Sensitivities to Initial Conditions of a Dynamic Snap-Through Buckling Structure


Animation of Snap ThroughDynamic snap-through (DST) buckling is a violent, highly nonlinear phenomenon that can devastatingly affect shallow-arch structures such as curved aircraft panels. As the future of flight research continues to focus on lighter, faster aircraft, hypersonic flight conditions will exacerbate the consequences of DST buckling due to harsher thermal and acoustic loading. DST buckling may occur periodically, quasi-periodically, and chaotically. This, combined with inherent large amplitudes of oscillation (i.e. increased stress magnitudes), automatically triggers high cycle fatigue (HCF) concerns. Another bothersome aspect of DST buckling is unpredictability. Systems demonstrating chaotic behavior by definition are nonlinear and have an extreme sensitivity to initial conditions. A useful design tool for engineers would be to map out the parameter space where specific harmonic loading characteristics (e.g. forcing amplitude, forcing frequency) may yield higher (safe) or lower (dangerous) predictability. To this end, experimental and numerical investigations a single-degree-of-freedom snap-through link model that is harmonically excited at various forcing frequencies and magnitudes were investigated. Experimental and numerical basins of attraction are created using automated stochastic interrogation techniques as to quantify the extreme sensitivity to initial conditions (i.e. position, velocity, or forcing phase). Additionally, an equation of motion is created for use in numerical models which show excellent agreement with experimental results.

Experiments - The Shallow Arch Link Model

Link Model

A simple single-degree-of-freedom link model was chosen for the following experiments (depiction shown in diagram). The lateral (structural) stiffness is achieved by attaching two opposing springs to the main slider, and the central hinge is transversely forced via harmonic displacement of the active spring.

Since this system is nonlinear, it is capable of having multiple stable (and unstable) equilibria. For example, this system has 3 equilibria: 2 stable (+/- θ well) and 1 unstable (θ=0°). As a result, these equilibria will compete for any system transients which gives rise to multiple steady state dynamic behaviors (e.g., chaos, harmonic snap-through, super/sub-harmonics, single-well oscillations, etc.)

Links to experimental videos:

Numerical Modeling - Basins of Attraction

The equation of motion was derived using the standard Lagrangian energy approach. Damping was incorporated into the model by using the Least Squares method in cooperation with experimental time series data to find the optimal energy dissipation for the numerical model.

A basin of attraction is defined as the region of phase space ultimately leading to one specific end behavior. For example, a chaotic basin of attraction is the complete set of all initial conditions (for a given set of forcing parameters) that lead to a chaotic response. For our purposes, 3 basins were defined using the following classification:

Basin Mutation

The time series shown is an example of stochastic interrogation. The gray bars are regions of system perturbation where the forcing frequency is randomly changed for a random duration. This stochastic perturbation results in a transient that leads to a specific basin (the color of which is shown above the time series). The red dots are Poincare points (periodic snap shots in phase with the forcing) which can be interpreted as different initial conditions leading to the same basin of attraction.

After compiling numerous numerical simulations, the animation to the right was created to demonstrate how the basins of attraction can mutate as a function of forcing frequency for a nonlinear structural system.

Links to other examples of possible pathological dynamic behaviors:

For a much more in depth discussion, a link to my M.S. thesis:

For more on identifying possible behaviors in nonlinear systems, a link to my International Journal of Bifurcation and Chaos paper:

Group Members Involved

Joshua Waite