# On Locating Unstable Equilibria and Probing Potential Energy Fields in Nonlinear Systems Using Experimental Data

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2020

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## Abstract

This study focuses on a series of data-driven methods to study nonlinear dynamic systems. First, a new method to estimate the location of unstable equilibria, specifically saddle-points, based on transient trajectories from experiments is proposed. We describe a system in which saddle-points (not easily observed in a direct sense) influence the behavior of trajectories that pass `close-by' them. This influence is used to construct a model and thus identify a more accurate estimate of the location using a number of refinements associated with linearization and regression. The method is verified on a rolling-ball model. Both simulations and experiments were conducted. The experiments consists of a small ball rolling on a relatively shallow curved surface under the influence of gravity: a potential energy surface in two dimensions. Tracking the motion of the ball with a digital camera provides data that compares closely with the output of numerical simulation. The experimental results suggest that this method can effectively locate the saddle equilibria in a system, and the robustness of the approach is assessed relative to the effect of noise, size of the local neighborhood, etc., in addition to providing information on the local dynamics. Given the relative simplicity of the experiment system used and a-priori knowledge of the saddle-points, it is a useful testing environment for system identification in a nonlinear context. Furthermore, a post-buckled beam model is used to test this method. Because in real world applications, continuous elastic structures are more common. The experiment results successfully capture both the stable and unstable configurations. However, the natural frequency provided by this regression method underestimates the natural frequency of the second mode. This is the result of low sampling rate in the experiment which leads to inaccurate estimation of velocity and acceleration from numerical differentiation. Simulation results from finite element method with higher sampling rate do not have this issue.

Then, a method to identify potential energy through probing a force field is presented. A small ball resting on a curve in a gravitational field offers a simple and compelling example of potential energy. The force required to move the ball, or to maintain it in a given position on a slope, is the negative of the vector gradient of the potential field: the steeper the curve, the greater the force required to push the ball up the hill (or keep it from rolling down). We thus observe the turning points (horizontal tangency) of the potential energy shape as positions of equilibrium (in which case the 'restoring force' drops to zero). We appeal directly to this type of system using both one and two-dimensional shapes: curves and surfaces. The shapes are produced to a desired mathematical form generally using additive manufacturing, and we use a combination of load cells to measure the forces acting on a small steel ball-bearing subject to gravity. The measured forces, as a function of location, are then subject to integration to recover the potential energy function. The utility of this approach, in addition to pedagogical clarity, concerns extension and applications to more complex systems in which the potential energy would not be typically known {\it a priori}, for example, in nonlinear structural mechanics in which the potential energy changes under the influence of a control parameter, but there is the possibility of force {\it probing} the configuration space. A brief example of applying this approach to a 1-D simple elastic structure is also presented. For multi-dimensional continuous elastic systems, it would be hard to derive the whole potential energy field. However, it is possible to learn the potential energy difference between different equilibria. This information could help us learn the global stability of the stable equilibria, \textit{i.e.}, how much energy is required to escape from the stable equilibria.

Finally, a case study using the two above-mentioned methods on short square box columns is presented. This case study relies on simulation from the finite element method. The buckling of short square box column is dominated by the local buckling of the panel on each side of the column. Hence, the buckling of short box columns shares strong similarities with the buckling of a rectangular panel under uni-axial load. The primary, secondary and tertiary

bifurcation of a series of square box columns with different height-to-width ratio is presented. Then, we focus on the column with height-to-width ratio of 1.4142, in which the primary and second bifurcation would happen almost simultaneously. And thus, the differences in the energy level between different stable equilibria are important. The simulation results show that after the secondary bifurcation, the energy `well' depth for these stable equilibria are similar initially. With the further increase of buckling load, the energy well for the second mode is deeper and the second mode becomes the more stable configuration. We also study the dynamic snap-through of the post-buckled column. The regression method is used to estimate the equilibria configuration and the natural frequencies with great accuracy. We notice an interesting phenomenon, there can be an energy exchange between different sides of the box column and hence, the real parts of the eigenvalue of the Jacobian matrix are positive if we only take the shape of one surface into account, whereas, if we take two next surfaces into the regression method, the real parts become negative.

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## Citation

Xu, Yawen (2020). *On Locating Unstable Equilibria and Probing Potential Energy Fields in Nonlinear Systems Using Experimental Data*. Dissertation, Duke University. Retrieved from https://hdl.handle.net/10161/21501.

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