Designing precise dynamical steady states in disordered networks

Elastic structures can be designed to exhibit precise, complex, and exotic functions. While recent work has focused on the quasistatic limit governed by force balance, the mechanics at a finite driving rate are governed by Newton's equations. The goal of this work is to study the feasibility, constraints, and implications of creating disordered structures with exotic properties in the dynamic regime. The dynamical regime offers responses that cannot be realized in quasistatics, such as responses at an arbitrary phase, frequency-selective responses, and history-dependent responses. We employ backpropagation through time and gradient descent to design spatially specific steady states in disordered spring networks. We find that a broad range of steady states can be achieved with small alterations to the structure, operating both at small and large amplitudes. We study the effect of varying the damping, which interpolates between the underdamped and the overdamped regime, as well as the amplitude, frequency, and phase. We show that convergence depends on several competing effects, including chaos, large relaxation times, a gradient bias due to finite time simulations, and strong attenuation. By studying the eigenmodes of the linearized system, we show that the systems adapt very specifically to the task they were trained to perform. Our work demonstrates that within physical bounds, a broad array of exotic behaviors in the dynamic regime can be obtained, allowing for a richer range of possible applications.