On the dynamics of a 2-DOF nonlinear vibratory system with bistable characteristic and circulatory forces

Abstract

We consider an autonomous damped 2-DOF mechanical system in which the two DOFs are coupled by a linear spring. A circulatory force is introduced into the system that may result in self-excited vibrations. A nonlinearity is added in the form of a cubic spring so that there are three fixed points, two of them are stable without the circulatory force, i.e., a bistable behavior. The basins of attraction for different values of the circulatory forces are numerically studied to see how the patterns of them change. Using a Poincaré map, the basins of attraction have three dimensions and they are cut with further cross-sections for the ease of visualization. The results are usually complicated maps with non-smooth boundaries, except when the circulatory force uncouples one DOF from the other. Special patterns of the maps are seen when in the nonlinear case a stable limit cycle is about to occur. Even in the case without stable limit cycle, initial conditions within the special range may lead to numerous cycles of periodic-like transient motion before asymptotically converging to a stable fixed point. Phase planes and bifurcation diagrams are also used, and multiple coexisting periodic solutions are found. Some interesting phenomena are also found in the supercritical region, including a restabilized motion that coexists with the associated period-doubled motion, and strong symmetric and asymmetric autonomous bursting-like motions.