-a-hill" analogy, an experiment is described in which a cart is constrained to roll along a track built in the shape of a desired potential energy well. Using gravity as the restoring force, the system can be used to mimic any system who's potential energy is (at least approximately) known. This experimental system is used to conduct a number of studies described below.; Nonstationary systems are systems with time-dependent control parameters. Experiments were performed on the effects of a nonstationary forcing frequency sweep upon bifurcation events, where the evolution was linear in time. An "almost-linear" oscillator (consisting of a parabolic track) was studied first as a baseline for resonance shifting and amplitude reduction, and then attention was turned to the twin-well Duffing system and the nonstationary effects upon the fold (the jump to and from resonance) and the flip (period-doubling) bifurcations. Experimental results in both cases confirmed the theory that the bifurcation event is delayed in time such that unstable (stationary) solution branches are followed temporarily. This "penetration" time scales with the magnitude of the evolution parameter.; Both free and forced oscillations were considered with an impact system, where a barrier was placed on the parabolic track to constrain the motion of the cart. Calculations revealed several relationships between natural frequency scaling and barrier location relative to both static equilibrium and initial energy. Forced results were analyzed for three barrier positions, all different in relation to static equilibrium of the unconstrained system, using bifurcation diagrams, power spectra, and time-delay coordinates. The results yielded rich subharmonic and chaotic windows embedded between multiple resonances.; Earlier theoretical work has revealed the possibility of indeterminacy in the post-critical outcome when the fold bifurcation coincides with fractal basin boundaries. A series of global bifurcations, resulting in manifold tangencies, organizes the series of events that can transform smooth basins into fractal basins. As a result, post-fold outcome can be classified as safe and determinate (low forcing levels), indeterminate ("medium" forcing levels), and unsafe and determinate (high forcing levels). Significant changes in the basins are observed, such as heavy erosion of resonant basins with escaping trajectories, that convey important information for the design or safety engineer. The experimental basins are obtained through a method of stochastic interrogation, where periodic but random perturbations are applied to the system to "stochastically" visit large regions of phase space.