# Underactuated fluidic control of a continuous multistable membrane

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Edited by John A. Rogers, Northwestern University, Evanston, IL, and approved January 29, 2020 (received for review November 12, 2019)

## Significance

Mechanical elements which exhibit several equilibria states and snapping instabilities are increasingly popular in the design of microelectromechanical systems, mechanical logic systems, origami structures, and especially in soft robotics. All previous research examined the actuation dynamics of either a single bistable element or multistable structures, which are an assembly of discrete bistable elements. In this work, we present analysis and experimental demonstration of underactuated fluidic control of a continuous multistable structure, enabling us to arbitrarily pattern such continuous structures by a single pressure inlet. These continuous multistable structures inherently have infinite possible stable patterns, thus offering an essential increase in versatility.

## Abstract

This work addresses the challenge of underactuated pattern generation in continuous multistable structures. The examined configuration is a slender membrane which can concurrently sustain two different equilibria states, separated by transition regions, and is actuated by a viscous fluid. We first demonstrate the formation and motion of a single transition region and then sequencing of several such moving transition regions to achieve arbitrary patterns by controlling the inlet pressure of the actuating fluid. Finally, we show that nonuniform membrane properties, along with transient dynamics of the fluid, can be leveraged to directly snap through any segment of the membrane.

A bistable or a multistable elastic element is a structure capable of transforming between different equilibrium deformation patterns, due to the stability transitions of its characteristic energy profile (Fig. 1) (1, 2). Bistable elements, such as curved elastic membranes, which exhibit snapping instabilities, are becoming increasingly popular in the design of switches and actuators in microelectromechanical systems (MEMS), for designing mechanical logic systems, origami structures, and energy-efficient soft robots (3⇓–5). A common way to fabricate such a system is by combining multiple discrete bistable elements, yielding the entire configuration as a multistable structure.

Pressurization of confined fluids is a leading method for the actuation of such bistable elements, yielding governing dynamics involving both viscous and elastic effects. While the interaction of fluids bounded by elastic structures was extensively studied in recent years (refs. 6⇓⇓⇓⇓⇓–12, as well as discussion in ref. 13), only a few researchers examined viscous flow interacting with bistable elasticity (e.g., refs. 14⇓⇓⇓–18). Previous relevant works involving bistability and viscous flow include the work by Hazel and Heil (19), who numerically studied the steady flow of a viscous fluid through a thin-walled elastic tube connected to two rigid tubes. When the pressure acting on the tube’s shell surpasses a critical value, the tube buckles and strongly modifies fluidic flow within the tube. Another recent relevant work by Gomez et al. (17) demonstrated passive control of viscous flow in a channel via an elastic arc positioned within the channel. By controlling the volumetric fluid flux, the bistable elastic arc can be made to snap between two deformation patterns, therefore modifying the channel’s viscous resistance by order of magnitude. Thus, the authors showed that bistability could be effectively used to replace externally controlled valves. Arena et al. (20) recently introduced a conceptual design for adaptive structures that utilize the instabilities of postbuckled membranes to obtain flow regulation and control. By tailoring the stress field in the postbuckled state and the geometry of the initial, stress-free configuration, the deformable section can snap through to close or open the inlet completely, thus providing a self-stimulating actuator that regulates the inlet flow without requiring external flow-regulating mechanisms.

So far, all previous works examined the actuation dynamics of either a single bistable element (21, 22) or multistable structures (23⇓⇓⇓–27), which are an assembly of discrete bistable elements. In this work, we present analysis and demonstration of fluidic control of a continuous, multistable structure. In contrast with discrete multistable configurations, such continuous structures inherently have infinite possible stable patterns. Importantly, in many of these works, each element has its own control input for inducing transitions between its bistable states (28, 29). Therefore, generating complex deformation patterns of such a multistable structure typically requires control of multiple inputs, which greatly complicates the system’s operation. We thus focus on achieving underactuated fluidic control, enabling to arbitrarily pattern such continuous structures by a single pressure inlet.

The configuration studied in this work is presented in Fig. 2, showing the experimental setup, including the membrane and a transition region separating between the two different equilibria states. The x coordinate denotes the streamwise direction, where the inlet is located at

The experimental setup consists of a rectangular channel, a pressure-flow controller, and a viscous fluid reservoir (Fig. 2). We used a rigid material for the side and bottom walls [poly(methyl methacrylate)] and soft membrane (latex) for the top wall. The dimensions of the fabricated channel were 16.6 × 15.8 × 800 mm (width × height × length), and the soft membrane was 35 mm wide and 0.52 mm thick, with an elastic modulus of *E* = 100 MPa. To obtain a geometric bistability, we clamped the 35-mm-wide membrane onto the shorter 15.8-mm-wide channel (Fig. 2). This created two stable deformation states, with centerline heights of 29.6 and 2.05 mm for the snapped-up and -down states. We connected the channel to a pressure controller (Elveflow OB1) at the inlet (at *x* = 0 mm) and kept the outlet (at *x* = 800 mm) open to atmospheric pressure. To actuate the channel, we used glycerol as a viscous fluid (ρ = 1.26 g/

Using this setup, in Fig. 3, we demonstrate the propagation of a single transition region in the channel. The initial state of the membrane is at a snapped-up state, and then we apply a Heaviside inlet pressure function of

To obtain insight regarding these results, we derived a theoretical model for the propagation of a single transition region in a multistable channel (Fig. 3). We considered a viscous fluid in a semi-infinite elastic channel with two stable cross-sectional shapes for given fluidic pressures p. For simplicity, and based on previous works such as refs. 25, 26, and 30, we adopted an approximated trilinear relation between the pressure induced by the internal flow and the channel cross-section. This approximation simplifies the relation between the fluidic pressure and the cross-sectional deformation to linear functions in both stable regions. This leads to a model of linearly elastic snapped-down channel at pressure below **2** into Eq. **1**, we get**2** into Eq. **4** yields**3** are similar to one-dimensional heat transfer problem involving a phase change, known as the Stefan problem (31, 32). Integration and determining the integration coefficients by applying boundary and initial conditions yields the self-similar result of**6** are substituted into Eq. **4**, yielding the additional relation**5**). While Eq. **7** is implicit in β, an approximate explicit solution can be obtained by regular asymptotic expansions (*SI Appendix*, section 1)*SI Appendix*, section 2. As is evident in Fig. 3, a good agreement is observed between the estimated location of the transition region and the experimental results.

Above, we analytically analyzed and experimentally demonstrated the emergence and motion of a single transition region and showed that when the inlet pressure returns to its nominal value, the membrane’s shape remains nearly unchanged. Thus, by sequencing several inlet pressures, any pattern of a snapped-down and -up regions along the channel can be created. The use of multiple moving transition regions for patterning is presented in Fig. 4. Fig. 4*A* presents various inlet pressure profiles, and the corresponding final patterns are presented in Fig. 4*B*. Fig. 4*C* focuses on the first pressure profile in Fig. 4*A* (marked by a blue line) and shows the temporal evolution of the patterning process. Initially, the channel is entirely at the snapped-up state. Then, we applied alternating positive- and negative-gauge inlet pressures (red line in Fig. 4, denoting inlet pressure) to generate moving snap-down and -up transition regions, thus patterning the equilibrium state of the continuously multistable membrane. Fig. 4 shows the evolution of the membrane shape and presents the location of the transition regions vs. time, as well as snapshots of the membrane shape at different time intervals.

We note that the shape of the transition regions, separating between the different cross-sectional equilibria states, resembles a single wrinkle (Fig. 5). Two possible geometric configurations of this wrinkle were observed and shown to be determined by the inducing flow field. In addition, some asymmetric wrinkles were occasionally observed, but were unstable and collapsed to the symmetric form (Fig. 4; at times *A*. A snap-up transition region is presented in Fig. 5*B*, which is similar to the inverse of the geometry presented in Fig. 5*A*. The transition-region shapes are nearly unchanged after the ending of the fluidic driven actuation, as presented in Fig. 5 *C* and *D*.

So far, we examined only membranes with constant properties. Arbitrary patterning of such membranes required sequencing of several transition regions and waiting for all transition regions to reach the required positions. However, we can exploit the transient dynamics of the fluidic pressure, along with nonuniform membrane properties, to immediately snap up or snap down any segment of the membrane. This concept is illustrated in Fig. 6, which presents numerical solutions (see code in ref. 33) of the transient flow in contact with a membrane with continuously varying properties. In Fig. 6, the same set of Eq. **3** were solved. However, in this case, the snapping pressure spatially varied according to *A1*–*A3* presents three different inlet pressure signals that vary in amplitude *B1*–*B3* presents the pressure field for different times (solid lines) along with snapping pressure distribution along the channel (

In Fig. 7, we experimentally demonstrate this concept, using a membrane with piece-wise spatially varying snapping pressure. At region *A*–*C*. We show that *D*, we demonstrate the combination of such signals to create a rather complex deformation pattern of the membrane. The effects of the membrane geometry on the snapping pressure are discussed in *SI Appendix*, Fig. S4 and section 3.

To conclude, in this work, we addressed the challenge of underactuated control of continuous multistable structures, which could play a vital role in the fields of soft robotics, MEMS, and meta-materials. We focused our study on a simple illustrative configuration composed of an slender elastic membrane, which is actuated by a viscous fluid. The membrane is able to concurrently sustain two different modes of stable cross-section shapes at different segments of the membrane. These different segments are shown to be separated by transition regions, and the location of these regions sets the stable equilibrium shape of the membrane. We theoretically analyzed and experimentally demonstrated the formation and motion of a single, and multiple, transition regions due to manipulation of the fluidic inlet pressure. We showed that sequencing of multiple transition regions enables one to achieve underactuated control of the membrane equilibria shape.

Detailed descriptions appear in *SI Appendix*, and related codes used in the work are available at Figshare, https://doi.org/10.6084/m9.figshare.11648022.v1.

## Footnotes

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^{1}To whom correspondence may be addressed. Email: ofekperetz{at}campus.technion.ac.il.

Author contributions: O.P., R.F.S., and A.D.G. designed research; O.P. and A.K.M. performed research; O.P. contributed new reagents/analytic tools; O.P. analyzed data; and O.P. and A.D.G. wrote the paper.

The authors declare no competing interest.

This article is a PNAS Direct Submission.

Data deposition: Related codes used in the work are available at Figshare (https://doi.org/10.6084/m9.figshare.11648022.v1).

This article contains supporting information online at https://www.pnas.org/lookup/suppl/doi:10.1073/pnas.1919738117/-/DCSupplemental.

Published under the PNAS license.

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