Monolithic binary stiffness building blocks for mechanical digital machines

Abstract We introduce two essential building blocks with binary stiffness for mechanical digital machines. The large scale fully compliant mechanisms have rectilinear and rotational kinematics and use a new V-shaped negative stiffness structure to create two extreme states of stiffness by static balancing. The use of a mechanical bistable switch allows us to toggle between near-zero-stiffness and high-stiffness states, effectively turning off and on stiffness. A stiffness reduction of 98.8 % and 99.9 % is achieved for linear and rotary motion over a range of 13.3 % (20 mm) and 0.4 rad (23 ∘ ) respectively. Stiffness states can be reversibly changed by toggling the mechanical switch, or irreversibly by actuating the main stage. These binary stiffness mechanisms could set the stage for a new type of mechanical logic, adaptive and programmable metamaterials and other types of digital mechanical devices. Practical mechanical digital machines and materials require miniaturized and easily micro-manufactured components. We have therefore carefully considered scalability by integrating all required structures into a planar and monolithic architecture. This allows miniaturization and fabrication with conventional surface-micro-machining and additive manufacturing such as photolithography, two-photon lithography and fused deposition modeling.


Introduction
Mechanisms with variable stiffness have the ability to change their stiffness over a range of values. This can be done actively and passively in a multitude of different ways. For example, electrostatically, piezoelectrically, thermally and mechanically [1]. They are commonly used in microelectromechanical system (MEMS) [1], vibration isolation [2] and variable impedance actuators [3] for example for human friendly robots [4] and robot fingers [5].
More recently variable stiffness is used to create metamaterials where controlled changes in local stiffness enable adaptive bulk properties. Examples are architectured materials with multiple Poisson's ratios [6], variable stiffness [7,8] as well as robotic materials with programmable properties [9][10][11]. These materials exploit variable stiffness created by geometry [6], phase * Correspondence to: Dept. Precision and Microsystems Engineering, The Netherlands.
Instead of continuously variable stiffness mechanisms, binary stiffness mechanisms may be used. A binary stiffness mechanism can only change its stiffness between two different values that ideally have an infinite ratio. Two extremely different states of stiffness can be created by static balancing [12,13]. It uses negative stiffness caused by preloading to counteract a positive stiffness elsewhere in the mechanism. Even though the upper limit of the stiffness is limited by bulk properties, the ratio between the upper and lower limit can be infinite if the lower limit is reduced to zero stiffness [14]. Zero stiffness can be achieved if negative stiffness is designed to be equal but opposite to the positive stiffness. Different techniques may be used such as preloading in postbuckling [15] or up to the critical buckling load [16], zerofree-length springs [17,18] and curved surfaces [19,20], see [21,22] for an extensive overview. Binary states of stiffness can thus be achieved by enabling and disabling static balancing by engaging and disengaging preloading, effectively turning the stiffness off and on.
Programmable zero stiffness based on multistable mechanisms [23][24][25] has been previously reported in [26]. However, the reported device is not of a digital nature, since the flexible programming inputs are analog and require a sustained holding https://doi.org/10.1016/j.eml.2020.101120 2352-4316/© 2020 Elsevier Ltd. All rights reserved. force to create both low or high stiffness. In addition, the zero stiffness (and zero force) monostable behavior is obtained in prebuckling which limits the low stiffness to small deflections, i.e. the zero-stiffness is instantaneous.
If binary stiffness mechanisms are used in large numbers, they may approach continuous properties and enable new types of digital mechanical machines and metamaterials [27]. Examples could include novel computing systems based on mechanical logic that could potentially be build on the molecular level [28,29] and provide more robustness in extremely harsh environments such as high radiation, because they lack a significant electromagnetic signature [30,28,31].
In most existing mechanical logic systems each Boolean state is associated with a predefined displacement of the mechanism, such as the location of a dial. Examples are rigid body logic [32], multi-stable micro-flexural additively manufactured logic gates [31] and micro-flexural NOT and AND gates [33,34]. All are examples of static logic, however in most modern computing systems dynamic logic is used because of its superior performance [35]. In dynamic logic a clocked signal is used to evaluate the state after input signals are applied, as opposed to static logic where the output is immediately available. An example of mechanical dynamic logic is rod logic [36]. It uses sliding rods that block or unblock each other to create logic states in terms of potential sliding motion evaluated by an oscillating mechanical signal [28]. We can regard the blocked and unblocked configurations as states of high and low stiffness such that one rod logical element is essentially a binary stiffness mechanism. A more recent example modulates the linear resonance frequency of an arch beam resonator electrothermally to perform various 2 bit and 3 bit logic operations [37].
Practical mechanical digital machines and materials require binary stiffness elements to be easily miniaturized and micromanufactured, since large numbers are needed in a small volume. It is therefore essential to avoid any form of assembly and account for micro-manufacturing techniques like photolithography and upcoming 3D printing technologies such as two-photon lithography [38]. Necessary properties are therefore a fully monolithic and planar architecture which can be created by exploiting the flexibility of materials in what is known as a compliant mechanism.
In this paper we present two fully compliant binary stiffness mechanisms with large deflection linear and rotary motion based on static balancing. A new V-shaped negative stiffness structure is introduced that maximizes the ratio between high and low stiffness. The preloading required for negative stiffness is controlled by a reversible bistable switch independent of primary mobility. Kinematics are dictated by two parallel or angled plate springs.
The method for obtaining binary stiffness is first explained with compression springs and extended to a fully compliant mechanism by replacing them with buckled beams. Final mechanism dimensions are iteratively tuned by hand. This semiautonomous process still relies on domain knowledge from the designer, but a fully automated process relying on machine learning and optimization could be used, as demonstrated in a different context [39]. Examples of design optimizations with this method include a supercompressible micro-structure [40] and an ultra-thin shell structure [41].
We have validated the designs by prototyping decimeter scale prototypes and performing mechanical tests. It is plausible that the presented designs can be scaled down to micro or nanoscale, because all required structures are planar and fully compliant and are integrated in a monolithic and planar device that avoids assembly. A scaled down version would rely on identical principles and only requires redesign for different material properties.
The mechanisms are stress free and in their high stiffness configuration when manufactured. Low stiffness is enabled when the bistable switch is engaged. This can be done irreversibly by actuating the main stage, or reversibly by toggling the bistable switch itself.

Method for binary stiffness
Binary variable stiffness is achieved by enabling and disabling static balancing. Static balancing is an approach to keep the potential energy in a system constant and consequently all net conservative forces and stiffness are zero [42]. It is achieved by adding an elastic structure with negative stiffness in parallel to the functional part with positive stiffness. If negative and positive stiffness exactly oppose, a net zero stiffness results. The required negative stiffness is a consequence of releasing elastic potential energy previously inserted by preloading. Therefore, we can toggle between balanced zero stiffness and unbalanced positive stiffness by engaging and disengaging preloading.
In its simplest form, negative stiffness is created by axially preloading an ordinary compression spring from its initial length L 0 to L p as shown in Fig. 1(a), note that L 0 = h 0 and L p = h p .
This works for both linear [20,13] and rotational motion [43] of the cart. The potential energy of the spring as function of cart displacement u is given by: ] T the cart location, h p the spring length at u = 0 and L 0 the initial length of the spring.
Force and stiffness at the cart are given by: 2 (2) E I , F I and K I are shown in Fig. 1(e) with k = 1, without preloading (h 0 = h p = 2) and with preloading (h 0 = 2, h p = 1).
It shows that for large values of u the stiffness K I converges to k independent of preloading, i.e. lim u→∞ K I The stiffness K I at u = 0 is given by: which shows the stiffness K I (u = 0) = 0 when h p = L 0 and becomes more negative as h p decreases , i.e.
This demonstrates the ability to turn negative stiffness on and off by controlling the preloading distance h p . A compliant analog is obtained by axially preloading a continuum plate in post buckling [13] as shown in Fig. 1(b), possibly with various boundary conditions such as pinned-pinned [20], clamped-pinned [15,44] and clamped-clamped [44].
A preloaded continuum plate is in theory able to reduce the combined stiffness to zero, because the amount of negative stiffness can be tuned by changing its geometry [15]. In practice, perfectly matching positive and negative stiffness is hard and affected by modeling errors, manufacturing tolerances and nonlinear effects. It is therefore desirable to maximize the high stiffness state, because stiffness states become more distinguishable. This is done by increasing the stiffness without preloading around  Fig. 1(c).
The potential energy of the V-shaped springs on a cart is given by: where k is spring stiffness, the initial length between the springs on the cart. Force F V and stiffness K V are computed by differentiation of E V as in Eq. (2). They are shown in Fig. 1 The stiffness K V at u = 0 is given by: which shows that unbounded negative stiffness can be obtained for 2 On the other hand, the stiffness without preloading (L 0 = L p ) is given by which is positive for any nonzero s. Eqs. (7) and (8) show that two springs placed in a V-shaped configuration can generate negative stiffness when preloaded yet have positive stiffness without preloading.
A compliant analog is shown in Fig. 1(d) which will be given exact dimension in an iterative design process. The advantage over a single or multiple vertical beams is the increased stiffness when not preloaded. Values for its negative and positive stiffness depend on geometry and preloading. Too much negative stiffness results in a combined system with negative stiffness. Too little negative stiffness and combined system stiffness is positive. Since we can continuously shift between these two behaviors, by changing for example V-shaped beam length or the geometry of the stage beams, an optimum can be found that results in zero combined stiffness.

Controlling the preloading
Preloading is easily controlled by hand in many macro-scale statically balanced mechanisms, for example during assembly [45] or by using linear stages [46,13]. On the micro-scale, preloading becomes a significant challenge because manual handling is difficult which makes assembly economically unjustifiable. Proposed monolithic solutions include shaking [47] and residual film stress [48], however these methods are irreversible. Reversible examples exist that make use of multistable compliant  mechanisms (MSCMs) [12,[49][50][51]. However, their direction of preloading is aligned with the intended mechanical mobility and can therefore not be controlled independently.
We propose to attach a MSCM to the negative stiffness device from Fig. 1(d). Since the direction of preloading and generated negative stiffness are perpendicular, preloading can be engaged and disengaged independently from the primary mobility. MSCMs are easily made fully compliant, because multistability is most often caused by snap-through of buckled beams [52]. This enables a monolithic system integration and hence miniaturization. Since only two stable states are required for binary stiffness, a bistable mechanism is used.
Although the above bistable mechanisms are passive devices, active implementations are equally common. These microrelays can switch between stable states by means of actuation, but do not require actuation to stay in either state. A wide variety of actuator-mechanism combinations is reported in literature, including rotary bistable mechanisms [62]. Most microrelays rely for their bistability on a set of at least two curved or chevron style beams centrally connected to prevent rotation. Actuation is most commonly done by chevron type thermal actuators [63][64][65][66][67], but also with bending beam thermal actuators [68,69], external electromagnetic actuation [70,71,62], a combination of thermal and electromagnetic [72] and electrostatic comb actuators [73][74][75][76].
In our large scale design a passive chevron type bistable mechanism is implemented. Maximizing the force required to switch equilibrium is important, because a higher threshold force means more negative stiffness can be created. To do this multiple chevron bistable beams are used in parallel. Also, lumped compliance is used instead of distributed compliance to further increase the severity of the bistability by thickening the beams in the middle [53]. The exact dimensions are determined by iterative design in combination with a parametric finite element model.

Modeling, design and fabrication
Parametric finite element models of the proposed mechanisms are built in Ansys APDL out of two-node beam elements (beam188) based on Timoshenko beam theory, a linear shape function and a rectangular cross section. The models are iteratively to create devices that approximate zero stiffness over a maximized range of motion. In practice this entails a design cycle in which a grid search is performed to determine dimensions, simulations ran and prototypes manufactured to validate simulations and build intuition for manufacturing and materials.
The model carefully takes into account the geometry of the flexible elements and models elastic boundary conditions. Supporting back and front plates are added to reinforce the frame, which is needed due to the gate allowing the main stage to exit the frame. The devices are grounded by constraining the motion of the lower left corner in all directions and the motion of the lower right corner in the vertical direction as shown in Figs. 2(a) and 2(f).
In order to compute post buckling behavior, imperfections are added to the bistable mechanisms [77]. A linear buckling analysis is first performed by adding a unit force on the bistable element. Next the first five normalized buckling mode shapes are added to the undeformed geometry with an amplitude of 0.01. Only then the bistable element is connected to the negative stiffness structure. An element plot and a deformation plot in post buckling are shown in Figs. 2(a), 2(b), 2(f) and 2(g).
All plate springs have a beam thickness of 0.7 mm and are 7 mm high. The plate springs of the stages have a length of 35 mm and are at an angle of 70 • for the rotational stage.
The angle α between the negative stiffness plate springs in the rectilinear case is 45 • and they are 75 mm long. In case of rotation the plate springs are 89 mm long and the angle α between them is 10 • . All negative stiffness plate springs have the shape of a cosine with an amplitude of 0.7 mm, making sure they will buckle outwards. Inwards buckling is prevented, because this may cause the V-shaped plate springs to touch each other and challenge performance by introducing for example friction and unwanted deformations. To save space, they are placed as far up as possible.
The systems are designed for a range of motion of ±10 mm and ±0.2 rad (±11.5 • ) for translation and rotation. A fillet of at least 0.7 mm is added to each corner.
In both cases an identical bistable switch is used. Its parameters are chosen such that it can stay in either configuration without effort, while it maintains the load provided by the negative stiffness from the V-shaped plate spring. To create sufficient transition force within material limits, three parallel beams are

Measurements
Force deflection measurements are done with an Instron 5966 tensile test bench. Transition from high to low stiffness is measured by actuating the linear stage bounded by a force of 40 N at a strain rate of 15 mm min −1 . Parasitic motion is accounted for by adding a follower beam between the clamp and the mechanism, see Fig. 3(a). Gravity is not accounted for.
By using a cable system, the rotational mechanism is measured on the same tensile test bench. Since cables cannot be loaded in compression, the combined system is preloaded in tension by a 10.2 N counterweight. Two cables run along grooves on an extended circular arm with a radius of 68.8 mm, see Figs. 3(b) and 3(c). Cable one (in red in Fig. 3(b)) runs from attachment one to the moving tensile grip that is attached to the force sensor. Cable two (in blue) runs from attachment two to the counterweight that preloads the cablesystem. A constant arm radius is assumed to compute the moments and angles from the measured forces and displacements.
Bistability is measured the same way, see Figs. 3(d) and 3(e).

Results
Measurement (EXP) and finite element simulation (FEA) results for the main stage, both linear and rotary, are shown in Fig. 4 and for the bistable switches in Fig. 5. All force displacement measurements are in good agreement with simulations.
The chronological photo sequences from Figs. 4(a)-4(f) and Figs. 4(g)-4(l) associate each mechanism state with a point on the force deflection graphs from Figs. 4(m) and 4(n). Three different mechanism configurations can be identified. The first (red S) is a stable configuration with high stiffness in which the device is manufactured and thus stress free, see Figs. 4(a) and 4(g). The second (yellow T) is unstable in which the bistable switch is toggling, see Figs. 4(b) and 4(h). In this instantaneous state of transition the device is neither on nor off, but at the onset of snap through to either configurations. In the third configuration (green C) the bistable switch is toggled and the V-shaped plate spring buckled, it is highly compliant and also stable as shown in Figs. 4(c)-4(f) and 4(i)-4(l). The configurations are also indicated in the force deflection characteristics of the bistable switch from Figs. 5(a) and 5(b).
A dramatic reduction in stiffness can be readily observed. Around the equilibrium position at (0, 0), translational stiffness is reduced by 98.8 % from 6.70 N mm −1 to 0.08 N mm −1 and rotational stiffness is reduced by 99.9 % from 10.20 N m rad −1 to 0.01 N m rad −1 . These states of stiffness differ by two and three orders of magnitude with a factor of 84 and 1020 respectively.
By pulling on the main stage a force and torque of 27.11 N and 0.79 N m are required to transition. However, if the bistable switch is directly actuated a force of 36.56 N and 29.81 N are required. By design the mechanisms cannot transition back from compliant to stiff by actuating the main stage, because the Vshaped plate spring cannot overcome the threshold force of the bistable switch. However, it is possible by directly actuating the bistable switch with −8.87 N and −4.82 N for the linear and rotary device, as can be seen in Figs. 5(a) and 5(b) In each case the low stiffness domain is in accordance with the intended range of motion. In the linear case, low stiffness stretches 20 mm and by taking into account a 150 mm frame width (without the clamp brackets) this results in a relative range of motion of 13.3 %. The range of motion in the stiff configuration is approximately 14 mm or 9.3 %. In the rotary case, the low stiffness domain stretches 0.4 rad (23 • ) and high stiffness approximately 0.32 rad (18 • ). Beyond these ranges stiffness ramps up quickly and a slight discrepancy between measurement and simulation is observed. These differences may be attributed to issues such as parasitic motion and contact between frame and stage. Contact is also clearly observed at the right side of the curves in Fig. 5.
The discrepancy observed between the measurements and simulations of the stiff parts of Fig. 4, but also that of the bistability in Fig. 5, may be attributed to frame geometry. The stiffness of the boundary conditions of a bistable mechanisms are extremely important. If boundaries are too compliant, the bistable behavior may even disappear. Since we have only coarsely approximated the frame geometry, it would be possible to have a discrepancy between the modeled and actual frame stiffness, but the simulations show good agreement with the experiments.

Conclusion
In this paper we have introduced two mechanisms with binary stiffness that are essential building blocks for mechanical digital machines. Extreme states of linear and angular stiffness are achieved by enabling and disabling static balancing by engaging and disengaging preloading. A new V-shaped negative stiffness plate spring is used successfully to maximize the ratio between the two states of stiffness by controlling its preloading with a bistable switch.
Linear stiffness is reduced by 98.8 % from 6.70 N mm −1 to 0.08 N mm −1 over an absolute range of motion of 20 mm and a relative range of motion of 13.3 %. Angular stiffness is reduced by 99.9 % from 10.20 N m rad −1 to 0.01 N m rad −1 over a range of 0.4 rad or 23 • . This is a difference in stiffness by two and three orders of magnitude with a factor 84 and 1020 respectively. This is a significant improvement over previously reported statically balanced mechanisms with stiffness reduction factors of 46, The presented mechanisms could set the stage for innovative mechanical logical systems, adaptive and programmable metamaterials and other types of mechanical digital machines. To enable miniaturization and micro-fabrication required for such machines we have carefully considered scalability. All required components are integrated in a single piece device that can be fabricated with conventional surface-micro-machining or upcoming micro 3D printing technologies. Although the introduced devices are passive, scaling down presents opportunities for actuator implementation by replacing the passive bistable switch with an active micro-relay commonly reported in scientific literature about MEMS.

Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.