Autonomous Actuation of Zero Modes in Mechanical Networks Far from Equilibrium

A zero mode, or floppy mode, is a nontrivial coupling of mechanical components yielding a degree of freedom with no resistance to deformation. Engineered zero modes have the potential to act as microscopic motors or memory devices, but this requires an internal actuation mechanism that can overcome unwanted fluctuations in other modes and the dissipation inherent in real systems. In this Letter, we show theoretically and experimentally that complex zero modes in mechanical networks can be selectively mobilized by nonequilibrium activity. We find that a correlated active bath actuates an infinitesimal zero mode while simultaneously suppressing fluctuations in higher modes compared to thermal fluctuations, which we experimentally mimic by high frequency shaking of a physical network. Furthermore, self-propulsive dynamics spontaneously mobilize finite mechanisms as exemplified by a self-propelled topological soliton. Nonequilibrium activity thus enables autonomous actuation of coordinated mechanisms engineered through network topology.

Figure 1

Zero mode actuation in a bead–spring model. (a) Single mass at offset $(x,y)$, with HM in $x$ and IZM in $y$ [38]. (b) Comparison of $T=0.01$ exact marginal densities of $x$ and $y$ at $\tau =0$ (black) with $\tau =2$ densities from simulation (blue) and approximated from Eqs. (1) and (2) (red dashed). Histogram for $\tau =2$ from 50 000 samples, others by quadrature (Supplemental Material [61]). (c) Variance ratio $r(\tau )=⟨y^2⟩/⟨x^2⟩$ from simulations (markers) with approximations for small $\tau$ (solid lines) and large $\tau$ (dashed lines) from Eqs. (3) and (4): 5000 samples per marker; 95% CIs smaller than markers (Supplemental Material [61]).

Figure 2

Active noise actuates an IZM while suppressing HMs in a mechanical network (Supplemental Material Videos 1 and 2 [61]). (a) A network of unit length, unit stiffness springs is designed to contain exactly one IZM (arrows). White nodes are pinned. (b) Histograms of node positions in highlighted area of (a) when actuated by thermal noise (left) and correlated noise with $\tau =10$ (right) of strength $T={10}^{-3}$. The IZM is more cleanly actuated by correlated noise while HMs are suppressed. (c) Thermal-relative amplitude $⟨u_i^2{⟩}_{\tau }/⟨u_i^2{⟩}_{\tau =0}$ of the 21 lowest eigenvalue modes $u_i$ of (a) from numerical integration at $0\le \tau \le 10$, with $T={10}^{-3}$ fixed to maintain constant intensity $\overline{⟨v_{\alpha i}(0)v_{\beta j}(t)⟩}$. The IZM (red) is barely affected, while HMs (blues) diminish. Grey areas are estimated 95% CIs; data from 20 realizations to $t=2\times {10}^4$ with $\delta t={10}^{-4}$ at each of 20 values of $\tau$ (Supplemental Material [61]).

Figure 3

Shaking actuates the IZM of a stiff mechanical network (Supplemental Material Video 3 [61]). (a) Photograph of network with isolated IZM (arrows) constructed from plastic. Joints are freely rotating pins, with boundary nodes fixed in position. (b) Actuating the network by high frequency shaking mobilizes the IZM, shown by node positional histograms for the subregion indicated in (a) computed by particle tracking over 14 min with distances rescaled by mean edge length. (See Supplemental Material [61] for experimental methods.)

Figure 4

Self-propulsive activity spontaneously mobilizes a complex mechanism (Supplemental Material Video 5 [61]). (a) Mechanical SSH model [11, 27], which has a periodic mechanism comprising progressive flipping of the masses from right to left and back again. Black nodes are fixed, blue nodes are mobile. (b) Endowing a 21-node chain with self-propulsive activity spontaneously mobilizes the mechanism. The progression speed depends on the propulsion $v_0$, seen through the time-dependent offsets $\mathrm{\Delta }{x}_i$ of mobile nodes from their pinning points. All bonds of strength $k=10$ with ${\gamma }_0=1$, integrated at $\delta t={10}^{-5}$ from initial random perturbation.