Sparse identification of nonlinear dynamics applied to the acoustic levitation of acoustically large objects

Many studies on acoustic radiation forces, especially those applied to acoustic levitation, focus on characterizing the behaviour of acoustic fields. However, the dynamic response of the levitated objects, particularly those larger than the wavelength limit, remains relatively underexplored. Here, we look to bridge this gap by deriving nonlinear equations of motion for a spherical object trapped under acoustic radiation forces while subject to external excitation. For such a contemporary scenario, the otherwise elemental Gorkov formulation fails to provide accurate results. Using Sparse Identification of Nonlinear Dynamical Systems (SINDy), first, we derive the corresponding nonlinear equation of motion from analytical time series data obtained through the Gorkov formulation and external excitation for acoustically small objects. This approach recovers the governing equation with less than 0.05% error in coefficient values when compared to the analytical solution. Second, we conduct experiments with the TinyLev levitator with external excitation applied via an external actuator to generate the required time series for an acoustically large object. SINDy is applied to reconstruct governing equations from experimental data, allowing for the study of how excitation amplitude affects acoustically large objects. All obtained coefficients change with excitation amplitude, and the coefficients in the dynamic equation of motion should not be treated as constants. Strong velocity-dependent terms emerged, indicating a complex relationship between viscosity and object response, which classical models do not predict. The bifurcation diagram obtained using the SINDy-derived equation of motion shows closer agreement with that obtained experimentally. These results demonstrate that SINDy can recover equations consistent with Gorkov's formulation and extend beyond it, providing a pathway to derive analytical expressions directly from data for levitating and manipulating objects beyond the Rayleigh limit.