Stability and Bifurcation in Discrete Mechanical Systems: An Experimental and Analytical Study

Abstract

The curse of dimensionality looms over many studies in science and engineering. Low-order systems provide conceptual clarity but often fail to reveal the extent of possible complexity, whereas high-order systems present a host of daunting challenges to the analyst, not least the classification and visualization of typical behavior. In this paper, we detail the behavior of systems that fall somewhere between a classification of low- and high-order. [Formula: see text][Formula: see text]We present both theoretical and experimental investigations into the nonlinear behavior of a couple of mechanical systems with three mechanical/structural degrees-of-freedom (DOF), with a special focus on bifurcation and multiple equilibria. Useful insight is provided by observation of transient trajectories as they meander about and between equilibria, especially revealing the influence of unstable equilibria, not normally accessible in an experimental context. For instance, the influences of index-1 saddles are mainly detected in three aspects: determining the systems capability to snap-through by generating accessible snap-though tubes, attracting nearby trajectories temporarily oscillating around it, and separating adjacent trajectories. Iso-potentials are 3D-printed to present the energy landscape. For these systems, the 3D configuration space allows considerable complexity, but is also somewhat amenable to geometric interpretation. By varying a mass/stiffness ratio as a control parameter, bifurcation structures and morphing potential energy landscapes exhibiting up to 11 equilibria are obtained. Finally, analytical and experimental studies reveal that parametric excitations can stabilize some unstable equilibria under the right amplitudes and frequencies.