Mechanical signaling cascades

Mechanical computing has seen resurgent interest recently owing to the potential to embed sensing and computation into new classes of programmable metamaterials. To realize this, however, one must push signals from one part of a device to another and do so in a way that can be reset robustly. We investigate the propagation of signals in a bistable mechanical cascade uphill in energy. By identifying a penetration length for perturbations, we show that signals can propagate uphill for finite distances and map out parameters for this to occur. Experiments on soft elastomers corroborate our results.

Figure 1

(a) The general shape of a bistable potential. The black line represents a symmetric potential with barrier height $E$. The blue (light gray) line represents an asymmetric potential biased to the left, where $\mathrm{\Delta }$ is the potential difference between the two states. We measure the barrier height $E$ with respect to the higher minimum. (b) An elastic bistable unit at equilibrium. The linear springs each have stiffness $k$. The torsional spring located at the point mass $m$ (filled circle) has torsional modulus $s$ and equilibrium angle ${\theta }_0$. The potential minima for the bistable units are located at $x=\pm d$. The potential barrier is located at $x=0$.

Figure 2

A finite length wire in the left (no signal) state. Each bistable element is at the $x=-d$ potential minimum. When the leftmost point mass is pushed over the energy barrier and into the $x=+d$ energy minimum, the interaction spring connecting adjacent point masses allow that transition to propagate along the wire. If the rightmost mass moves to the right, the entire wire is in the right (signal) state, and we say that the signal is fully propagated along the wire. If only a portion of the bistable elements transition to the $x=+d$ energy minimum, we say that the signal only propagated a finite distance.

Figure 3

For all three plots, the wire length parameters are $h=1, a=1$, and $d=1/4$.(a) Scaling relationship between the potential barrier height $E$ and the length of the wire when varying the beam stiffness. The beam stiffness $b$ ranges from 0 to 5. The interaction spring stiffness $k$ is set to 1, and the torsional modulus $s$ is set to 0. Each boundary point (white circle) corresponds to a simulated wire with a specific choice of length $L$ and characteristic energy $E$. (b) Scaling relationship between the potential difference between minima $\mathrm{\Delta }$ and the length of the wire when varying the torsional modulus. The torsional spring stiffness $s$ ranges from 0 to 0.3. The interaction spring stiffness $k$ and beam stiffness $b$ are both set to 1. Each boundary point (white circle) corresponds to a simulated wire with a specific choice of length $L$ and torsional modulus $s$. (c) Scaling relationship between the interaction spring stiffness and the length of the wire. The interaction spring stiffness $k$ ranges from 0 to 5. The beam stiffness $b$ is set to 1 and the torsional modulus $s\mathrm{is}$ set to 0. Each boundary point (white circle) corresponds to a simulated wire with a specific choice of length $L$ and interaction spring stiffness $k$.

Figure 4

3D models of wire components. (a) The initial state of a wire with symmetric bistable beam elements. (b) Wire slotted into a holding device (frame) that compresses all bistable beam elements to buckle them uniformly. (c) Interaction springs of varying thickness. Different spring stiffnesses are achieved through changing the thickness of the spring. Individual interaction springs are printed specifically for measuring their stiffness.

Figure 5

A wire before (a), during (b), and after (c) a signal is fully propagated through manual displacement of the first bistable beam in the wire. Comparison between a wire with symmetric (d) and asymmetric (e) bistable beam elements when mounted in the frame. (f)–(h) Recording the force-displacement data for a single bistable beam. (i) The interaction spring mounted and ready for force-displacement measurements. All beams shown have an end-to-end distance of 24 mm. All interaction springs have a rest length of 11.2 mm. The length of the full wires shown in (a)–(e) is approximately 10 cm.

Figure 6

Comparison between signal propagation experiments and numerical simulations. (a) Qualitative comparison between simulations and experiments for varying interaction spring stiffness. The wire parameter values used for simulations are $h=0.012m, d=0.0055m, a=0.0112m, b=120N/m$, and $s=0N/m$. (b) Direct comparison between simulations and experiments for varying minima difference $\mathrm{\Delta }$ of the bistable beam potential. In simulations, this was done by varying the torsional stiffness on each bistable element. In experiments, this was done by changing the angle of the beam at its fixed endpoints [see Fig. 5 for example]. The wire parameter values used for simulations are $h=0.012m, d=0.0043m, a=0.0112m, b=230N/m$, and $k=30N/m$.

Figure 7

Sending a signal along a wire of length 20 with $h=1, a=1$, and $d=1/4$. The $x$ axis plots the run time for the simulation. For each choice of run time, we vary the interaction spring stiffness $k$ from 0 to 5 and plot the displacement of the final beam in the wire. A displacement of $-0.25$ (purple/dark gray region) corresponds to finite propagation and a displacement of 0.25 (light blue/light gray region) corresponds to full propagation.

Figure 8

Re-creations of the plot from Fig. 3 with increasing values of the mass. The interaction spring stiffness $k$ ranges from 0 to 5. The beam stiffness $b$ is set to 1, and the torsional modulus $s$ is set to 0. A displacement of $-0.25$ [purple (dark gray) region] corresponds to finite propagation, and a displacement of 0.25 [light blue (light gray) region] corresponds to full propagation. For all plots, the coefficient on the damping term is equal to 1. The first plot with $\text{mass}=0$ is what was used in Fig. 3.