Signal Propagation in Resettable Mechanical Logic

doi:10.21203/rs.3.rs-2440784/v1

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Signal Propagation in Resettable Mechanical Logic

https://doi.org/10.21203/rs.3.rs-2440784/v1

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Unconventional computing, such as mechanical¹ and microfluidic logic circuits², quantum gates³, and mechanical metamaterials⁴ create opportunities for embedded computation, which overcome the power⁵, package size, and environmental limitations of conventional electronics. Emerging micro-manufacturing capabilities⁶ with environmentally robust materials enable mechanical logic circuits miniaturization. Kinematically, bistable logic propagates binary signals through cascading gate displacement transitions. Energetically, the inter- and intra- node compliances are tuned for re-programmable signal propagation. Applications need computational architectures which integrate resettable signal propagation^7–10, logical operation^11–16, and signal storage^17–19. While many researchers explore aspects of these elements^{1, 20–23}, none consider energetic limits and propagation dynamics to evaluate and advance the field. Here, we show a generalized model and metrics, validated by experimental results, that enables the design of scale-independent, resettable, mechanical logic circuits. By studying propagation energy flows, we identified non-dimensional operating regimes in which signals propagate and resettable logic is possible. We provide deterministic design methods to evaluate future divergent topologies for displacement-based mechanical logic structures. Our results demonstrate the framework for designing densely integrated mechanical computation systems which harvest available ambient energy to propagate computational cascades. This logic responds to multi-dimensional environmental inputs and thus enables re-programmable, powerless, and embedded computation.

Physical sciences/Engineering/Mechanical engineering

Physical sciences/Mathematics and computing/Information technology

Many important applications require micro, nano, and molecular scale embedded logic. Conventional electronics are limited by their heat dissipation⁵, irreversibility²⁵, power consumption, total package size including energy storage, and environmental sensitivity to heat, chemicals and electromagnetic radiation^26,27. Three key insights enable mechanical computation as a viable alternative. First, miniaturization is now possible due to advances in micro and nano additive manufacturing and conventional microfabrication. Second, mechanical systems can access a wide range of material and metamaterial properties, which can be selected for environmental robustness in contrast to the limitations of silicon. Third, unlike electrical energy stored in batteries, elastic energy is innately stored in the pre-loaded structure itself, generating powerless self-propelled computational cascades. A swell of research is investigating the fundamentals of mechanical signal propagation^7–10, logical operation^11–16 and information storage^17–19, as well as generating new design topologies^9,10,24, materials, metamaterials, and performance metrics. However, while previous work shows signals propagating through logic gates without signal degradation⁹, resettable systems fundamentally limit the distance and dynamics of signal propagation. Forward-only systems rely on a unidirectional energy release by each element to maintain signal propagation. However, designing resettable systems necessitates symmetric energy profiles. Thus, it is important to consider the energetic limits and dynamic response of signal propagation in resettable mechanical logic.

In this work, we provide a scale-independent model and non-dimensional metrics, validated by multi-scale experimental results, to understand signal propagation in resettable, binary mechanical logic. Our model generalizes over a range of designs and scales by focusing on energy flow manipulation for logic signal transduction. Mechanical bi-stability mimics binary digital states and the requisite energy well equivalence ensures logic performs the same for both forward and reverse signals. The benefits of mechanical logic are magnified at smaller scales²³, where signal pulses of sufficient energy are more easily accessible or can be designed into miniature unpowered structures. We identify first law limits to signal propagation through bistable elements as well as AND gates and link the scale-independent parameters to simulated and measured performance limitations for logical operations. Future work building on these deterministic methods could enable mechanical computation with inputs keyed to specific, multi-domain environmental temperature, biological, chemical, magnetic, vibration, or acceleration signals with applications such as powerless rare-event detection on space missions^28,29, extreme temperature data storage³⁰, authentication tagging of high value items ^31,32, disposable health sensors³³, and smart materials³⁴.

We demonstrate a general, parameterized model for mechanical logic, independent of the actualized geometry or materials producing compliance (Fig. 1a). Bistable elements propagate a soliton along the element chain (Fig. 1b) by transitioning between unstable to stable energy states. However, pulse propagation can only occur in a limited design region within the full parameter space (Fig. 1c), thus parameter constraints guide the balance of intra- and inter- element compliance. Compliance enables the necessary bistability for reversible designs with equal energy at states 1 and 0, in ways that buckling cannot. See Methods for detailed descriptions, equations, and variable definitions.

Tuning the intra- and inter- element stiffnesses is essential to pulse propagation. Intra-element transverse stiffness, k_bt, accesses bistable behavior by modulating the potential energy well of the bistable element (Fig. 1d). The optimal transverse stiffness k_bt is dependent on compression and flexure performance and must avoid exceeding the negative stiffness generated by Y-axis compression. Inter-element coupling stiffness, k_bc, controls pulse and propagation behavior. The pulse stiffness ratio, ${\text{r}}_{\text{k}\text{p}\text{u}\text{l}\text{s}\text{e}}={\text{k}}_{\text{b}\text{c}}/{\text{k}}_{\text{b}\text{F}\text{m}\text{a}\text{x}}$, non-dimensionalizes the coupling stiffness ${\text{k}}_{\text{b}\text{c}}$ by the characteristic bistable element stiffness ${\text{k}}_{\text{b}\text{F}\text{m}\text{a}\text{x}}$. A weak pulse stiffness ratio generates short pulses across few nodes (95% pulse width Fig. 1c) while increasing the pulse stiffness ratio leads to a strong coupling linkage that extends the pulse length towards infinity.

The pulse potential energy oscillates between two equilibrium states, the unstable (UE) and stable (SE) energy locations in Fig. 1b. The two equilibria approach degenerate functional equivalency as the pulse stiffness ratio increases (Fig. 1c) and pulse characteristic energy rises without limit. The pulse will eventually settle into a local stable equilibrium if the potential energy step between each bistable element nodes in a chain, E_step, is 0 (Fig. 1d). Pulse propagation is only possible if the gate characteristic energy is successively decremented for each downstream node. Inter-node SE must be sufficiently low, E_step=E_min, for the soliton kinetic energy to reach the next SE state. With each nodal advance the pulse potential energy transforms into kinetic energy. For practical operation given manufacturing variability, E_step=1.5E_min ensures the step energy is consistently above the minimum threshold.

The equilibrium differential approximates the energy cost to advance the pulse by one node, normalized by pulse potential energy, e_ue (Fig. 1c). At high pulse stiffness ratios, the variation energy falls without limit relative to total pulse energy. Discrete soliton behavior is only clearly observable in the design region r_kpulse = [0.5–3.5] (Fig. 1c). The lower bound occurs when the maximum displacement between adjacent nodes is insufficient to pull the downstream element over its peak restoring force, halting the pulse propagation. The upper propagation bound is practically defined by increasing pulse width and stored energy, which lead to increasingly difficult pulse triggering. Pulse engineering balances tradeoffs between small pulse energy, short travel and compactness (low r_kpulse) versus high pulse energy, long travel and wide pulse widths (high r_kpulse).

The AND gate is composed of two input ports and one output port, each at a binary 1 or 0 state (Fig. 2a). Asynchronous inputs to the linkages (Fig. 2b-c) suppress output translation by rotation, generating the AND gate critical nonlinearity, cosine length attenuation. Synchronous input actuation suppresses rotation and directly displaces the output (Fig. 2d-e).

Signal propagation through the AND gate requires consideration of two dynamic output scenarios: the conjunct and disjunct transitions (Fig. 2b-e). The state label is a function of input1, input2 and the output. A disjunct transition changes from equivalent to opposite input states 000→010 or the degenerate cases 111→010, 111→100, 000→100. The conjunct transition changes from opposite to equivalent input states, such as 010→111. Signal propagation continues through the AND gate into the bistable element output chain for each dynamic case. Bistable elements downstream of the AND gate output backpropagate complex restrictions on the AND gate elasto-mechanics.

The AND gate contains two energy storage mechanisms–differential and compressive (see Methods section ‘AND gate design’). Differential motion between the two input ports acts as an energetic source in conjunct transitions. The AND gate nondimensional differential stiffness ratio r_kad is the differential stiffness k_ad normalized by the maximum differential stiffness possible in the structure given the pulse energy at the inputs. Differential motion acts as an energetic sink in disjunct transitions, limiting the AND gate efficiency. Compressive motion, subject to the nondimensional compressive stiffness k_ac, acts as a coupling linkage between the input and output ports. Impedance matching between the compressive stiffness and the output coupling linkage stiffness enables the pulse to propagate out of the AND gate.

Our Eulerian dynamics code simulates the AND gate, compliance link network, damping, and mass of each body. The two transitions (Fig. 2b-e) show the pulse structure, energy profiles and nodal locations. The Methods section details the design frameworks for the AND gate combined with bistable elements. Pulse energy cyclically oscillating between bistable element nodes and coupling linkages is visible in the pulse propagating out of the output port (Fig. 2c,e).

The AND-gate impedance plot (Fig. 2f) captures the competing dynamic between AND-gate k_{ac and downstream chain element impedances}. The conjunct and disjunct transitions define compressive stiffness limits, depending on the gate energy efficiency η_a, which is the ratio of output versus active input port pulse energy. The general operating region is defined by satisfying both the conjunct and disjunct requirements, where the AND gate impedance is tuned to pulse propagation. At high gate energy efficiencies, small energy margins exist for imperfect transmission mechanics, thus the operational space shrinks toward an impedance match with the output port coupling stiffness, k_ap3. The minimum disjunct transition stiffness needed to pull the output through pulse transition and leave the AND gate sets the lower bound. The maximum conjunction transition stiffness sets the upper bound, which still allows the pulse center to pull the input through transition and enter the AND gate. The gate differential stiffness k_ad controls the energy released by the AND gate during the conjunct transition, boosting incoming pulse energy to raise the upper bound for the pulse to enter, but plays little role in the minimum stiffness at the output which lets the pulse leave. The gate differential stiffness k_ad also drains energy from the overall theoretical peak efficiency, setting an upper bound at η_aMax. The optimal tradeoff, for all designs, occurs around r_kad = 0.15 with η_a = 63% gate efficiency. A ratio of k_ac/k_ap3 = 1.4 ensures an impedance match.

Using our deterministic design methodology, we simulated, designed, fabricated, and tested logic gates in the pulse propagating regime at both the macro and micro scales. First, we set the energy and size scale for each design, then determined topology and flexure behavior parameters. Tuning the bistable element transverse stiffness activates bistable behavior, while tuning the coupling stiffness ensures cascading propagation.

The linear bistable element chain design is bipolar, resettable, scalable from macro to micro length scales, and compatible with additive manufacturing approaches. Nodal displacement data was extracted from high-speed camera footage using tracking markers on the macro-scale chain (Fig. 3a) (Supplementary Video S1). The micro-scale two-photon polymerization printed chains demonstrate scalability and manufacturing capability (Fig. 3b and Supplementary Video S2). The actualized, generalized, and deformed bistable element design (Fig. 3c-e) consists of a rigid shuttle guided along the X-axis via a symmetric 4-bar parallelogram linkage, using flexure cross-pivots to approximate pin joints. The assembled layers include tuned kinematics, energetics, preload layers, and the coupling linkage. Figure 3f demonstrates agreement between the experimental and simulated time variant, node displacements, capturing both the primary bipolar signal, as well as secondary resonances, and vibrations. The force profile (Fig. 3g) confirms the actual device energetics closely match predicted performance, capturing the cross pivot kinematic modifications (see Methods).

The macro and micro AND gate logic chains (Fig. 4a-b) are fabricated in the unstressed state, energized by displacing the preload locks, and actuated by sequential input displacements (see Methods and Supplementary Videos S3 and S4). The instantiated, generalized, and deformed designs provide degenerate and symmetric bi-stability, critical for reversible logic (Fig. 4c-e). Discrete compliant elements linking rigid bodies ensure independent compliance tuning for development. In Fig. 4f, AND gate input 1 is pre-actuated, then input 2 actuation initiates the bistable node cascade, progressively displacing each downstream node based on tuned masses, stiffnesses, and damping ratios to adjust to the characteristic resonance for each node. Measurements confirm the AND gate geometric nonlinearity (Fig. 3g). Two tests tracked the output port displacement while cycling input 1 between limits, with input 2 fixed at 0 (red) and 1 (blue) successively. The 000→100 and 010→111 state transitions correctly avoid and engage the output port respectively. The ideal AND-gate behavior is compared to the propagation modified AND-gate behavior to show the scale of the output change.

Combining the AND gate with a simple NOT gate encapsulates functionally complete logic²⁴. Furthermore, we demonstrate micro-gate manufacturing using ceramics³⁶ with enhanced mechanical and thermal properties, compared to conventional polymers (see Methods and Supplementary Information).

We demonstrate first principles, non-dimensional modeling methods, scale independent performance metrics, and manufacturable designs for resettable, binary mechanical logic. Generalized energy flow models quantify the design space and parameter limits for mechanical logic, inclusive of diverse topologies and sources of compliance. We show actualized scale-independent, macro, and micro compliant designs to propagate mechanical logic signals. The scale-independent insight provides a roadmap to further miniaturization of mechanical logic elements, triggered by equivalently microscale environmental transduction energy, obviating typical power storage demands. Comparing experimental results from fabricated logic chains with Euler simulations validates nodal dynamics capabilities. Our kinematics and energetics modeling approach provides a paradigm framework identifying the limits of logic propagation, a result of resettable requirements. Connecting logic circuits to multi-domain input transducers enables embedded self-energizing computation for many important applications.

Capital letters represent non-dimensional (n.d.) values. The first subscript indicates the logic element: a for AND gate and b for bistable element. A second subscript indicates a for axial, c for coupling, and t for transverse. Variables are standard where L is length, K is stiffness, X and Y are displacements, F is force, and E is energy.

Bistable element design

A bistable element and coupling linkage form a repeating unit, comprised of kinematics and energetics layers. (Fig. S1a) The kinematics layer determines the motion of the bearing, while the energetics layer fine tunes the stiffness values. The kinematics layers, which constrain motion, are the same for each bistable element in a chain ensuring equivalent kinematic performance. Coupling linkages, designed with 6 DOF compliance, connect bistable elements without over-constraint. Design details are elaborated in depth in the supplemental information section ‘Layered design of bistable elements’.

Bistable element general model

The general model of a bistable element is parametrically defined by: lever arm characteristic length l_b, axial stiffness k_ba, y-axis displacement y_b, transverse stiffness k_bt, x-axis displacement x_b, and coupling stiffness k_bc (Fig. S1b). Degeneracy is required in the bistable states to ensure equivalent response to signal propagation of either value; symmetry is the simplest approach to ensuring degeneracy. Symmetry requires that the unstressed structure hold the orientation of the unstable equilibrium and be energized via preload y_b after fabrication. These conditions hold regardless of the means of compliance, structure scale, or bistable displacement orientation relative to the chain orientation since the coupling stiffness simply represents the compliance between relative node locations independent of the actual axis these motions. Thus, the general model represents a wide range of designs if equivalents for l_b, k_ba, y_b, k_bt, and k_bc are identified.

General model parameters are nondimensionalized (n.d.) to capture scale independent behavior, Eq. (1), where E is the n.d. energy, F is the n.d. force, K is the n.d. stiffness, and X_b, Y_b, R_b, are the n.d. displacements or lengths. Normalization parameters are generated through combinations of the axial stiffness and lever arm.

$$E= \frac{e}{\frac{1}{2}{k}_{ba}{l}_{b}^{2}}, F= \frac{f}{{k}_{ba}{l}_{b}}, K= \frac{k}{{k}_{ba}}, {X}_{b},{Y}_{b},{R}_{b}=\frac{{x}_{b,}{y}_{b},{r}_{b}}{{l}_{b}}$$

The bistable element potential energy, E_b, is a function of displacements X_b, Y_b, R_b, Eq. (2).

$${E}_{b}= {K}_{bt}{X}_{b}^{2}+{\left[{Y}_{b}-(2-4{R}_{b})(1-\sqrt{1-{X}_{b}^{2}})\right]}^{2}$$

The energy derivative is the force generated by the bistable element bearing on the node, Eq. (3).

$${F}_{bx}=\left[4{(1-2{R}_{b})}^{2}\left(1-\frac{1-\frac{{Y}_{b}}{2-4{R}_{b}}}{\sqrt{1-{X}_{b}^{2}}}\right)-{K}_{bt}\right]{X}_{b}$$

The force derivate is the bistable element bearing stiffness on the node, reduced to Eq. (4).

$${K}_{bx}=\left[4{(1-2{R}_{b})}^{2}\left(1-\frac{1-\frac{{Y}_{b}}{2-4{R}_{b}}}{{1-{X}_{b}^{2}}^{3/2}}\right)-{K}_{bt}\right]$$

A range of important parameters are defined from these expressions, including Eq. (5) the n.d. scale of the potential energy change from bistable equilibrium to unstable equilibrium, Δ_EbMax, Eq. (6) the n.d. minimum potential energy at bistable equilibrium, E_bMin, Eq. (7) the n.d. relative bistable element potential energy, ΔE_b, Eq. (8) the n.d. maximum restoring force exerted on the node by the bistable bearing, F_bxMax, Eq. (9) the n.d. nodal displacement associated with generating the maximum restoring force, X_bFMax, Eq. (10) the n.d. characteristic stiffness of the bistable profile, K_bFmax, and Eq. (11) the n.d. bistable equilibrium distance, X_be.

$\varDelta {E}_{bMax}= \frac{{\left(2{Y}_{b}-\frac{{K}_{bt}}{1-2{R}_{b}}\right)}^{2}}{\left(4-\frac{{K}_{bt}}{{\left(1-2{R}_{b}\right)}^{2}}\right)}$	(5)
${E}_{bMin}= {Y}_{b}^{2}-\varDelta {E}_{bMax}$	(6)
$\varDelta {E}_{b}= {E}_{b}-{E}_{bMin}$	(7)
${F}_{bxMax}={\left(1-2{R}_{b}\right)}^{2}{\left[{\left(4-\frac{{K}_{bt}}{{\left(1-2{R}_{b}\right)}^{2}}\right)}^{2/3}- {\left(4-\frac{2{Y}_{b}}{\left(1-2{R}_{b}\right)}\right)}^{2/3}\right]}^{3/2}$	(8)
${X}_{bFMax}=\sqrt{1-{\left(\frac{4-\frac{2{Y}_{b}}{\left(1-2{R}_{b}\right)}}{4-\frac{{K}_{bt}}{{\left(1-2{R}_{b}\right)}^{2}}}\right)}^{2/3}}$	(9)
${K}_{bFMax}=\frac{{F}_{bxMax}}{{X}_{bFMax}}$	(10)
${X}_{be}=\sqrt{1-{\left(\frac{4-\frac{2{Y}_{b}}{\left(1-2{R}_{b}\right)}}{4-\frac{{K}_{bt}}{{\left(1-2{R}_{b}\right)}^{2}}}\right)}^{2}}$	(11)

Propagation Regimes

The general model shows signal propagation only occurs in certain compliance regimes. In these regimes, a soliton can propagate along a chain of bistable elements (Fig. 1b). We define a decision hierarchy to simplify propagation regime discovery. First, choose the element energy and size scale via k_ba and l_b selection, which then determine explicit values for all following terms. Second, choose Y_b the scale of bistable energy storage, which determines the amount of bistable element axial compression, as well as the approximate angle of the lever arm at equilibrium, θ_b, via Eq. (12). Generally, Y_b ≈ 0.2 is an upper bound as this corresponds to ≈ 26° rotation, which is large scale motion for compliant bearings.

$${\theta }_{b}\approx {\text{cos}}^{-1}(1-Y/2)$$

Third, we chose the compliant bearing approach and set the effective cross-pivot rolling radius R_b to describe the scale of rolling, which occurs in the bearing during large range rotation (Supplemental Information section ‘Cross-pivot rolling radius’). Fourth, the transverse stiffness modulates the depth of the bistable potential energy well. Ideally, K_bt should approach 1 to keep all stiffnesses at the same scale and simplify fabrication, but bistability disappears when K_bt > K_btMax, defined in Eq. (13). A practical value of K_bt = K_btMax/2 balances the need for high bistable element stiffness to allow for manufacturing feasibility without suppressing bistability.

$${K}_{btMax}= 2Y\left(1-2{R}_{b}\right)$$

Fifth, the coupling stiffness is used to engineer the pulse properties, including the propagation speed, pulse size, and pulse energy. The pulse properties are governed by r_kPulse which describes the balance between the nondimensionalized coupling linkage stiffness, K_bc, with the nondimensionalized characteristic stiffness of the bistable element profile, K_bFMax. The coupling stiffness captures the ability of nodes to affect neighbors while the characteristic stiffness captures the ability of nodes to resist neighbors. r_kPulse also captures a lower limit to pulse propagation which occurs when the coupling stiffness is too low to pull the rising node over the peak restoring force. A simple force balance is created with the edge nodes approximated at bistable equilibria, even though the finite compliance of the chain always results in the node being closer than this. The rising node n is then controlled by the balance of the three dominant forces, i) the coupling stiffness dragging it up from the prior n-1 node at the positive equilibrium, ii) the bistable restoring force pulling it back down towards the initial negative equilibrium, and iii) the coupling stiffness of the following n + 1 node at the negative equilibrium, shown in Eq. (14).

$$\stackrel{n-1}{\underset{\left({x}_{be}+ {x}_{bFMax}\right){k}_{bc}}{⏞} }- \stackrel{n+1}{\underset{\left({x}_{be}- {x}_{bFMax}\right){k}_{bc}}{⏞} }- \stackrel{n}{\underset{{F}_{bxMax}}{⏞} }\ge 0$$

The simplified expression compares the coupling stiffness and characteristic profile stiffness in Eq. (15), producing a simple constraint r_kPulse > ½ to propagate the pulse.

$$2{k}_{bc}-{k}_{bFMax}\ge 0$$

Pulse propagation

The minimum energy step is used to engineer the pulse chain for pulse propagation. The node ‘energy’ is calculated as the potential energy of a pulse in the unstable equilibrium centered on the node, assuming no energy decrement across the pulse. This allows the measurement to be a spot analysis of the pulse characteristics at the node. The pulse chain energy profile is engineered by calculating the energy and minimum energy step of the n’th nodes. The n + 1 nodal energy decrements from the n node by the minimum energy step, scaled by a factor around 1.5x to ensure fabrication variation does not halt pulse operation. The energy of each node is tuned by adjusting k_a for the bistable element, while holding the nondimensional parameters constant. This process tunes the energy for each node, resulting in bistable elements with the same kinematics but successively less potential energy. The process is repeated with a calculation of the minimum energy step for the n + 1 node being used to determine the energy of the n + 2 node and so on.

Pulse propagation – steady state velocity

A pulse propagates along a bistable element chain at an approximately steady state velocity, determined by the energy flow from the pulse potential energy per nodal step into energy dissipation e_d, where the unknown is the time increment for the pulse to advance one node, dt. The velocity of the i’th node in the profile [d_xp]_i/dt, and the distance each node travels [d_xp]_i, are used to calculate the dissipated energy, Eq. (16).

$${e}_{d}={b}_{b}{\sum }_{i}\left(\frac{{\left[d{x}_{p}\right]}_{i}}{dt}{\left[d{x}_{p}\right]}_{i}\right)={e}_{step}$$

The known energy decrement between nodes, e_step, leads to the time increment between nodes, Eq. (17).

$$dt=\frac{{b}_{b}}{{e}_{step}}{\sum }_{i}{\left({\left[d{x}_{p}\right]}_{i}\right)}^{2}$$

AND gate design

Figure S3a shows the detailed structure of the AND gate, composed of several linkages, which freely rotate relative to one another. The l_a1, l_a2 linkage from the input ports to the first rotary joint is structured in a V to allow for the two input ports to expand away from one another along the Y-axis. This ensures the structure does not lock when it rotates, since the rotation forces l_a1 to increase slightly when the input ports are guided by linear shuttles. Fig. S3b defines and labels the geometry. The gate inputs and output are connected to bistable elements, which properly constrain the AND gate shuttle motion. The bistable elements are connected on separate levels, however right-angle linkages could maintain an alternative planar design. Rigid linkages provide high stiffness to reject unwanted state changes via back-driving. Compressive compliance at the output interface retains the basic structure of bistable element nodes separated by compliant couplings to enable propagation. The AND gate kinematics are initially tuned based on the assumption that the two inputs reach the full equilibrium location, however these displacements are attenuated by finite compliance of the structure and coupling linkages. The central AND gate lever arms are modified beyond the static operation constraint to compensate for this effect and ensure full output propagation despite incomplete input state transitions.

A five-dimensional design space (three geometric and two stiffness parameters) represents the AND gate, where only a small regime enables pulse propagation. Kinematic analysis ensures the rotational nonlinearity produces AND operator performance in ideal conditions. The basic AND gate dimensional parameters are k_ad, k_ac, l_a1, l_a2 and l_a3. An additional parameter captures the energy transfer behavior of the AND gate, r_aE, which describes the ratio of the pulse energy out of the gate, e_aOut, compared to the theoretical maximum energy output, e_aOutMax. Eq. (18) shows the nondimensionalized (n.d.) terms. The n.d. differential motion stiffness, K_ad, is normalized to the compressive stiffness, simplifying elasto-mechanic calculations. The n.d. compressive stiffness, K_ac, is normalized to the coupling linkage stiffness of the output port, k_acp3, as this output port stiffness determines scales the requirements on the compressive stiffness. All displacements in the AND gate, X_a = [x_ap1, x_ap2, x_ap3, x_ap3o, x_ad, x_ac, x_as] and linkage lengths L_a = [l_a1, l_a2, l_a3] are normalized by the equilibrium displacement of the bistable elements, x_be.

$${K}_{ad}=\frac{{k}_{ad}}{{k}_{ac}}, {K}_{ac}=\frac{{k}_{ac}}{{k}_{acp3}}, {X}_{a}, {L}_{a}=\frac{{x}_{a}, {l}_{a}}{{x}_{be}}, {r}_{aE}=\frac{{e}_{aOut}}{{e}_{aOutMax}}$$

The lever arm lengths, l_a1, l_a2, and l_a3, are set by the requirements to ensure AND gate nonlinear behavior as well as the design maximum values set for the two linkage angles, θ_aaMax, θ_abMax. The stiffness and energy terms, K_ad, K_ac, and r_aE, are set by the requirements of pulse propagation. The AND gate nonlinear response to the two inputs (p1 and p2) is captured in the output port (p3) uncompressed displacement, x_ap3o.

The input motion X_ap1, X_ap2, is redefined as differential, X_ad, and shared, X_as, terms, shown in Eq. (19) and Eq. (20).

$${X}_{as}=\frac{{X}_{ap1}+{X}_{ap2}}{2}$$

$${X}_{ad}={X}_{ap2}-{X}_{ap1}$$

Eq. (21) determines the compression displacement X_ac by comparing the output motion, X_ap3 to the uncompressed displacement, X_ap3o.

$${X}_{ac}={X}_{ap3}-{X}_{ap3o}$$

Eq. (22) demonstrates the uncompressed displacement, calculated geometrically, and determined by the differential and shared motion.

$${X}_{ap3o}={X}_{as}+{L}_{a2}\left(\sqrt{1-{\left(\frac{{X}_{ad}}{{L}_{a1}}\right)}^{2}}-1\right)+{L}_{a3}\left(\sqrt{1-{\left(\frac{{L}_{a2}{X}_{ad}}{{L}_{a1}{L}_{a3}}\right)}^{2}-1}\right)$$

The AND gate nonlinear behavior constrains the output port to the same location, regardless of the input port states such that the 000 and 010 states are equivalent. Three constraints are supplied to set the three AND gate linkage sizes when the nonlinear behavior constraint is combined with the design maximum values set for the two linkage angles, θ_aaMax, θ_abMax. The requirement on the horizontal linkage L_a1 is simply constrained by the θ_aa rotation angle, shown in Eq. (23).

$${L}_{a1}= \frac{2}{\text{s}\text{i}\text{n}{\theta }_{aaMax}}$$

The requirements on L_a2 Eq. (24) and L_a3 Eq. (25) capture the nonlinear complexity.

$${L}_{a3}=\frac{\frac{1}{\sqrt{\frac{1}{{L}_{a1}^{2}\left(\sqrt{\frac{1}{4}-{L}_{a1}^{-2}}-\frac{1}{2}\right)+1}+{L}_{a1}^{2}-2}}+\frac{{\left[\text{cos}{\theta }_{abMax}\right]}^{-1}}{\frac{1}{{L}_{a1}\left(\sqrt{\frac{1}{4}-{L}_{a1}^{-2}}-\frac{1}{2}\right) }+{L}_{a1}}+\text{tan}{\theta }_{abMax}}{\text{tan}{\theta }_{abMax}[2+{L}_{a1}sin{ \theta }_{abMax}]}$$

$${L}_{a2}=\frac{1}{2} \sqrt{\frac{1-2{L}_{a3}}{{L}_{a1}^{2} \left( \sqrt{\frac{1}{4}-{L}_{a1}^{-2}-\frac{1}{2}}\right)+1}+{L}_{a3}^{2}}+\frac{1}{2}-\frac{{L}_{a3}}{2}$$

The next design steps define gate elastomechanics based on the two internal stiffnesses. The force of the AND gate on each port’s node, f_a = [f_ap1, f_ap2, f_ap3] is defined in each case as positive in the + x-axis direction and nondimensionalized to F_a = [F_ap1, F_ap2, F_ap3] by the characteristic gate force, k_ac x_be, as shown in Eq. (26).

$${F}_{a}=\frac{{f}_{a}}{{k}_{\alpha }{x}_{be}}$$

Differential stiffness between the input ports generates equal and opposite forces on the input gates, while compression stiffness generates a return force, modified by gate kinematics, captured by the term α_aF as shown in Eq. (27).

$${\alpha }_{aF}=\frac{{L}_{a2}{X}_{ad}}{{L}_{a1}^{2}}\left[{\left[{\left(\frac{{L}_{a3}}{{L}_{a2}}\right)}^{2}-{\left(\frac{{X}_{ad}}{{L}_{a1}}\right)}^{2}\right]}^{-\frac{1}{2}}+{\left[1-{\left(\frac{{X}_{ad}}{{L}_{a1}}\right)}^{2}\right]}^{-\frac{1}{2}}\right]$$

Eq. (28) is the nodal force at the first input port, F_ap1.

$${F}_{ap1}={K}_{ad}{X}_{ad}+\left(\frac{1}{2}+ {\alpha }_{aF}\right){X}_{\alpha }$$

Eq. (29) is the nodal force at the second input port, F_ap2.

$${F}_{ap2}= -{K}_{ad}{X}_{ad}+\left(\frac{1}{2}-{\alpha }_{aF}\right){X}_{ac}$$

Eq. (30) is the nodal force at the output port, F_ap3.

$${F}_{ap3}= -{X}_{ac}$$

Propagation through the AND gate requires the definition of several stiffness and energetics terms. The maximum possible differential stiffness, k_adMax, is defined by the total energy flow of the pulse into charging the differential mode accounting for the energy lost to dissipation during the traversal of the pulse between nodes, e_aDis, and the minimum value of the pulse energy at the inputs, e_aIn, as shown in Eq. (31).

$${k}_{adMax}=\frac{{e}_{aIn}-{e}_{aDis}}{2{x}_{be}^{2}}$$

The differential stiffness ratio captures the extent to which the pulse energy is routed to the differential mode as shown in Eq. (32). Figure 2f shows that high efficiency AND gates exist in the r_kad = 0.15 regime, where only 15% of the pulse energy is drained to the differential mode.

$${r}_{kad}=\frac{{k}_{ad}}{{k}_{adMax}}$$

The energy ratio, r_aE, is the ratio of the output energy over the available energy after accounting for losses, and it determines the pulse energy at the gate output, which is generally set around 0.8 to ensure propagation despite fabrication variation, and the theoretical maximum energy output of the gate, e_aOutMax, calculated as shown in Eq. (33). The theoretical maximum energy output considers energy dissipation and storage in the differential mode. This theoretical value provides an energetics-based system constraint, but as shown in Fig. 2f, it is often not the limiting factor.

$${e}_{aOut}={r}_{aE}\stackrel{{e}_{aOutMax}}{\underset{({e}_{aIn}-{e}_{aDis}-2{k}_{ad}{x}_{be}^{2} }{⏞} }$$

The overall gate efficiency η_a is the ratio of the gate energy output over input, and thus will always be a lower than r_aE. Kinematic modifications are necessary to ensure propagation, as the preceding analysis assumes all ports are at the bistable equilibrium locations. Tuning a parameter to overcompensate the gate output ensures the incomplete input transitions still produce sufficient output displacement to drive pulse propagation. In this work, l_a1 is the preferred term for modification and is slightly reduced to amplify the gate response to input motions. All nodes in the profile are then displaced beyond the threshold x_trans/x_be ≈ 0.98 to add less energy to the profile than lost by advancing by a node. Propagation switches from energetically unfavorable to favorable once the output port passes through this threshold. The AND gate static force balance captures the unstable equilibrium when the output port reaches the transition displacement, accounting for the input ports being connected to the tail end of a pulse profile rather than locked at equilibrium displacements. Eq. (34) shows where functions f_bxp1(…), f_bxp2(…) are the nodal bistable element forces at the input 1 and 2 locations; f_ap1(…), f_ap2(…) and f_ap3(…) are functions calculating the nodal AND gate forces at the input 1, 2 and output; k_cp1 and k_cp1 are the coupling stiffnesses of the linkages to the input 1 and 2 ports; The coupling stiffness scaling terms account for the extended chain of nodes connected to the port node. We assume the output port node experiences no forces from the chain, since forces on the output port switch direction, passing through 0 during transition. The modified l_a1 term is extracted from the system of equations, producing the value used in the fabricated AND gate to ensure pulse propagation.

$given {x}_{p3}={x}_{trans}$ ${k}_{cp1}\left(-{x}_{be}-{x}_{p1}\right)(1-\frac{{k}_{cp1}}{2{k}_{cp1}+{k}_{bxp1}{ x}_{be}}+{f}_{bxp1} {x}_{p1}+{f}_{ap1}\left({x}_{p1}, {x}_{p2}, {x}_{p3}, {l}_{a1trans}\right)=0$ ${k}_{cp2}\left({x}_{be}-{x}_{p2}\right)(1-\frac{{k}_{cp2}}{2{k}_{cp2}+{k}_{bxp2}{ x}_{be}}+{f}_{bxp2} {x}_{p2}+{f}_{ap2}\left({x}_{p1}, {x}_{p2}, {x}_{p3}, {l}_{a1trans}\right)=0$ ${f}_{ap3 }\left({x}_{p1}, {x}_{p2}, {x}_{p3}, {l}_{a1trans}\right)=0$ $find {x}_{p1}, {x}_{p2}, {l}_{a1trans}\to {l}_{a1trans}$ (34)

Circuit synthesis

The bistable element chain Fig. 3a and AND gate chain Fig. 4a used the following parameters: Y_b = 0.1, K_bt = 0.065, K_bc = 0.06, initial k_ba = 37643 N/m, l_b = 35mm, r_EstepAct = 1.5, which produced a 1.96% energy drop per node. The AND gate was designed with maximum angular deviation for θ_a,θ_b = 10°, r_kad = 0.15 which produced k_ad = 154N/m, k_ac = 1876N/m, l_a1 = 108.6mm, l_a2 = 322.2mm, l_a3 = 322.2mm.

Fabrication

Our sequential printing method enables additive manufacturing of large and long overhanging features without support materials/structures using two-photon lithography (TPL). To print the model shown in Fig. S2, the overhanging geometries are divided into shorter substructures to reduce unwanted layer movement while printing and to increase the precision and quality of fabricated structures. The sub-structures are printed in an order ensuring the last printed feature is connected to a previously grounded/stable structure. We limited the maximum layer height of substructures to 12µm to avoid obscuring the TPL laser beam in the subsequent printing of features located on the lower layers.

Data availability

Data and designs are available from the corresponding authors upon reasonable request.

Code availability

The codes that support the findings of this study are available from the corresponding authors upon reasonable request.

Acknowledgements We gratefully acknowledge funding from the LLNL Laboratory Directed Research and Development Program (19-ERD-018 ) and AFOSR (FA9550-22-1-0008) for supporting this research work. This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344, IM number LLNL-JRNL-843587.

Author contributions R.P., A.P., and J.H. conceived the research problem. A.F. and L.B. fabricated and tested the micro and macro scale logic gates. F.S., R.P. and H.J. performed the finite-element simulations and analyzed the data. R.P., F.S., and H.J. wrote the manuscript and supplementary information. J.C., A.P., and J.H. managed the research work. All authors discussed the results and commented on the manuscript.

Competing interests The authors declare no competing interests.

Additional information

Supplementary information The online version contains supplementary material available at doi.org/…

Correspondence and request for materials should be addressed to John Cortes ([email protected]) or Hilary Johnson ([email protected])

Peer review information Nature thanks X.X. and other, anonymous, reviewer(s) for their contribution to the peer review of this work.

Reprints and permissions information is available at http://www.nature/reprints.

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SupplementaryInformationv3.docx
Supplementary Information
SupplementaryVideoS1.mp4
Macro bistable element propagation
SupplementaryVideoS2.mp4
Micro bistable element propagation
SupplementaryVideoS3.mp4
Macro AND gate propagation
SupplementaryVideoS4.mp4
Micro AND gate propagation

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Signal Propagation in Resettable Mechanical Logic

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Abstract

Figures

Introduction

Characterizing Bistable Element Propagation

Characterizing And Gate Propagation

Signal Propagation Experimental Validation

Conclusion

Methods

Bistable element design

Bistable element general model

Propagation Regimes

Pulse propagation

Pulse propagation – steady state velocity

AND gate design

Circuit synthesis

Fabrication

Declarations

References

Additional Declarations

Supplementary Files

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\(\varDelta {E}_{bMax}= \frac{{\left(2{Y}_{b}-\frac{{K}_{bt}}{1-2{R}_{b}}\right)}^{2}}{\left(4-\frac{{K}_{bt}}{{\left(1-2{R}_{b}\right)}^{2}}\right)}\)	(5)
\({E}_{bMin}= {Y}_{b}^{2}-\varDelta {E}_{bMax}\)	(6)
\(\varDelta {E}_{b}= {E}_{b}-{E}_{bMin}\)	(7)
\({F}_{bxMax}={\left(1-2{R}_{b}\right)}^{2}{\left[{\left(4-\frac{{K}_{bt}}{{\left(1-2{R}_{b}\right)}^{2}}\right)}^{2/3}- {\left(4-\frac{2{Y}_{b}}{\left(1-2{R}_{b}\right)}\right)}^{2/3}\right]}^{3/2}\)	(8)
\({X}_{bFMax}=\sqrt{1-{\left(\frac{4-\frac{2{Y}_{b}}{\left(1-2{R}_{b}\right)}}{4-\frac{{K}_{bt}}{{\left(1-2{R}_{b}\right)}^{2}}}\right)}^{2/3}}\)	(9)
\({K}_{bFMax}=\frac{{F}_{bxMax}}{{X}_{bFMax}}\)	(10)
\({X}_{be}=\sqrt{1-{\left(\frac{4-\frac{2{Y}_{b}}{\left(1-2{R}_{b}\right)}}{4-\frac{{K}_{bt}}{{\left(1-2{R}_{b}\right)}^{2}}}\right)}^{2}}\)	(11)