Harnessing Oscillatory Dynamics for Reprogrammable Mechanical Functionality

TL;DR

The work tackles the challenge of truly reprogrammable mechanical systems by moving beyond fixed bistable elements to a hybrid approach in which symmetry breaking is achieved kinematically via a global boundary actuation. Arrays of oscillating pendula are made into reprogrammable mechanical bits whose state is dictated solely by the timing of the actuation window $\Delta t_w$ relative to each pendulum’s natural period $T_i$, enabling rapid writing of arbitrary configurations. By leveraging desynchronization, all $2^N$ configurations become accessible, enabling functions such as a reprogrammable linear spring and a mechanical piano that plays user-defined note sequences within a few oscillation periods; a Python-based algorithm further optimizes $\Delta t$ and the pendulum periods to minimize programming time. The framework generalizes to other oscillatory systems and holds promise for scalable, dynamics-driven reprogrammable matter across mechanical, fluid, chemical, and electronic domains.

Abstract

A long-standing goal in the field of "mechanical computing" is the creation of truly reprogrammable mechanical structures, where the function of each unit can be dynamically defined, modified, and accessed on demand, much like rewriting data on a hard drive. Prior efforts have largely focused on bistable building blocks, which mimic binary states, but robust and efficient methods for programming large arrays of such units remain limited. In this study, we introduce a new approach for defining and reconfiguring the state of mechanical bits. Specifically, we investigate arrays of pendula whose boundary conditions break symmetry, effectively transforming them into mechanical bits. When actuation times are short compared to the natural oscillation periods, the state of each pendulum can be controlled solely by adjusting the timing of global boundary conditions. This mechanism enables rapid reprogramming, arbitrary information writing, and even the construction of a "mechanical piano" capable of generating user-defined note and chord sequences within only a few oscillation cycles. Because it integrates seamlessly with diverse functionalities, our strategy establishes a scalable framework for reprogrammable mechanical systems and can be readily generalized to other oscillatory systems like membranes or beams.

Figures (16)

• Figure 1: Mechanical bit.a) Schematic of the globally convex energy landscape of a freely oscillating pendulum. b) Schematic of the energy landscape when the indenter is lowered, breaking the symmetry of the system. c) Experimental time evolution of the pendulum angular motion (left axis) together with the indenter displacement $u_{y}$ (right axis). The pendulum has a natural period $T = 0.72$ s. Insets show snapshots of the pendulum locked in state $s = 0$ at $t=7$ s and in state $s = 1$ at $t=17$ s.
• Figure 2: State programming via desynchronization.a) Temporal evolution of $s(t)$ for three pendula with $T_{\mathrm{min}} = 0.59$ s, $T_{\mathrm{mid}} = 0.61$ s, and $T_{\mathrm{max}} = 0.64$ s. Gray windows indicate intervals $\Delta t_w = 0.1$ s during which all three pendula occupy the desired state. b) Experimental temporal evolution of the angular positions $\theta(t)$ of the three pendula. c) Experimental setup of the reprogrammable linear spring. The system is shown in state 101 with three elastomeric spheres positioned on the left side of each pendulum. d) Experimental force–displacement curves (continuous lines) for three hemispherical shells with thickness-to-radius ratio $d/R = 1$, 0.3, and 0.2 under pendulum indentation. Curves are obtained by taking the mean of three tests. Dashed lines indicate linear fits. e) Experimental force–displacement curves obtained by further lowering the indenter with the system programmed in all 8 states (continuous lines). Predictions from the summed responses of the indented shells are shown as dotted lines. f) Numerical prediction of $\Delta t_{\max} = \max(\Delta t)$ as a function of the pendula natural periods for $\Delta t_w = 0.1$ s. Blue and red markers indicate the set used in a) and the optimal set that minimizes $\Delta t _{max}$, respectively.
• Figure 3: Playing five-notes songs.a) Diagram of all possible transitions between the five states, with colored arrows indicating the transition times $\Delta t$. b) The system sorted into state 10000 by lowering the indenter. c) A further 2 mm downward displacement of the indenter brings the pendulum in state $s=1$ into contact with its switch, switching on the red LED. d) Chorus of "Oh When the Saints" (34-notes sequence). e) Experimental demonstration of a five-note sequence from "Oh When the Saints". Top: notes and rest times; middle: indenter displacement over time with snapshots showing the pendula activating the corresponding LEDs; bottom: pendulum angles $\theta$ over time.
• Figure 4: Accessing all 32 states.a) Transition times $\Delta t$ for the 32 distinct state changes, using pendulum periods chosen to minimize the maximum transition time $\Delta t_{\max}$. b) Dependence of $\Delta t_{\max}$ on the observation window $\Delta t_{\mathrm{w}}$. c) Chorus of "Smoke on the Water" (12-chords sequence). d) Experimental demonstration of a three-chords sequence from "Smoke on the Water". Right axis: indenter displacement over time (black lines). Left axis: pendulum angles $\theta$ over time (colored lines). Insets: snapshots from Supplementary Video S5.
• Figure S1: Design of the pendula.a) Schematic of a pendulum used for laser cutting. b) Experimentally measured periods $T$ for pendula of varying lengths $L$ (circular markers). The dashed line shows the linear fit described by Eq. (\ref{['eq1']}).