Nonlinear Behavior of Systems with Multiple Equilibria

This study describes the nonlinear behavior of a number of various systems with multiple equilibria, from discrete mechanical systems to high-dimensional continuous structures. All these systems are capable of exhibiting sophisticated potential landscapes, including multiple equilibria with different stability properties, whereas their nonlinearities are somewhat sensitive to the geometric conditions. Similar behavior and equivalent relationships are developed for various systems.

First, numerical and experimental investigations are presented for systems with three mechanical/structural degrees-of-freedom (DOF). Considering moderate complexity between low-order and relatively high-order systems, the three-DOF system is able to exhibit a visible configuration space. Useful insights are provided by observations of the iso-potentials and experimental transient trajectories meandering within them. Hyperboloidal passable tubes are found around index-1 saddles on the iso-potential shapes, enabling possible transitions between stable equilibria. As a result, transient trajectories have a tendency to slow down, temporarily oscillate, and separate from each other in the vicinity of these saddles. These phenomena have been verified by experiments, which imply possible existence of an unstable equilibrium in dynamics. Bifurcation structures and morphing potential landscapes are revealed by varying key geometric parameters of the system. Parametric excitation shows a possibility to stabilize unstable equilibria in dynamics under the right amplitudes and frequencies - this is exposed both theoretically and experimentally. A practical four-member pyramidal lattice frame is used as an example of a complex system, adequately modeled by three-DOF. Though significantly different from the mass-spring system, the pyramidal frame presents similar universal features and behavior as the discrete system, with its three dominant modes.

Static nonlinear behavior of higher-order structures is then investigated. For geodesic lattice domes with rigid joints, the complete load-displacement relationship and multiple equilibrium configurations are exhibited both in experiment and in simulation. Multiple snaps are observed, when the system discontinuously pops from one stable equilibrium configuration to another. Symmetry breaks as a result of the equilibrium path bifurcation. Experimental result shows the sensitivity of the structure due to minor perturbations. Geometric parameters have a qualitative influence on the system’s nonlinearity. Furthermore, for a shallow arch structure, the geometric conditions to maintain a stable snapped-though equilibrium position (in addition to the nominally unloaded configuration) are studied. Critical stability boundaries are generated in the parameter space. When the boundary is crossed, the stable inverted equilibrium disappears, and as a result, the structure will snap back to its initial configuration spontaneously. A set of 3D-printed arches on both sides of the critical boundary are produced for verification purposes. The results have also been extended to thermal conditions. Finally, as a real high-dimensional system, the buckling and post-buckling behavior of a cylindrical shell is taken into consideration. Various initial imperfections are tested. While the structure is sensitive to some initial imperfection shapes (e.g., a post-buckled deformation, a dimple imperfection, etc.), some other initial imperfection shapes (e.g., an axisymmetric half ‘sine’ wave) hardly have any influence on the buckling behavior. Therefore, the sensitivity of the system can be reduced by applying prescribed initial imperfections in certain `sensitive’ shapes. Lateral probing tests under varying axial loads exhibit a view of the underlying potential landscape and implies an upper limit of the critical buckling force.

In contrast to previous studies on known systems, in the last part, a quadratic regression method is developed to locate unstable equilibria for an unknown system from its transient responses. The method shows great accuracy and efficiency for various systems with two or three mechanical/structural DOFs, especially in locating the `most important’ index-1 saddle point. It is also beneficial in identifying systems with adjacent equilibria or saddle point ghosts. The method also shows a robustness against noise, if a proper zero-phase filter is applied.