# Dynamics of reconfigurable straw-like elements

Dotan Ilssar1,2<EMAIL_ADDRESS>Author to whom correspondence should be
addressed. Michael Pukshansky1 Yizhar Or1 Amir D. Gat1<EMAIL_ADDRESS>1Faculty of Mechanical Engineering, Technion – Israel Institute of Technology,
Haifa 3200003, Israel
2Mechanics & Materials Lab, Department of Mechanical and Process Engineering,
ETH Zürich, Zürich 8092, Switzerland

###### Abstract

In this paper, we discuss the dynamic modeling of fluid-filled straw-like
elements consisting of serially interconnected elastic frusta with both
axisymmetric and antisymmetric degrees of freedom, assuming planar motion.
Under appropriate conditions each sub-structure has four stable equilibrium
states. This gives the system under investigation the ability to remain stable
in a large number of complex states, which is a vital ability for myriad of
applications, including reconfigurable structures and soft robots. The
theoretical model explains the dynamics of a single straw-like element in a
discrete manner, considering inertial, damping, and gravitational effects,
while taking into account the nonlinear elasticity of the elastic frusta, and
assuming hydrostatic behavior of the entrapped fluid. After identifying the
geometric and elastic parameters of the theoretical model based on relatively
simple experiments, the model is validated compared to numerical simulations
and experiments. The numerical simulations validate the theoretical elasticity
of the elastic frusta, whereas the overall dynamic behavior of the system and
the influence of unmodeled fluidic effects are examined experimentally. It is
demonstrated both theoretically and empirically that straw-like elements
cannot be adequately modeled using simple uniaxial deformations. In addition,
the experimental validation indicates that the suggested model can accurately
capture their overall dynamics.

††preprint: APS/123-QED

##  I Introduction

Reconfigurable metamaterials capable of changing their forms and mechanical
properties are an emerging field in mechanics and soft robotics. Recently,
instability-based reconfigurable metamaterials have gained great importance
thanks to their unique characteristics and performance [1]. For example,
hierarchical structures composed of repeated multistable elements in different
configurations show extreme properties such as large zero Poisson’s ratio
deformations in the elastic regime [2, 3, 4], as well as multiaxial complex
stable states [5]. The various stable equilibria of instability-based
metamaterials also lead to exotic dynamic behaviors, allowing for instance to
alter the propagation characteristics of elastic waves [6]. Another important
dynamic property achievable in arrays of multistable elements is energy
localization. Indeed, recent studies demonstrated the creation of propagating
wave fronts along one- and two-dimensional arrays of bistable elements [7, 8,
9, 10, 11]. Finally, the ability to employ a single input to bring a
configuration of discretely [12, 13] and continuously [14] interconnected
bistable elements from one stable equilibrium state to another, was achieved
by embedding these locally bistable structures with viscous fluid. By these
means, a change of pressure at the inlet creates a pressure wave that folds or
deploys different sections of the structure in order, allowing it to be
brought to any designated stable state, while passing through undesired
configurations.

The present paper deals with modeling and investigating the dynamic behavior
of locally multistable mechanical elements inspired by ‘bendy straws’, to pave
the way towards inflatable reconfigurable structures utilizing these elements
as building blocks. Thanks to their large number of stable equilibria,
structures consisting of such straw-like elements can be brought to myriad of
complex operative configurations, by governing the pressure of the fluid
trapped inside them. A recent study started shedding light on the static
behavior of straw-like elements by showing empirically and theoretically that
their multistability originates in internal stresses [15]. A similar local
multistability was demonstrated in an origami implementation of ‘bendy
straws’, where pop-through defects gave rise to a single non-axial stable
state of each unit-cell [16]. In another recent article, the authors produced
a two dimensional array of uncoupled straws restricted to deform uniaxially
[17]. This configuration has an unprecedented number of stable states which
provide a high flexibility while determining its reconfigurable form and local
stiffness. Nevertheless, no available model describing the static and dynamic
behaviors of straw-like elements was found in the literature. Such model can
enable to theoretically design the properties and operating conditions of
these elements, to achieve designated behaviors.

Indeed, here we start with theoretical modeling of a single straw-like element
consisting of serially connected elastic conical frusta, whose dynamics is
governed by an externally applied pressure at a reservoir supplying
incompressible fluid through a slender channel. Considering a degenerated two-
dimensional representation of the system, each constituent frustum has both
axisymmetric and antisymmetric degrees of freedom (DOFs), where due to a
nonlinear elastic behavior, under appropriate geometrical and loading
conditions each frustum has four distinct stable equilibria. The proposed
theoretical model considers inertial and gravitational effects assuming a
nearly uniform pressure field, as well as a simplified description of the
dissipative forces accounting for both fluidic effects and structural damping.
The model also considers the nonlinear stiffness of the elastic frusta
utilizing an extension of the well-known formulation by Almen and Laszlo [18],
and the fluidic effects related to the flow inside the channel which is
assumed laminar at high Reynolds numbers. Following the theoretical
derivation, we introduce an efficient system identification process based on
measurements and relatively simple experiments, used to determine the
geometrical parameters of the system, as well as the physical parameters
related to the solid. This process is utilized to calibrate the theoretical
model so it captures the behavior of an experimental demonstrator that was
designed and manufactured for model verification. The nonlinear quasi-static
theoretical behavior of a single elastic frustum is then investigated and
validated compared to finite element simulations. These simulations are
further exploited as a basis for a numerical justification to disregard the
deformations of half of the constituent frusta, which significantly reduces
the number of DOFs. Finally, the experimental demonstrator is utilized for
verifying the overall dynamic behavior of the theoretical model, and to assess
unmodeled fluidic effects by calibrating the parameters which are affected by
them. The highly correlated analytical and numerically calculated quasi-static
behaviors of the system predict that when the inertial effects are weak, each
frustum should pass through a partially-snapped state, while switching between
fully-snapped configurations. This claim is experimentally confirmed as under
negative external gauge pressure, the different sections of the demonstrator
are folded according to this theoretically predicted response. Nevertheless,
it is shown that while deploying the system by applying positive external
gauge pressure, due to lower damping reflected in high inertial forces, each
segment can be immediately brought from one fully-snapped state to another.

##  II Model derivation

The system under investigation consists of a liquid-filled straw which is
sealed on one end, where its second end is connected to a reservoir whose
pressure is externally dictated, see Figure 1 (a). Here, we present the
formulation of a simplified model capturing the governing dynamics of this
system, starting with descriptions of the inertial and dissipative effects as
well as the influence of external and gravitational forces. Next, we
supplement the model with the strain energy function of the system, derived
based on a quasi-static analysis of a single elastic frustum, which is the
basic building block of the system.

Figure 1: (a) Schematic layout of the system under investigation, consisting
of a liquid-filled straw-like element which is connected through a slender
channel to a liquid reservoir with externally dictated static pressure. (b, c)
Top and section views of a constituent multistable cell, composed of two
elastic conical frusta.

###  II.1 Formulation of the dynamic and external effects

As illustrated in Figure 1, we describe the straw as a serial interconnection
of $N$ liquid-filled cells, each consisting of two elastic frusta. For
simplicity we consider a two-dimensional problem where the motion of the
system and its constituent building blocks are bounded to the
$\hat{X}-\hat{Z}$ plane of the global coordinate system whose origin is
located at the middle of the straw’s base, see Figure 1 (a). Under this
simplification, the deformation of each multistable cell is formulated in
terms of ${\zeta_{n,1}}$ and ${\varphi_{n,1}}$ denoting the axisymmetric and
antisymmetric DOFs of the frustum closer to the inlet, as well as
${\zeta_{n,2}}$ and ${\varphi_{n,2}}$ describing the corresponding DOFs of the
frustum that is farther from the inlet. Namely, under the assumption of small
angles, the axial position of each material point in the kth frustum of the
nth multistable cell, with respect to the cell’s middle surface, is

${z_{n,k}}\left({{r_{n,k}},\,{\theta_{n,k}}}\right)\approx\left({{\zeta_{n,k}}-{r_{i}}{\varphi_{n,k}}\cos{\theta_{n,k}}}\right)\frac{{{r_{o}}-{r_{n,k}}}}{{{r_{o}}-{r_{i}}}}.$
(1)

Here, ${r_{n,k}}\in\left[{{r_{i}},\,{r_{o}}}\right]$ and
${\theta_{n,k}}\in\left[{0,\,2\pi}\right)$ are the local radial and tangential
coordinates of the relevant frustum, where $r_{i}$ and $r_{o}$ respectively
describe the inner and outer radii of all frusta. The representation in (1)
neglects radial deformations, thus keeps the radial cross-sections of the
frusta as well as their bases, undistorted. Next, the channel connecting the
straw to the liquid reservoir whose dictated time-dependent static pressure is
denoted ${p_{{\rm{external}}}}\left(t\right)$, is considered to be slender
compared to both the straw and the reservoir, where its radius is denoted
$\,{r_{ch}}\,$. Thus, referring to the fluid in both the straw and the
reservoir as semi-infinite media, their pressure fields are approximated
uniform, whereas assuming incompressible laminar flow at high Reynolds
numbers, the fluid inside the channel is described by the steady Bernoulli
equation. It was previously shown that under these conditions, the flow
regimes throughout filling and depletion of a closed vessel are radically
asymmetric due to boundary layer separation, giving rise to an internal jet
when the fluid flows into the vessel [19]. In this case, the flow velocities
in the vessel being filled are significantly higher than those at the other
end of the channel filling it, where the dynamic pressure can be neglected.
Therefore, while inflating the straw, the dictated pressure is referred to as
the stagnation pressure of the closed system. However, during deflation,
boundary layer separation occurs in the liquid reservoir, thus the stagnation
pressure is taken as the uniform pressure inside the straw. Consequently,
denoting the density of the fluid by $\rho$, the uniform pressure inside the
straw is taken as

$p\left(t\right)={p_{{\rm{external}}}}\left(t\right)-\frac{\rho}{{2{\pi^{2}}r_{ch}^{4}}}{\left({\sum\limits_{n=1}^{N}{\frac{{{\rm{d}}{V_{n}}}}{{{\rm{d}}t}}}}\right)^{2}}{\rm{sgn}}\left({\sum\limits_{n=1}^{N}{\frac{{{\rm{d}}{V_{n}}}}{{{\rm{d}}t}}}}\right),$
(2)

where $V_{n}$ is the volume of the fluid residing inside the nth multistable
cell. As discussed above, for simplicity the radial deformations of the
different frusta are assumed to be negligible, meaning that their radial
dependency is represented by the linear function given in (1). Integration
over the volume of the nth multistable cell utilizing this assumption yields
the following expression, describing in terms of the generalized coordinates
${\zeta_{n,k}}$ and ${\varphi_{n,k}}$, the volume of the fluid enclosed within
this cell:

$V_{n}=\pi\frac{\left(r_{o}^{2}+r_{i}^{2}+r_{o}r_{i}\right)\left(\zeta_{n,1}+\zeta_{n,2}\right)}{3}.$

Assuming that the inertial and body forces of the solid are negligible
compared to those of the entrapped fluid due to low thickness of the straw,
the effects of the solid’s mass are disregarded. However, for practical
reasons such as the need of a relatively heavy body to seal the tip of the
straw, the mass and moment of inertia of this sealing denoted
${m_{{\rm{tip}}}}$ and ${I_{{\rm{tip}}}}$ are taken into account. Further, in
agreement with the assumption claiming that the pressure field inside the
straw is uniform, the motion of the fluid trapped in each cell is treated as a
quasi-rigid body, whose instantaneous mass and moment of inertia are
determined by the DOFs of its constituent frusta. Thus, the contribution of
each multistable cell to the total kinetic energy is computed considering the
translation of its center of mass and the rotation around this point,
considering its varying volume and shape. Summing the kinetic energies of the
sealing as well as those related to all cells leads to the following
expression, describing the overall kinetic energy of the system:

$\begin{array}[]{l}{\cal
T}=\sum\limits_{n=1}^{N}{\left\\{{\frac{{\rho{V_{n}}}}{2}\left[{{{\left({\frac{{{\rm{d}}{{\bar{X}}_{n}}}}{{{\rm{d}}t}}}\right)}^{2}}+{{\left({\frac{{{\rm{d}}{{\bar{Z}}_{n}}}}{{{\rm{d}}t}}}\right)}^{2}}}\right]+\frac{{{I_{n}}}}{2}{{\left[{\frac{{{\rm{d}}\left({{{\bar{\Phi}}_{n}}+{{\bar{\varphi}}_{n}}}\right)}}{{{\rm{d}}t}}}\right]}^{2}}}\right\\}}\\\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,+\frac{{{m_{{\rm{tip}}}}}}{2}\left[{{{\left({\frac{{{\rm{d}}{{\bar{X}}_{{\rm{tip}}}}}}{{{\rm{d}}t}}}\right)}^{2}}+{{\left({\frac{{{\rm{d}}{{\bar{Z}}_{{\rm{tip}}}}}}{{{\rm{d}}t}}}\right)}^{2}}}\right]+\frac{{{I_{{\rm{tip}}}}}}{2}{\left({\frac{{{\rm{d}}{{\bar{\Phi}}_{{\rm{tip}}}}}}{{{\rm{d}}t}}}\right)^{2}}.\end{array}$
(3)

Here,

${\bar{X}_{n}}=\sum\limits_{j=1}^{n-1}{\left({{\zeta_{j,1}}+{\zeta_{j,2}}}\right)\sin{{\bar{\Phi}}_{j}}}+\left({{\zeta_{n,1}}+{{\bar{z}}_{n}}}\right)\sin{\bar{\Phi}_{n}}+{\bar{x}_{n}}\cos{\bar{\Phi}_{n}},$

and

${\bar{Z}_{n}}=\sum\limits_{j=1}^{n-1}{\left({{\zeta_{j,1}}+{\zeta_{j,2}}}\right)\cos{{\bar{\Phi}}_{j}}}+\left({{\zeta_{n,1}}+{{\bar{z}}_{n}}}\right)\cos{\bar{\Phi}_{n}}-{\bar{x}_{n}}\sin{\bar{\Phi}_{n}},$

are the horizontal and vertical coordinates of the nth multistable cell’s
center of mass, with respect to the global coordinate system. Furthermore,

${\bar{\Phi}_{n}}={\varphi_{0}}+\sum\limits_{j=1}^{n-1}{\left({{\varphi_{j,1}}+{\varphi_{j,2}}}\right)}+{\varphi_{n,1}},$

is the orientation of the cell’s middle surface around the $\hat{Y}$ axis,
where ${\varphi_{0}}$ is the inclination angle of the straw’s base, relative
to the horizon. Similarly,

${X_{{\rm{tip}}}}=\sum\limits_{j=1}^{N}{\left({{\zeta_{j,1}}+{\zeta_{j,2}}}\right)\sin{{\bar{\Phi}}_{j}}},$
${Z_{{\rm{tip}}}}=\sum\limits_{j=1}^{N}{\left({{\zeta_{j,1}}+{\zeta_{j,2}}}\right)\cos{{\bar{\Phi}}_{j}}},$

and

${\Phi_{{\rm{tip}}}}={\varphi_{0}}+\sum\limits_{j=1}^{N}{\left({{\varphi_{j,1}}+{\varphi_{j,2}}}\right)},$

represent the horizontal, vertical and rotational coordinates of the straw’s
sealed end. Next, $\bar{x}_{n}$, $\bar{z}_{n}$ and $\bar{\varphi}_{n}$ are the
translational and angular coordinates of the nth multistable cell’s center of
mass, with respect to the local coordinate system of its constituent frustum
which is farther from the straw’s base, see Figure 1 (c). Assuming
${\varphi_{n,1}},\,{\varphi_{n,2}}\ll 1$ and that the frusta do not deform
radially, these local coordinates are given by the following approximated
expressions:

$\bar{x}_{n}=-r_{o}r_{i}\frac{\varphi_{n,1}+\varphi_{n,2}}{4\left(\zeta_{n,1}+\zeta_{n,2}\right)},$
$\begin{array}[]{l}{\bar{z}_{n}}\approx\frac{{2\left({r_{o}^{2}+3r_{i}^{2}+2{r_{o}}{r_{i}}}\right)\left({\zeta_{n,2}^{2}-\zeta_{n,1}^{2}}\right)+{r_{o}}r_{i}^{2}\left({{r_{o}}+2{r_{i}}}\right)\left({\varphi_{n,2}^{2}-\varphi_{n,1}^{2}}\right)}}{{8\left({r_{o}^{2}+r_{i}^{2}+{r_{o}}{r_{i}}}\right)\left({{\zeta_{n,1}}+{\zeta_{n,2}}}\right)}}\end{array},$

and
${\bar{\varphi}_{n}}={\rm{atan}}2\left({{{\bar{z}}_{n}},{{\bar{x}}_{n}}}\right)$.
The last component in the expression of the kinetic energy to be derived is
the moment of inertia of the nth multistable cell around its center of mass.
Under the underlying assumptions this expression is given by

$\begin{array}[]{l}{I_{n}}\approx\pi\rho\frac{\left(r_{o}^{2}+6r_{i}^{2}+3r_{o}r_{i}\right)\left(\zeta_{n,1}^{3}+\zeta_{n,2}^{3}\right)}{30}\\\
\,\,\,\,\,\,\,\,\,+\pi\rho
r_{i}^{2}\frac{\left(r_{o}^{2}+r_{i}^{2}+3r_{o}r_{i}\right)\left(\zeta_{n,1}\varphi_{n,1}^{2}+\zeta_{n,2}\varphi_{n,2}^{2}\right)}{20}\\\
\,\,\,\,\,\,\,\,\,+\pi\rho\frac{\left(r_{o}^{5}-r_{i}^{5}\right)\left(\zeta_{n,1}+\zeta_{n,2}\right)}{20\left(r_{o}-r_{i}\right)}-\rho{V_{n}}\left({\bar{x}_{n}^{2}+\bar{z}_{n}^{2}}\right).\end{array}$

The Rayleigh dissipation function emanating from structural damping and the
interaction between the solid and the entrapped and ambient fluids, is
considered linearly dependent on the time derivatives of the DOFs, thus is
taken as

${\cal
D}=\sum\limits_{n=1}^{N}{\sum\limits_{k=1}^{2}{\left[{\frac{{{c_{\zeta}}}}{2}{{\left({\frac{{{\rm{d}}{\zeta_{n,k}}}}{{{\rm{d}}t}}}\right)}^{2}}+\frac{{{c_{\varphi}}}}{2}{{\left({\frac{{{\rm{d}}{\varphi_{n,k}}}}{{{\rm{d}}t}}}\right)}^{2}}}\right]}}.$
(4)

Here, for simplicity the damping coefficients of all axisymmetric and
antisymmetric DOFs, denoted ${c_{\zeta}}$ and ${c_{\varphi}}$, are considered
identical.

Finally, we formulate the virtual work considering the uniform pressure
applied to the straw’s sealed end assuming the terminal radius equals to
$r_{i}$, as well as the horizontal and vertical localized forces $F_{X}$ and
$F_{Z}$ and bending moment $M_{Y}$, exerted on this end. Moreover, the
gravitational forces acting on the sealing as well as on the fluid trapped
inside the straw’s cells are also taken into account. Considering all the
effects mentioned above, the total virtual work is given by:

$\begin{array}[]{l}\delta{\cal W}=-\rho
g\sum\limits_{n=1}^{N}{{V_{n}}\delta{{\bar{Z}}_{n}}}+\left({{F_{X}}+\pi
r_{i}^{2}p\sin{{\bar{\Phi}}_{{\rm{tip}}}}}\right)\delta{\bar{X}_{{\rm{tip}}}}\\\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,+\left({{F_{Z}}+\pi
r_{i}^{2}p\cos{{\bar{\Phi}}_{{\rm{tip}}}}-{m_{{\rm{tip}}}}g}\right)\delta{\bar{Z}_{{\rm{tip}}}}\\\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,+{M_{Y}}\delta{\bar{\Phi}_{{\rm{tip}}}},\end{array}$
(5)

where $g$ is the gravitational acceleration and $\delta\bullet$ is a variation
in $\bullet$.

###  II.2 Adding the effects of the solid’s elasticity

To complete the formulation of the system’s dynamics, it is necessary to
derive the potential energy related to the elasticity of the frusta, and to
the direct influence of the uniform pressure field on them. Thus, in this
section a quasi-static analysis of a single elastic conical frustum is carried
out, while omitting the subscript $n,k$ for brevity. This results in the
potential energy of a general frustum, which is then used to describe all
constituent frusta. The latter leads to the overall potential energy of the
straw assuming the creases connecting the different frusta are extremely thin.
Namely, the contribution of these creases to the strain energy is negligible,
as well as the tangential moments they apply to the bases of the frusta.

The analysis is based on an article by Almen and Laszlo [18] who modelled the
deformations of a Belleville washer undergoing axial loading, and some more
recent studies that corrected and modified this model, yet still consider only
axial deflections [20, 21]. Thus, the underlying assumptions of these papers
are adopted. These assumptions claim that the frustum is thin, its material is
linear, and that the deformations of its cross-sections are minor, meaning
that the bending and elongation along the radial direction are disregarded.
Instead, each radial cross-section is considered rotating around a neutral
point, located on its natural axis at a radial location ${r_{{\rm{neutral}}}}$
which should be determined. Consequently, the radial and axial components of
the stress are neglected. It should be noted that since the model derived here
considers deformations that deviate from axisymmetry, ${r_{{\rm{neutral}}}}$
is initially considered dependent on the tangential coordinate $\theta$. The
rotations of the cross-sections result in direct tangential bending stresses,
and additional tensile stresses caused due to radial displacements which
stretch each material point tangentially. These two stress components apply
internal moments which balance those caused due to external loads, leading to
the elastic behavior of the frustum.

The first contribution to the internal moment to be formulated is the one
caused due to the tangential stretching of frustum. For this sake, the
tangential strain of a general infinitesimal material point is calculated as
function of the frustum’s base angle taken as
$\psi={\psi_{0}}+\Delta{\psi_{1}}+\Delta{\psi_{2}}\cos\theta$, see Figure 2
(b), where $\theta$ is the tangential coordinate revolving around the axis of
the frustum denoted $\hat{z}$, see Figure 2 (a). This form allows the frustum
to undergo both axisymmetric and antisymmetric deflections related to
$\Delta{\psi_{1}}$ and $\Delta{\psi_{2}}$ respectively, from the stress-free
axisymmetric state where the base angle is constant and equals to $\psi_{0}$.
Employing the above notations and assuming small base angles such that
$\psi\ll 1$, the local tangential strain is given by [18]

$\varepsilon_{\theta,1}=\frac{\left(r_{\rm{neutral}}-\chi\cos\psi\right){\rm{d}}\theta}{\left(r_{\rm{neutral}}-\chi\cos\psi_{0}\right){\rm{d}}\theta}-1\approx\frac{\chi\left(\psi^{2}-\psi_{0}^{2}\right)}{2\left(r_{\rm{neutral}}-\chi\right)},$

where $\chi\left(\theta\right)$ is a cross-section dependent coordinate which
is tangent to the frustum’s face, whose origin is located at
${r_{{\rm{neutral}}}}\left(\theta\right)$, see Figure 2 (b). Under the
assumptions that the material is linear and that the radial and axial stress
components are negligible, Hooke’s law relates the tangential stress and
strain by ${\sigma_{\theta,1}}=E{\varepsilon_{\theta,1}}$, where $E$ is the
solid’s Young modulus. Utilizing this stress component, the contribution of a
general infinitesimal material point’s tangential stretching, to the internal
radial moment per unit length around the cross-section’s neutral point is
given by

${M_{r{\rm{,1}}}}=\left[{\chi\sin\psi\hat{z}\times{\sigma_{\theta,1}}h\left({-\hat{\theta}}\right)}\right]\cdot\hat{r}\approx\frac{{Eh{\chi^{2}}\psi\left({{\psi^{2}}-\psi_{0}^{2}}\right)}}{{2\left({{r_{{\rm{neutral}}}}-\chi}\right)}},$

where $h$ is the thickness of the frustum.

Figure 2: Schematic layout of an elastic conical frustum. (a, b) Top and
section views of the entire frustum, (c) Top view of an arbitrary
infinitesimal material point, (d) Cutout of an arbitrary cross-section.

As mentioned above, the second contribution to the internal radial moment per
unit length is related to the tangential bending of the frustum. To formulate
this contribution, first the base angle dependent local change of the
tangential curvature is computed under the assumptions mentioned above, and is
taken as [22]

$\Delta{\kappa_{\theta}}=\frac{{\sin\psi}}{{{r_{{\rm{neutral}}}}-\chi\cos\psi}}-\frac{{\sin{\psi_{0}}}}{{{r_{{\rm{neutral}}}}-\chi\cos{\psi_{0}}}}\approx\frac{{\psi-{\psi_{0}}}}{{{r_{{\rm{neutral}}}}-\chi}}.$

Next, the desired expression is computed by integrating the resultant
tangential stress multiplied by $z$, over the thickness of the frustum [23].
Considering the solid behaves according to Kirchhoff plate theory, and
recalling that the axial and radial stresses are nulled, the tangential stress
component is given by ${\sigma_{\theta,2}}=-Ez\Delta{\kappa_{\theta}}$, thus
the corresponding radial moment per unit length is
${M_{r{\rm{,2}}}}\approx{{E{h^{3}}\Delta{\kappa_{\theta}}}\mathord{\left/{\vphantom{{E{h^{3}}\Delta{\kappa_{\theta}}}{12}}}\right.\kern-1.2pt}{12}}$.
Summation of ${M_{r{\rm{,1}}}}$ and ${M_{r{\rm{,2}}}}$, yields the overall
internal radial moment per unit length.

The next stage is calculating the radial positions of the cross-sections’
neutral points. This is done by balancing the internal normal forces
$N_{\theta}$ acting on the radial faces of an arbitrary infinitesimal sector
due to the overall tangential stress $\sigma_{\theta}$ computed above. Namely,
the normal forces which are shown in Figure 2 (c) for an infinitesimal
material point, are integrated over the two faces of the sector, and projected
on the radial direction. From static considerations, these two terms should
cancel each other, which leads to an expression which indicates that in every
cross-section the neutral point resides in
${r_{{\rm{neutral}}}}\approx{{\left({{r_{o}}-{r_{i}}}\right)}\mathord{\left/{\vphantom{{\left({{r_{o}}-{r_{i}}}\right)}{\ln\left({{{{r_{o}}}\mathord{\left/{\vphantom{{{r_{o}}}{{r_{i}}}}}\right.\kern-1.2pt}{{r_{i}}}}}\right)}}}\right.\kern-1.2pt}{\ln\left({{{{r_{o}}}\mathord{\left/{\vphantom{{{r_{o}}}{{r_{i}}}}}\right.\kern-1.2pt}{{r_{i}}}}}\right)}}$.

To balance the internal moment formulated above, the influence of the external
loading applied to the frustum is derived. This is done while considering a
distributed load given by
$q\left(\theta\right)=\left({{F\mathord{\left/{\vphantom{F{2\pi{r_{i}}}}}\right.\kern-1.2pt}{2\pi{r_{i}}}}}\right)-\left({{M\mathord{\left/{\vphantom{M{\pi
r_{i}^{2}}}}\right.\kern-1.2pt}{\pi r_{i}^{2}}}}\right)\cos\theta$,
representing the interaction with the neighboring frusta which introduce both
normal force and bending moment denoted $F$ and $M$, to the frustum’s edges.
Moreover, the direct influence of the entrapped fluid’s uniform pressure is
also taken into account. Consequently, under the assumption of small angles
enabling to refer to both the pressure and the applied loads as purely
uniaxial, the resulting shear force per unit length acting on an arbitrary
radial cutout is given by

${S_{z}}\approx\frac{{2{r_{i}}q\left(\theta\right){\rm{+}}\left[{{{\left({{r_{{\rm{neutral}}}}-\chi}\right)}^{2}}-r_{i}^{2}}\right]p}}{{2\left({{r_{{\rm{neutral}}}}-\chi}\right)}},$

see Figure 2 (d). Finally, a moment balance considering the internal moments
as well as the shear force couple acting on the general infinitesimal material
point in Figure 2 (c), is applied after projection on the axis perpendicular
to the radial coordinate. Integration of the expression obtained by these
means along the radial dimension of the frustum, and elimination of small
terms which are dependent on the differential radius and angle ${\rm{d}}r$ and
$\rm{d}\theta$ leads to the quasi-static behavior of the frustum. The latter
is then converted to terms of the axisymmetric and antisymmetric DOFs by their
relations with the base angles given by
$\zeta\approx\left({{r_{o}}-{r_{i}}}\right)\left({{\psi_{0}}+\Delta{\psi_{1}}}\right)$
and
$\varphi\approx{{\left({{r_{i}}-{r_{o}}}\right)\Delta{\psi_{2}}}\mathord{\left/{\vphantom{{\left({{r_{i}}-{r_{o}}}\right)\Delta{\psi_{2}}}{{r_{i}}}}}\right.\kern-1.2pt}{{r_{i}}}}$,
see Figure 2 (b). The resulting formulation consists of terms dependent on
$\cos j\theta$ where $j=0,\cdots,3$, which can be separated into four
equations thanks to orthonormality. Here, the equations related to the terms
multiplying $\cos 2\theta$ and $\cos 3\theta$ can be balanced by considering
higher harmonics of the distributed load ${q\left(\theta\right)}$ but are
disregarded due to irrelevance. Yet, the other two equations which are related
to the leading harmonics, yield the following closed form expressions of the
axial force and bending moment applied externally by the neighboring frusta:

$F\approx
4{C_{1}}\zeta\left({2{\zeta^{2}}-2\zeta_{0}^{2}+3r_{i}^{2}{\varphi^{2}}}\right)+2{C_{2}}\left({\zeta-{\zeta_{0}}}\right)-{C_{3}}p,$
(6a) $M\approx
r_{i}^{2}\varphi\left[{{C_{1}}\left({12{\zeta^{2}}-4\zeta_{0}^{2}+3r_{i}^{2}{\varphi^{2}}}\right)+{C_{2}}}\right],$
(6b)

where ${\zeta_{0}}\approx\left({{r_{o}}-{r_{i}}}\right){\psi_{0}}$ is the
load-free deflection of the frustum, whereas

${C_{1}}=\frac{\pi
Eh}{8\left(r_{o}-r_{i}\right)^{3}}\left(\frac{r_{o}+r_{i}}{2}-\frac{r_{o}-r_{i}}{\ln\left(r_{o}/r_{i}\right)}\right),$
${C_{2}}=\frac{\pi
Eh^{3}\ln\left(r_{o}/r_{i}\right)}{12\left(r_{o}-r_{i}\right)^{2}},$

and

${C_{3}}=\pi\frac{r_{o}^{2}-2r_{i}^{2}+r_{o}r_{i}}{3},$

are constant coefficients.

Combining the integrations of equations (6a) and (6b) with respect to the
axisymmetric and antisymmetric DOFs, followed by retrieving the general
subscript $n,k$ yields the potential energy of a general frustum, given by:

$\begin{array}[]{l}{{\cal V}_{n,k}}\approx
2{C_{1}}\zeta_{n,k}^{4}+\left({{C_{2}}-4{C_{1}}\zeta_{0,k}^{2}}\right)\zeta_{n,k}^{2}\\\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,-\left({2{C_{2}}{\zeta_{0,k}}+{C_{3}}p}\right){\zeta_{n,k}}+\frac{{3{C_{1}}r_{i}^{4}}}{4}\varphi_{n,k}^{4}\\\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,+\frac{{{C_{2}}-4{C_{1}}\zeta_{0,k}^{2}}}{2}r_{i}^{2}\varphi_{n,k}^{2}+6{C_{1}}r_{i}^{2}\zeta_{n,k}^{2}\varphi_{n,k}^{2}+Const.\end{array}$
(7)

Here, assuming all odd frusta are identical as well as all even frusta,
${\zeta_{0,k}}$ denotes the load-free deflection of the kth frusta in each nth
multistable cell. The overall potential energy of the system in Figure 1 (a)
is computed by the summation of those corresponding to all frusta, thus is
given by ${\cal V}=\sum\limits_{n=1}^{N}{\sum\limits_{k=1}^{2}{{{\cal
V}_{n,k}}}}$. Therefore, applying the Hamilton’s principle (e.g. [24]) on this
summation alongside the kinetic energy, dissipation function, and virtual work
derived in the previous section yields a system of $4N$ ordinary differential
equations, governing the dynamics of the system under investigation.

##  III Results

The theoretical model derived above is examined based on the experimental
demonstrator presented in Figure 3. This figure displays three of the
demonstrator’s stable states alongside their corresponding theoretical forms
achieved numerically from the model, considering body forces where the
gravitational acceleration is taken as $g={\rm{9.795}}$
${{\rm{m}}\mathord{\left/{\vphantom{{\rm{m}}{{{\rm{s}}^{\rm{2}}}}}}\right.\kern-1.2pt}{{{\rm{s}}^{\rm{2}}}}}$.
The demonstrator consists of a water-filled truncated off-the-shelf toy straw
with nine multistable cells, clamped horizontally such that $\varphi_{0}=90$
$\rm{deg}$. In agreement with the theoretical system, the fixed end of the
straw is connected to a water reservoir through a narrow channel, whereas its
free end is sealed with a 3D printed part. The estimated mass and moment of
inertia of this sealing with respect to the center of the face which is
farthest from the inlet are taken as ${{m_{{\rm{tip}}}}}=13.45$ gr and
${{I_{{\rm{tip}}}}}=2986.91$ ${\rm{gr}}\cdot{{\rm{mm}}^{2}}$, considering the
entrapped fluid. The motion of the demonstrator is governed by an ELVEFLOW OB1
MK3 piezoelectric pressure controller having a settling time of down to 35 ms,
connected to the reservoir through a second channel which is filled with air.
The two channels as well as the water reservoir and the connections between
the different elements constitute a complex channel system whose effective
geometry is to be determined empirically, see section III.1. Finally, the
motion of the system is captured by tracking a ${38.5\times 38.5}$
${\rm{mm}^{2}}$ checkerboard with an ${8\times 8}$ grid glued to the free end
of the straw, utilizing a GoPro HERO6 BLACK camera, capturing 240 frames per
second in a resolution of 1080p. Here, since there is an offset of
approximately 8.8 mm between the checkerboard’s plane and the face of the
ninth multistable cell which is considered the sealed end, this offset is
subtracted from the measurements in post processing.

To examine the theoretically predicted behavior of the system based on the
experimental demonstrator, the model’s physical and geometrical parameters
should be computed. Indeed, the upcoming section presents the algorithm used
to efficiently estimate most parameters of the system, alongside a numerically
justified assumption made in order to simplify the model, for reducing the
computational effort of its simulations. Next, since the fitted parameters are
sufficient to describe the quasi-static behavior of the system, the
theoretical elastic behavior of a single frustum is validated and
investigated, utilizing finite element simulations. Finally, by estimating the
remaining parameters, the overall dynamic behavior of the system is examined
utilizing the experimental setup.

Figure 3: (a,c,e) The experimental demonstrator in three zero gauge pressure,
statically stable states. (b,d,f) The corresponding theoretically predicted
states.

###  III.1 Estimation of the model’s parameters

The first stage of the parameter estimation process is finding the effective
geometry of the channel system. For this, a set of 38 experiments where a
${V=90}$ ${\rm{cm}^{3}}$ vessel was filled through the channel system while
the latter was detached from the straw, was executed. Assuming the flow is
mostly steady, the flow rate is approximated as ${\rm{d}}V/{\rm{d}}t\approx
V/t_{f}$, where $t_{f}$ denotes the time taken to fill the vessel. Under this
assumption, the filling times $t_{f}$ and the known pressure differences
$\Delta
p\left(t\right)\buildrel\Delta\over{=}{p_{{\rm{external}}}}\left(t\right)-p\left(t\right)$
spanning between 5 kPa and 90 kPa, are used to calculate the proportion
coefficient that according to (2) quadratically relates the volumetric flow
rate to the pressure difference. Indeed, Figure 8 in Appendix A shows that the
coefficient achieved by curve fitting considering the theoretical Bernoulli
model captures well the pressure-flow rate relation, thus it is utilized to
compute the effective radius of the channel which is given by $r_{ch}$=0.52 mm
while taking the fluid’s density as $\rho=1000$ ${\rm{kg/m^{3}}}$.

As mentioned above, the experimental demonstrator includes a straw with nine
multistable cells, meaning that under the assumption of a two-dimensional
motion, the demonstrator should be described by 36 DOFs. However, since the
computational effort needed to simulate a large number of equations is high,
the DOFs of the odd frusta are disregarded meaning that they are considered
rigid, which leaves the model with 18 DOFs. This assumption which is employed
in the remaining steps of the parameter estimation process as well as the
analyses presented in section III.3, is supported by the observations that
${\zeta_{0,1}}\approx 1.66{\left|{{\zeta_{0,2}}}\right|}$ (see next
paragraph), and that the thickness of the odd frusta is higher than this of
the even ones, both significantly increase the stiffness of the odd frusta.
This simplification is further justified, based on numerical analyses, as
discussed in section III.2.

The next stage is determining the geometrical parameters as well as the Young
modulus of the solid. For this, first a caliper having a resolution of 0.01 mm
was used to measure the inner and outer radii of the frusta, according to
which $r_{i}=$11.1 mm and $r_{o}=$14.65 mm. Moreover, it was utilized to
measure the axial lengths of 63 folded cells and 26 deployed cells, showing
that the distance between the bases of a single folded cell is 1.5 mm whereas
the corresponding value in a deployed cell is 5.8 mm. Further, the pressure
values causing each active frustum to lose stability from its folded and
deployed states while the straw is in a predetermined configuration, were
assessed following a series of experiments. In these experiments, the pressure
in which a frustum loses stability from a folded state was achieved by
deploying the entire straw except for the frustum under investigation,
followed by elevating the pressure in increments of 0.5 kPa until the frustum
post-buckled to a different state. A similar procedure was executed to examine
the pressure causing each frustum to lose stability from its deployed state.
In this case all other frusta were folded, and the negative pressure value
causing the frustum under investigation to post-buckle into a different state
was evaluated. These experiments showed high consistency, where all active
frusta except for the one farthest from the base lost stability from their
deployed and folded states in approximately -13.5 kPa and 22.5 kPa,
respectively. Thus, for simplicity the properties of all active frusta are
considered identical. The main cause for the asymmetry between the pressure
values leading to instability stems from the elastic properties of the frusta.
This can be seen analytically from Eq. (6a) and graphically from Figures 4 and
5 (both thoroughly discussed in section III.2), which show that if the
thickness of a frustum is not infinitely small its elastic behavior is not
symmetrical around $\zeta_{n,k}=0$. Namely, $\zeta_{0,k}$ serves as a stable
equilibrium state, whereas the absolute value of the second stable
axisymmetric equilibrium is smaller than $\left|{{\zeta_{0,k}}}\right|$.
Furthermore, Figure 5 clearly shows that among the axisymmetric stable
equilibria, the one with the smaller deflection loses stability under a lower
pressure variation. Consequently, since the absolute pressure value that leads
to instability at the snap-up state is smaller than the corresponding value of
the snap-down state, $\zeta_{0,2}$ must be negative. Therefore, according to
the abovementioned axial lengths of the folded and deployed cells,
${\zeta_{0,1}}+{\zeta_{0,2}}=1.5$ mm. Finally, to compute the missing
parameters, the values specified above are used in a quasi-static analysis
geared to find the pressure values that lead to the instabilities of the first
active frustum. This analysis numerically describes the system according to
the theoretical model while dropping all time-derivatives, yet it considers
the effect of gravity, excluding the negligible torque originating in the
offset between the sealing’s center of mass, and the straw’s closed end.
Utilizing the nonlinear algebraic system achieved by these means, the stable
states corresponding to the ones examined experimentally to assess the
stability of the first active frustum, are computed for different pressures
values. For each pressure, the eigenvalues of the $2\times 2$ Jacobian which
considers only the DOFs of the first active frustum and the corresponding
equations are calculated to examine stability, where a non-positive eigenvalue
indicates on instability. According to this analysis, the parameters leading
to instability at the formerly mentioned pressure values and meet the
geometrical constraints discussed above are: ${\zeta_{0,1}}=3.78$ mm,
${\zeta_{0,2}}=-2.28$ mm, $h=0.508$ mm and $E=0.92$ GPa. Due to the model’s
simplifying assumptions, these should be referred to as effective values,
rather than the correct physical values.

The last two parameters to be calculated are the damping coefficients of the
different DOFs, recalling that for simplicity the coefficients related to all
axisymmetric DOFs are considered identical, as well as those related to the
antisymmetric DOFs. The calibration of these parameters is thoroughly
discussed in section III.3, as part of the experimental verification of the
theoretical model’s overall dynamics.

###  III.2 Numerical verification and investigation of the system’s elastic
properties

To start shedding light on the behavior of the system under investigation,
this section begins with examining the elastic nature of a single active
frustum while it is not subjected to pressure. This is done based on the
theoretical formulation, and finite element analyses. Indeed, Figure 4 shows
the strain energy function of a single active frustum based on Eq. (7)
utilizing the parameters specified above. This figure also presents the
frustum’s equilibrium states achieved by nulling Eq. (6a) and Eq. (6b), where
their stability is classified based on the eigenvalues of the corresponding
Jacobians. Namely, a certain equilibrium state is stable if both eigenvalues
of the $2\times 2$ partial derivative matrix of Eq. (6a) and Eq. (6b) with
respect to the frustum’s DOFs, are positive around this state. Figure 4
further shows several cutouts of the theoretical strain energy function where
either the axisymmetric or the antisymmetric DOF is fixed. These are presented
alongside the corresponding cutouts achieved from a finite element scheme
devised in COMSOL Multiphyics, where the Poisson ratio which is absent from
the theoretical model is taken arbitrarily as $\nu=0.33$. In this scheme, the
frustum is discretized by second order rectangular shell elements modelled
according to Reissner–Mindlin shell theory which considers shear deformations.
These elements are distributed uniformly such that there are 15 elements in
the radial direction and 400 elements along the circumference. The curves
extracted from the finite element scheme are computed utilizing a built-in
function applied on the results of numerous stationary simulations, in which
the displacement of the frustum’s small base along the $\hat{z}$ axis is
dictated, whereas its large base is bounded to a plane in a way that allows
free rotations in the tangential direction, in agreement with the model’s
boundary conditions.

Figure 4: (a) The theoretical potential energy of a single active frustum
under zero gauge pressure according to (7), alongside the stable and unstable
equilibria. (b) and (c) Comparison between the theoretical values (orange
dashed curves) and the numerically calculated values (black solid curves) of
the potential energy, at the cutouts described by the orange dashed lines in
panel (a).

Figure 4 shows a very good agreement between the theoretical and the
numerically obtained results in moderately small values of the DOFs. Yet,
significant values of $\zeta_{n,2}$ and $\varphi_{n,2}$ lead to a weaker
correlation as they result in significant base angles, which are assumed to be
small in the model derivation. Nevertheless, the model manages to capture the
multistability of the frustum, and its range of validity seems adequate to
capture most reasonable deformations. Figure 4 further shows that when the
frustum is not subjected to pressure it has nine equilibrium states, four of
which are stable and correspond to folded and deployed axisymmetric states, as
well as two antisymmetric states, see graphical demonstration in Figure 5.
These stable states provide a non-pressurized straw the capability of staying
stable in a large number of equilibrium states where each active frustum can
be in any of these four states. Finally, as formerly mentioned, Figure 4 shows
that in the realistic case where the thickness of the frustum is not
infinitely small, its potential energy and equilibrium states are not
symmetrical around $\zeta_{n,k}=0$.

To complete the picture of the quasi-static behavior of a single active
frustum, Figure 5 shows its analytically obtained equilibrium states while it
is subjected to different pressure values. This figure shows four bifurcation
pressure values, separating five regions with different number of stable
states. The first region corresponds to small absolute pressure values thus is
described qualitatively by Figure 4, meaning that in this region the frustum
has four stable states. Conversely, in high positive gauge pressure values the
frustum can stay stable only in a deployed state, and similarly in high
negative pressure values the only stable state is the folded state. At the two
supplementary regions achieved in moderate pressure values (positive or
negative), the frustum has two stable equilibria, corresponding to both
deployed and folded states. Finally, from Figure 4 and Figure 5 one should
conclude that in slow motions where the inertial forces are weak, it is
energetically preferable for the frustum to exhibit antisymmetrical motion,
while switching between folded and deployed states. As seen in the upcoming
section, in some cases it might pass through a stable antisymmetric
equilibrium.

Figure 5: Three projected views describing the dependency of the equilibrium
states of a single elastic frustum, on the pressure applied on it directly.
The red curves describe the unstable branches, whereas the blue curves
represent the stable branches, corresponding to the states drawn on the
$\zeta_{n,2}-\varphi_{n,2}$ plane.

The last analysis executed which deals with the system’s elastic properties
examines the assumption that all odd frusta can be considered rigid. This
analysis utilizes a finite element scheme established in COMSOL Multiphysics,
which is similar to the one discussed above. Yet, in this case the analysis
describes a complete multistable cell composed of two interconnected frusta
with identical thicknesses, whose unstressed axisymmetric deflections are
$\zeta_{0,1}$ and $\zeta_{0,2}$. In agreement with the model’s assumptions,
the bases of both frusta are allowed to rotate freely around the tangential
direction where their large bases are forced to have identical translational
motion. Further, the displacements between the small bases are dictated and
denoted ${\zeta_{{\rm{dictated}}}}$ and ${\varphi_{{\rm{dictated}}}}$, where
in similar to the scheme discussed above, each frustum is discretized by
second order rectangular shell elements, dividing it to 15 radial and 400
tangential evenly distributed segments. Based on this scheme, Figure 9 in
Appendix B compares the idealized deformations of the cell represented only by
those of the frustum with the smaller unstressed deflection, and the
deformations of this frustum in the more realistic case where the second
frustum is not considered rigid. In the latter case, the deflections of the
relevant frustum are computed based on the relative deformations of its bases,
utilizing line integration to describe only their rigid body motion. Figure 9
shows that in moderate values of the DOFs which suit most practical
deformations, the deviations between the realistic and the idealized cases are
small, implying that the model simplification which truncates half of its
DOFs, is indeed applicable.

###  III.3 Experimental validation and investigation of the overall dynamics

To validate and examine the overall dynamic behavior of the theoretical model,
the latter is simulated by a numerical scheme of its truncated form which
consists of 18 coupled ordinary differential equations. These describe both
axisymmetric and antisymmetric DOFs of all nine active frusta. In agreement
with the experimental setup, each simulation starts from a stable state
achieved by solving the static form of the governing equations which neglects
all time dependencies, see discussion in section III.1. Next, the dynamic part
of the simulation is executed by numerical integration of the non-degenerated
equations under a pressure excitation given by
${p_{{\rm{external}}}}\left(t\right)={p_{ss}}\left[{1-\exp\left({-50\,\,{{\mathop{\rm
s}\nolimits}^{-1}}\cdot t}\right)}\right]$, bringing the input pressure to
$97\%$ of the desired value denoted $p_{ss}$ after 70 $\rm{ms}$, in agreement
with the rising time measured experimentally. The geometrical and physical
parameters utilized in all numerical simulations are those specified above.
However, recalling that the damping coefficients ${c_{\zeta}}$ and
${c_{\varphi}}$ which are affected by both structural and fluidic effects are
yet to be quantified, these are manually fitted to best describe each
experiment. Furthermore, to examine the assumption claiming that the fluidic
effects related to the flow inside the water reservoir and the straw can be
neglected, the effective radius of the channel is also readjusted, followed by
comparison to its pre-calibrated value.

Figure 6 (a,b,c) compares the experimentally measured and theoretically
simulated dynamic responses of the demonstrator’s free end, while being folded
by an external steady-state gauge pressure of $p_{ss}=-25$ kPa. Here, the
experiment and corresponding simulation start from an initial state where the
first and fourth multistable cells are deployed, and all others are folded.
The manually fitted damping coefficients considered in the simulation are
$c_{\zeta}=620$ $\rm{N\cdot s/m}$ and $c_{\varphi}=7\times 10^{-4}$
$\rm{N\cdot m\cdot s/rad}$, whereas the effective radius of the channel is
taken as $r_{ch}=0.43$ $\rm{mm}$, which is $82.5\%$ of its pre-calibrated
value. Similarly, Figure 7 (a,b,c) compares the measured and theoretical
responses of the system’s free end throughout deployment by an external
steady-state gauge pressure of $p_{ss}=35$ kPa. Here, the system begins from
an initial state where the first and fourth cells are folded, and all others
are deployed. The damping coefficients utilized in this case are
$c_{\zeta}=0.1$ $\rm{N\cdot s/m}$ and $c_{\varphi}=3\times 10^{-3}$
$\rm{N\cdot m\cdot s/rad}$, whereas the effective radius of the channel is
taken as $r_{ch}=0.315$ $\rm{mm}$, which is $60.5\%$ of its pre-calibrated
value. Further, to analyse the internal motion of the system, Figure 6 (d,e,f)
and Figure 7 (d,e,f) show the numerically simulated theoretical responses of
the different DOFs in time, and on the configuration space alongside the
potential functions of the different frusta considering the steady-state
pressure. Finally, the video in the supporting information, whose caption
appears in Appendix D, shows the theoretically simulated and the
experimentally obtained responses, corresponding to Figure 6 and Figure 7.

Figure 6 shows that the model well describes the dynamic behavior of the
system which folds throughout two steps, where in each step one of the
deployed frusta snaps through an antisymmetric stable state, into a fully
folded state, see panels (d,e,f). This figure further shows an overall high
quantitative correlation between the experiment and the theory, as the model
manages to capture both the magnitude and time scales of the rotational and
horizontal motion of the straw’s free end. However, the theoretical motion
along the vertical axis is in a lesser agreement with the corresponding
experimental response, which is reflected in a strong overestimation of the
magnitude throughout the first snap-through post-buckling. One reason for this
overestimation is that even though the torque applied to the first active
frustum is higher than this applied to the fourth active frustum, in practice
the first one to post-buckle is the latter due to non-uniformity in the
physical properties of the different frusta. Other possible explanations of
the deviations are unmodeled effects such as longitudinal curvatures of the
unloaded straw, originating from internal stresses. Finally, according to
Figure 6 (d) showing the theoretical dynamic responses of the active frusta on
the configuration space, the folding of the initially deployed frusta hardly
deviates from the upper bifurcation branch presented in Figure 5. This
indicates that the behavior of the system is governed by its potential energy,
which implies on weak inertial effects.

Figure 6: (a,b,c) Comparison between the experimentally measured and
theoretically simulated motion of the straw’s free end along (a) the
horizontal and (b) the vertical axes, as well as (c) its orientation, while
being folded due to a steady-state external pressure of -25 kPa. (d) The
theoretical dynamic responses of all frusta on the configuration space, and
the potential energy of each frustum when subjected to the steady-state static
pressure. (e,f) The theoretical time responses of (e) the axisymmetric and (f)
the antisymmetric DOFs of all frusta. Figure 7: (a,b,c) Comparison between the
experimentally measured and theoretically simulated motion (as achieved
originally and after a delay of 0.77 s) of the straw’s free end along (a) the
horizontal and (b) the vertical axes, as well as (c) its orientation,
throughout deployment with an external steady-state pressure of 35 kPa. (d)
The theoretical dynamic responses of all frusta on the configuration space,
and the potential energy of each frustum when subjected to the steady-state
static pressure. (e,f) The theoretical time responses of (e) the axisymmetric
and (f) the antisymmetric DOFs of all frusta.

Next, Figure 7 shows that the significantly lower damping coefficient related
to the axisymmetric DOFs, achieved when deploying the system leads to a
radically different dynamic behavior. In this case the system’s inertial
forces are much more significant compared to those achieved while the system
is being folded. Consequently, each frustum snaps from a folded state to a
deployed state with only a slight antisymmetric motion and without passing
through an intermediate state. Figure 7 further shows that the model
accurately describes the magnitude of both translational and rotational
motions of the straw’s closed end, as well as the frequency and decay rates of
the oscillations occurring after each snap-through post-buckling. However,
even though the model also manages to capture the time-period between both
snaps, it significantly underestimates the time until the first one.
Nevertheless, several repetitions of the same experiment show almost identical
dynamic responses, where in each repetition the initial time period is
shortened, see Figure 10 in Appendix C. The latter implies that this deviation
originates from unmodeled effects such as cohesion forces acting between the
folded frusta, or non-negligible stiffness of the thin creases connecting
them, which weaken in each cycle. As seen in Figure 7, a delay of 0.77 s in
the theoretical dynamic response compensates on the initial period
underestimation, thus leads to a good correlation with the measurements. Yet,
it can be seen that despite the effect of gravity, throughout the first snap,
the frustum that underwent post-buckling bent upwards since it is not
perfectly axisymmetric. Further, as already mentioned, since the fourth active
frustum is weaker than the first one, in contrast to the theoretically
predicted response, it was the first one to undergo post-buckling.

Recalling that the theoretical responses presented in Figure 6 and Figure 7
are achieved utilizing different damping coefficients and equivalent radii of
the channel, the model can be adjusted to different configurations and
responses merely by fitting the parameters that are affected by fluidic
effects. Thus, to reduce the number of parameters to be calibrated,
theoretical modeling of the fluid in both the straw and the water reservoir
should be added to the model derived in section II. A possible fluidic effect
can be thin film damping related to the fluid trapped between closely adjacent
frusta in partially- or fully-folded multistable cells. Other possible effects
can originate from inertial forces of the fluid in both the water reservoir
and the straw, which increase the resistance to folding and deployment of the
latter, and consequently reduce the channel’s effective radius.

##  IV Conclusion

The large number of complex stable equilibrium states attainable by straw-like
elements make them promising reconfigurable structures. Thus, the ability to
predict their dynamics has a great potential for countless practical
applications, spanning from deployable space structures to soft robots.
Indeed, we derived a comprehensive model describing the dynamics of fluid-
filled straw-like elements, consisting of interconnected multistable building
blocks, each of which has up to four stable equilibria. After the theoretical
derivation, we followed an algorithm geared to efficiently estimate most
parameters of the system, for sake of adjusting the model to a specific
experimental setup. Next, we validated the theoretical quasi-static behavior
of the constituent building blocks compared to finite element analyses, and
numerically studied this behavior to gain physical insights regarding the
system’s dynamics when inertial effects are weak. Finally, we experimentally
showed that by calibrating a few parameters which are influenced by fluidic
effects, it is possible to capture various dynamic behaviors achieved under
different conditions. The latter implies that the theoretical model captures
well all the dominant effects of the highly complex system studied here,
except for those related to the fluid which we modelled in a simplistic
manner. Nevertheless, either a simple calibration, or a rigorous supplementary
derivation of these effects can adjust the model to different configurations
that exhibit distinct fluidic phenomena, making the suggested model universal.
The theoretical model can be further explored to gain a deeper understanding
of the dynamics of straw-like elements, and it can be leveraged to describe
structures with higher complexity, consisting of hierarchical configurations
of such elements.

##  Appendix A Identification of the channel’s effective radius

Figure 8 which is discussed in section III.1, shows the experimentally
obtained relation between the time-averaged flow rate in the channel system,
and the dictated pressure difference between its ends, while the straw was
detached from the system. The raw data presented in this figure is utilized to
calculate the effective radius of the channel by the shown fitted steady
Bernoulli formulation.

Figure 8: The measured relation between the dictated pressure difference among
the channel system’s ends, and the time-averaged flow rate in absence of the
sealed straw, alongside the fitted Bernoulli model.

##  Appendix B The validity of the rigidization of the odd frusta

Figure 9 mentioned in section III.2, displays the deformations of the frustum
with the smaller unstressed axisymmetric deflection, under dictated relative
displacements between the bases of a whole multistable cell. The results are
based on finite element analyses, and the simplification which claims that the
deformations of the frustum that has a larger unstressed axisymmetric
deflection, can be disregarded.

Figure 9: The deformations of a single active frustum under dictated relative
displacements between the bases of a whole multistable cell, according to
finite-element analyses, and based on the idealization under which the frustum
with the larger unstressed axisymmetric deflection can be considered rigid.
The presented comparison is along (a) the horizontal and (b) the vertical
cutouts of Figure 4.

##  Appendix C Comparison between similar dynamic experiments

Figure 10 shows four repetitions of the same dynamic deployment experiment,
used to examine the claim that the period between the first and second snaps
hardly changed, see section III.3.

Figure 10: The orientation of the straw’s free end, the period between the
pressure elevation and the first snap, and the time between the two snaps, in
four repetitions of the experiment in which the first and fourth active frusta
are deployed by a steady-state external input pressure of 35 kPa.

##  Appendix D Supplementary video

The video that appears as supplementary information, shows the comparison
between the theoretically predicted and the experimentally obtained dynamics
of the system throughout folding and deploying of two distinct frusta. These
responses correspond to Figure 6 and Figure 7, presented and discussed in
III.3

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