Dicovering the emergent nonlinear dynamics of acoustically levitated cube clusters

Abstract

The complex behavior of many natural and engineered systems emerges from the interaction of a small number of effective degrees of freedom. Discovering the physical basis of the interactions between these degrees of freedom directly from experimental observations has been a longstanding challenge, particularly with respect to predicting the long-time dynamics of dynamical systems with unknown equations of motion. Here, we introduce a data-driven approach that is able to produce a generative model for the long-time dynamical behavior of systems with a weakly attracting manifold. We apply this method to an experimental dynamical system with two degrees of freedom: acoustically levitated pairs of cube-shaped particles, which cluster by sharing a single edge. In the acoustic trap, the center-of-mass of the cube cluster oscillates vertically about the levitation plane, while also oscillating about their flexible hinge-like connection. Depending on their initial condition, the hinge dynamics evolve about three distinct nonlinear dynamical attractors persisting for hundreds of cycles. In order to capture the underlying physics, we develop a numerical fitting procedure and extract a minimal nonlinear dynamical model that captures both the long-time dynamics of the cluster as well as the convergence onto the dynamical steady state. This dynamical model uncovers the nonlinear, non-reciprocal coupling between the center-of-mass motion and the hinge degree of freedom that stabilizes the dynamical attractors, which we subsequently confirm by independent finite-element methods. Our results demonstrate a novel data-driven method for the discovery of nonlinear models with long-timescale stable predictions.

Figures (6)

• Figure 1: Schematic of the fitting procedure. We show sample data collected from an experimental system consisting of time-series data with two degrees of freedom $x$ and $y$. The instantaneous acceleration of the system is then reconstructed from linear and nonlinear combinations of the system variables by linear regression, resulting in an initial model $\mathbf{C_0}$ which fits instantaneous accelerations well but fails to capture long-time dynamical behavior. The dynamical equation is then refined by integrating the fitted dynamics forward, and modifying the coefficients to reconstruct long-time features of the original data. Finally, the coefficients are aggregated from repeat experiments, and only the most consistent terms are included in the final fit.
• Figure 2: An acoustically levitated cube dimer is a granular hinge with an emergent elastic degree of freedom arising from particle geometry. (a) Diagram of experimental setup. The transducer emits ultrasound in air at $f_0 = 45.650$ kHz and wavelength $\lambda \approx 7.5$ mm. The reflector is placed at a resonant distance to generate a pressure standing wave with a single node. We apply a square wave modulation to turn off the field for some time $t_{\text{off}}$, injecting energy into the system. (b) Image of cube cluster dynamics over time, with subsequent images having a time difference of $\Delta t = 4$ ms. The levitation plane defines $y=0$ (see Supplementary Information for details of the measurement), and we define $\theta$ as difference of the upper contact angle $\alpha$ and the lower contact angle $\beta$ divided by $\pi$. (c) Restoring vertical force about the levitation plane due to the primary acoustic force. Experimental data (orange) is compared to the results of a Lattice-Boltzmann method (LBM) simulation (black) (d) Restoring torque about the contact point of the cluster. Experimental results (blue) are compared to the results of a LBM simulation (black).
• Figure 3: External modulation controllably excites both oscillatory hinge modes. (a) Image sequence showing field modulation. Prior to modulation ($t<0$), the cluster is stably levitated in the acoustic field. At $t=0$ the field is turned off and the cubes fall under gravity. At $t=13$ms, the field is turned on again and the clusters experience restoring forces, giving rise to oscillations in both degrees of freedom. (b) (Left) Smoothed traces of $y$ and $\theta$ between the cubes before, during, and after the modulation. (Right) Discrete Fourier transform of the $y$ and $\theta$ signals. (c) Ratio between $\omega_y$ and the two slow and fast angle components, plotted in light green and dark green respectively. (d) Plot of the probability density function of the phase difference between the $y$ and $\theta$ oscillations (examples for phase difference of 0 and 1 plotted in insets).
• Figure 4: Phase locking in oscillations leads to three distinct dynamic memories persisting over hundreds of cycles (a) Sample trajectories for three distinct attractor types (100 cycles of each), projected onto the $\theta$,$\dot{\theta}$,$y$ axes. The $y=0$ hyperplane is shown in purple. (b) Example of the Poincaré section, together with resulting Poincaré map. We define the Poincaré section as the $y=0$ hyperplane (purple), and take the Poincaré map to be the point in each cycle where the data first crosses the section in the $\dot{y}<0$ direction (from above in this projection). Purple box: resulting Poincaré map for 60 cycles of data (color corresponding to time after drop for the first 30 cycles. Cycles 30-60 are plotted in light blue). (c) Sample trajectories of $\theta(t)$ (dark) and $y(t)$ (light) for the three classes shown in (a). (d) Sample flows on the Poincaré maps for the first $10$ ($20$) points in each trajectory, separated by attractor class. $10$ trajectories from each class are included. A Savitzky-Golay filter of order $2$ and window length $5$ is applied to the Poincaré map to smooth the flows. (e) Aggregated Poincaré maps for the last $60$ points in each trajectory, colored by attractor class. $10$ trajectories from each class are included.
• Figure 5: Discovering the governing nonlinearity via model fitting. (a) Schematic of our parameter fitting procedure. We first find the best linearly fitting coefficients $\mathbf{C_0}$ which minimizes the error $\norm{\mathbf{\ddot{X}} - \mathbf{\Lambda}_X \mathbf{C_0}}$ between experimental acceleration $\mathbf{\ddot{X}}$ and polynomial combinations of our state variables $\mathbf{\Lambda}_X$. These coefficients $\mathbf{C_0}$ are then used as initial conditions for gradient descent over the loss function $\mathcal{L}$ which has three terms, each penalizing different deviations of simulated, integrated dynamics from the data time-series. (b) Example optimization trajectory starting from initial $\mathbf{C_0}$ and reaching final $\mathbf{C}$. (c) Z-scores for second order fits of $\ddot{y}$ and $\ddot{\theta}$ after optimization over 10 large attractor (class 1) trajectories. Discarding terms with a z-score lower than $3$, and including physical terms for viscous damping, we attain two nonlinear equations for $\ddot{y}$ and $\ddot{\theta}$. (d) Normalized coefficient distributions for the eleven significant terms used in the reduced model.