Magneto-origami structures: engineering multi-stability and dynamics via magnetic-elastic coupling

Miura-ori pattern is a kind of degree-4 vertex crease pattern with a pair of collinear creases and reflectional symmetry. Two Miura-ori cells with kinematic compatibility can be assembled into a SMO structure [41, 42]. The bottom and the top Miura-ori cells are characterized by crease lengths ${a}_{k}$ and ${b}_{k},$ and a sector angle ${\gamma }_{k},$ where $k$ takes 'A' or 'B', denoting the bottom cell A and the top cell B, respectively (figure 1(a)). They have to satisfy the following kinematic compatibility conditions

$$\begin{eqnarray}&&{b}_{{\rm{A}}}={b}_{{\rm{B}}}=b,\,{a}_{{\rm{B}}}\,\cos \,{\gamma }_{{\rm{B}}}={a}_{{\rm{A}}}\,\cos \,{\gamma }_{{\rm{A}}}.\end{eqnarray} \tag{ 1 }$$

Folding of the SMO structure is a one degree-of-freedom (DoF) mechanism that can be described by the folding angles ${\theta }_{{\rm{A}}}$ or ${\theta }_{{\rm{B}}},$ they are not independent to each other but satisfy

$$\begin{eqnarray}&&{\theta }_{{\rm{B}}}=\arccos \left(\cos \,{\theta }_{{\rm{A}}}\tan \,{\gamma }_{{\rm{A}}}/\tan \,{\gamma }_{{\rm{B}}}\right).\end{eqnarray} \tag{ 2 }$$

The outer dimensions of the SMO structure, i.e. the length ${L}_{{\rm{M}}},$ the width ${W}_{{\rm{M}}},$ and the height ${H}_{{\rm{M}}},$ can be expressed in terms of ${\theta }_{{\rm{A}}}$ (here and in what follows, the subscript 'M' denotes the Miura-ori)

$$\begin{eqnarray}&&\begin{array}{l}{L}_{{\rm{M}}}=\displaystyle \frac{2b\,\cos \,{\theta }_{{\rm{A}}}\,\tan \,{\gamma }_{{\rm{A}}}}{\sqrt{1+{\cos }^{2}{\theta }_{{\rm{A}}}{\tan }^{2}{\gamma }_{{\rm{A}}}}},\\ {W}_{{\rm{M}}}=2{a}_{{\rm{A}}}\sqrt{1-{\sin }^{2}{\theta }_{{\rm{A}}}{\sin }^{2}{\gamma }_{{\rm{A}}}},\\ {H}_{{\rm{M}}}={a}_{{\rm{B}}}\,\sin \,{\theta }_{{\rm{B}}}\,\sin \,{\gamma }_{{\rm{B}}}-{a}_{{\rm{A}}}\,\sin \,{\theta }_{{\rm{A}}}\,\sin \,{\gamma }_{{\rm{A}}}.\end{array}\end{eqnarray} \tag{ 3 }$$

Folding of the structure can also be characterized by the dihedral angles between facets ${\rho }_{ki}$ and the dihedral angles at the connecting creases ${\rho }_{C},$ where $k=A,\,{\rm{B}}$ and $i=1,2,3,4$ (figure 1(a)). They can also be expressed as functions of ${\theta }_{{\rm{A}}}$

$$\begin{eqnarray}&&\begin{array}{l}{\rho }_{{\rm{C}}}={\theta }_{{\rm{B}}}-{\theta }_{{\rm{A}}},\,{\rho }_{k1}={\rho }_{k3}=\pi -2{\theta }_{k},\\ {\rho }_{k2}=2\arccos \displaystyle \frac{\sin \,{\theta }_{k}\,\cos \,{\gamma }_{k}}{\sqrt{1-{\sin }^{2}{\theta }_{k}{\sin }^{2}{\gamma }_{k}}},\\ {\rho }_{k4}=2\pi -{\rho }_{k2},\,k={\rm{A}},{\rm{B}}.\end{array}\end{eqnarray} \tag{ 4 }$$

Based on whether the bottom cell A nests into or bulges out from the top cell B, the SMO structure could exhibit two qualitatively different configurations. For clarity, the configurations with ${\theta }_{{\rm{A}}}\gt 0$ and ${\theta }_{{\rm{A}}}\lt 0$ are denoted as nested-in and bulged-out, respectively (figure 1(a)).

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An SMO structure possesses 12 vertices, 20 creases, and 8 facets that can be employed for magnet integration. Figure 1(b) displays three different designs for embedding magnets. In Design I, two magnets are placed at the top vertex '2' and the bottom vertex '11'; in Design II, four magnets are positioned at four coplanar vertices ('4', '6', '7', '9'); in Design III, eight magnets are embedded at the facet centers. In addition to changing the magnet arrangement, the design can be further enriched by adjusting the magnetic polarization. In Design I, the two magnets can be set with their like poles or opposite poles facing each other, giving rise to the repulsive-magnet and the attractive-magnet layouts, respectively (figure 1(c)). In Design II, the four coplanar magnets can make up 16 different pole layouts, which can be deducted into 5 by considering the symmetry among the four magnets and the identity among certain pole-pole relationships (figure 1(d)). In Design III, the 8 magnets could constitute variegated 3D pole layouts; for demonstration purpose, a few examples are given in figure 1(e).

During folding (${\theta }_{{\rm{A}}}$ varies between $-\pi /2$ and $\pi /2$), the outer dimensions of the SMO structure would experience significant changes [41, 42], which would, therefore, alter the relative distances among the magnets and change the system's stability profile. This will be detailed in section 3.

The crease pattern of a Kresling-ori [33] is made up of a row of $n$ identical parallelogram panels, each parallelogram panel is defined by three length parameters: ${a}_{{\rm{K}}}$ and ${b}_{{\rm{K}}}$ are the side lengths, ${l}_{{\rm{K}}}$ is the length of the long diagonal (here and in what follows, the subscript 'K' denotes the Kresling-ori) (figure 2(a)). Construction of the structure from the flat crease pattern is achieved by folding and rolling it into a polygonal prism such that points 'A' and 'B' overlap with 'A^*' and 'B^*'. Geometries of the folded structure can be described by the circumradius of the basal polygon ${R}_{{\rm{K}}},$ the height ${H}_{{\rm{K}}},$ and the relative rotation angle $\alpha$ of the top polygon with respect to the frame fixed to the bottom polygon (figure 2(b)). Variation in the polygon circumradius ${R}_{{\rm{K}}}$ would scale the size of the origami; the height ${H}_{{\rm{K}}}$ or the rotation angle $\alpha$ are used to characterize the structure configuration.

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To quantify the aspect ratio of the structure and the degree of transformation during folding, an angle ratio $\lambda$ is defined, which is the ratio of the angle $\angle {\rm{CAD}}$ to half the internal angle of the basal polygon ${\theta }_{{\rm{K}}}.$ The value of $\lambda$ is bounded by $0.5\leqslant \lambda \leqslant 1.0$ [33], where the lower bound indicates a limiting case that the structure height ${H}_{{\rm{K}}}=0,$ constructed from two offset polygon bases connected by triangular panels, and the upper bound indicates a limiting case that the parallelogram panels become rectangular panels such that ${H}_{{\rm{K}}}={b}_{{\rm{K}}}={l}_{{\rm{K}}}\,\sin (\lambda {\theta }_{{\rm{K}}}).$ With a smaller value of $\lambda ,$ the structure height ${H}_{{\rm{K}}}$ and the rotation angle $\alpha$ vary less during folding. $\lambda \gt 1$ indicates a change in chirality, i.e. anticlockwise rotation of the top polygon will result in an expansion of the structure. Negative and positive chirality are mathematically identical; hence, situations with $\lambda \gt 1$ are not considered here. With parameters ${R}_{{\rm{K}}},$ $\lambda ,$ and $n,$ the remaining lengths and angles (${a}_{{\rm{K}}},\,{b}_{{\rm{K}}},\,{\theta }_{{\rm{K}}}$ and ${l}_{{\rm{K}}}$) can be determined from the crease pattern (figure 2(a)), the folded structure, and the basal polygon (figure 2(b)):

$$\begin{eqnarray}&&\begin{array}{l}{a}_{{\rm{K}}}=2{R}_{{\rm{K}}}\,\sin \left(\pi /n\right),\\ {b}_{{\rm{K}}}={\left({{l}_{{\rm{K}}}}^{2}+{{a}_{{\rm{K}}}}^{2}-2{l}_{{\rm{K}}}{a}_{{\rm{K}}}\cos (\lambda {\theta }_{{\rm{K}}})\right)}^{1/2},\\ {\theta }_{{\rm{K}}}=\pi (n-2)/2n,\,{l}_{{\rm{K}}}=2{R}_{{\rm{K}}}\,\cos \left({\theta }_{{\rm{K}}}(1-\lambda )\right).\end{array}\end{eqnarray} \tag{ 5 }$$

Note that each vertex in the Kresling-ori crease pattern has one DoF. Hence, for $0.5\leqslant \lambda \leqslant 1.0,$ overlapping of the vertices 'A', 'B', and 'A^*', 'B^*' could generate an 'open' or a 'closed' polygonal prism via rigid folding. However, if these overlapped vertices are fixed together (e.g. by gluing), the relative position of each vertex with respect to its neighbors will be fully defined, and the generated structure will be kinematically rigid at either the 'open' or the 'closed' configuration. That is, if the triangular panels are truly rigid, the expanded polygonal prism cannot be contracted, and vice versa. Only if the panels or the creases are deformable, the structure is possible to exhibit snapping transformations between the 'open' and the 'closed' configurations (figure 2(c)). Applying a force on the top of the structure and allowing free rotation, the structure will twist and contract; inversely, applying a pull force on the top, the structure will twist and expand. Note that such folding of the Kresling-ori structure is no longer rigid.

To determine the structure configuration, i.e. the height ${H}_{{\rm{K}}}$ and the rotation angle $\alpha ,$ the following vector loop equation and kinematic constraint equations (figure 2(d)) have to be satisfied [33]:

$$\begin{eqnarray}&&\begin{array}{l}{{\bf{R}}}_{{\rm{AB}}}+{{\bf{R}}}_{{\rm{BO}}}+{{\bf{R}}}_{{\rm{OC}}}+{{\bf{R}}}_{{\rm{CA}}}=0,\\ {\rm{Constraint}}\,1:\,{{\bf{R}}}_{{\rm{BC}}}\cdot {{\bf{R}}}_{{\rm{CA}}}=-{l}_{{\rm{K}}}{a}_{{\rm{K}}}\,\cos (\lambda {\theta }_{{\rm{K}}}),\\ {\rm{Constraint}}\,2:\,\parallel {{\bf{R}}}_{{\rm{CA}}}\parallel ={l}_{{\rm{K}}}.\end{array}\end{eqnarray} \tag{ 6 }$$

Constraint 1 implies that the angle between ${{\bf{R}}}_{{\rm{BC}}}$ and ${{\bf{R}}}_{{\rm{CA}}}$ is a constant $\lambda {\theta }_{{\rm{K}}};$ Constraint 2 indicates that the fold $\overline{{\rm{AC}}}$ is rigid so that its length remains a constant ${l}_{{\rm{K}}}$ (in this paper, $\parallel \cdot \parallel$ denotes the Euclidean length). In addition, the length of the fold $\overline{{\rm{AB}}}$ is mathematically assumed to be variable, so that an additional DoF is introduced that allows the structure to be foldable between the open and the closed configurations. Detailed procedures for determining the folding kinematics are briefly summarized as:

• Step 1:  
  Solve the initial rotation angle ${\alpha }_{0}$ via the vector loop equation and Constraint 1;
• Step 2:  
  Solve the initial height ${H}_{{\rm{K0}}}$ via Constraint 2;
• Step 3:  
  Solve the final rotation angle ${\alpha }_{{\rm{End}}}$ via Constraint 2 and with ${H}_{{\rm{K}}}={H}_{{\rm{K}} \mbox{-} \mathrm{End}}=0;$
• Step 4:  
  Let ${H}_{{\rm{K}}}$ varies between ${H}_{{\rm{K}}0}$ and ${H}_{{\rm{K}} \mbox{-} \mathrm{End}},$ with step $h,$ i.e. ${H}_{{\rm{K}}i}={H}_{{\rm{K}}0}-ih,$ $i=1,\ldots ,\,{\rm{End}};$ solve ${\alpha }_{i}$ via Constraint 2 at each ${H}_{i};$
• Step 5:  
  With ${\alpha }_{i}$ and ${H}_{{\rm{K}}i},$ calculate the strain in the crease $\overline{{\rm{AB}}}$ based on ${\varepsilon }_{\overline{{\rm{AB}}}}=\parallel {{\bf{R}}}_{\mathrm{AB}\_i}-{{\bf{R}}}_{\mathrm{AB}\_0}\parallel /\parallel {{\bf{R}}}_{\mathrm{AB}\_0}\parallel ,$ $i\,=1,\mathrm{...},\,{\rm{End}}.$

The value of ${\varepsilon }_{\overline{{\rm{AB}}}}$ is a direct measure of the structure's non-foldability and bi-stability; however, Pagano et al [33] has pointed out that the assumed strain ${\varepsilon }_{\overline{{\rm{AB}}}}$ cannot properly reflect the observed deformation of a paper Kresling-ori structure, rather, with a relief cut along the fold $\overline{{\rm{AB}}},$ bending of the triangular panels $\overline{{\rm{ABC}}}$ and $\overline{{\rm{ACD}}}$ can better describes the facet deformations. Thus, two virtual folds, $\overline{{\rm{CV}}}$ and $\overline{{\rm{AV}}^{\prime} },$ are mathematically added across each kinematically rigid panel (figure 2(e)) to capture the panel bending [32, 33, 42]. In this way, the rigid triangular panels $\overline{{\rm{ABC}}}$ and $\overline{{\rm{ACD}}}$ are divided into four rigid triangular facets: $\overline{{\rm{BCV}}},$ $\overline{{\rm{ACV}}},$ $\overline{{\rm{DAV}}^{\prime} },$ and $\overline{{\rm{CAV}}^{\prime} }.$ Note that the triangular facets $\overline{{\rm{DAV}}^{\prime} }$ and $\overline{{\rm{CAV}}^{\prime} }$ can be considered as rotations of $\overline{{\rm{BCV}}}$ and $\overline{{\rm{ACV}}};$ they are mathematically redundant. In what follows, only the virtual fold $\overline{{\rm{CV}}}$ is examined.

The position of the virtual fold $\overline{{\rm{CV}}},$ i.e. the position of the virtual vertex V, can be set arbitrarily on the crease $\overline{{\rm{AB}}}.$ During folding, its spatial position can be determined based on the following constraints [33]:

$$\begin{eqnarray}&&\begin{array}{l}{\rm{Constraint}}\,3:\,\parallel {{\bf{R}}}_{{\rm{VA}}}\parallel +\parallel {{\bf{R}}}_{{\rm{VB}}}\parallel ={b}_{{\rm{K}}},\\ {\rm{Constraint}}\,4:\,\parallel {{\bf{R}}}_{{\rm{VC}}}\parallel ={l}_{{\rm{v}}};\\ {\rm{Constraint}}\,5:\,\parallel {{\bf{R}}}_{{\rm{AB}}}\parallel ={\left({\parallel {{\bf{R}}}_{{\rm{VA}}}\parallel }^{2}+{\parallel {{\bf{R}}}_{{\rm{VB}}}\parallel }^{2}-2{{\bf{R}}}_{{\rm{VA}}}\cdot {{\bf{R}}}_{{\rm{VB}}}\right)}^{1/2}.\end{array}\end{eqnarray} \tag{ 7 }$$

Constraint 3 indicates that the lengths of sides $\overline{{\rm{VA}}}$ and $\overline{{\rm{VB}}}$ sum to the length of the crease $\overline{{\rm{AB}}};$ Constraint 4 asks that the length of the virtual fold $\overline{{\rm{VC}}}$ remain constant ${l}_{{\rm{v}}}$ during folding, where ${l}_{{\rm{v}}}$ is determined from the open configuration; Constraint 5 is the Cosine rule of the triangle $\overline{{\rm{ABV}}}$ that determines the out-of-plane deflection, where the length $\parallel {{\bf{R}}}_{{\rm{AB}}}\parallel$ is obtained from the strain ${\varepsilon }_{\overline{{\rm{AB}}}},$ $\parallel {{\bf{R}}}_{{\rm{VA}}}\parallel$ and $\parallel {{\bf{R}}}_{{\rm{VB}}}\parallel$ remain fixed. By solving the constraint equation (7), the position of the virtual vertex ${\rm{V}},$ as well as the dihedral angles at folds $\overline{{\rm{BC}}},$ $\overline{{\rm{VC}}},$ and $\overline{{\rm{AC}}}$ (i.e. ${\rho }_{{\rm{BC}}},$ ${\rho }_{{\rm{VC}}},$ and ${\rho }_{{\rm{AC}}}$) can be determined.

The Kresling-ori structure shown in figure 2(b) possesses 8 vertices and 8 sides on the top layer, as well as 8 vertices and 8 sides on the bottom layer that can be employed for embedding magnets. Figure 2(f) shows a design example, in which 8 magnets are embedded on the top-layer vertices, and 8 magnets are embedded in the bottom-layer side centers. As the two easiest pole layouts, the top-layer and bottom-layer magnets can be set with their like poles or opposite poles facing each other, giving rise to the repulsive-magnet layout and the attractive-magnet layout (figure 2(f)). Other designs and magnetic pole layouts are also possible but are not considered in this research.

During folding, in addition to the height change, the top layer would rotate with respect to the bottom layer, characterized by the rotation angle $\alpha .$ Thus, folding would not only alter the distance between the top-layer and bottom-layer magnets but also change their correspondence. For example, the top-layer magnet that is closest to the bottom-layer magnet on side $\overline{{\rm{AD}}}$ would experience three shifts during folding, from magnet 'B', to '${{\rm{P}}}_{1}$', to '${{\rm{P}}}_{2}$', and to '${{\rm{P}}}_{3}$' (figure 2(c)). Such changes on the distance and correspondence among magnets would therefore alter the stability profile of the system, which will be detailed in the following section.

It is worth noting again that in the magnetic-elastic coupled origami structures, by employing electromagnets, the magnetic poles can be easily reversed by switching the direction of electric currents. However, in what follows, permanent magnets will be used for proof-of-concept analyses and experiments.

To analyze how the magnets contribute to the overall potential energy, a reliable magnetic force model has to be determined in advance. According to the Gilbert model [43], the magnetic forces between magnets originate from the magnetic charges near the poles. If the magnetic poles are small enough, they can be represented as point magnetic charge. The magnitude of a magnetic pole $q$ (in A m) can be expressed as $q={B}_{r}S/{\mu }_{0},$ where ${B}_{r}$ is the residual flux density (in T), ${\mu }_{0}=4\pi \times {10}^{-7}\,{\rm{H}}\,{{\rm{m}}}^{-1}$ is the permeability of vacuum, and $S$ is the cross-section area (in ${{\rm{m}}}^{2}$) of the magnet. Generally, the force between two identical magnetic poles separated by distance $r$ (in ${\rm{m}}$) is given by

$$\begin{eqnarray}&&{F}_{{\rm{Mag}}}(r)=\displaystyle \frac{{\mu }_{0}{q}^{2}}{4\pi {r}^{2}}=\displaystyle \frac{{B}_{r}^{2}{S}^{2}}{4\pi {\mu }_{0}{r}^{2}}.\end{eqnarray} \tag{ 9 }$$

For bar or cylindrical magnets, by assuming point-like magnetic poles on the end surfaces, the force between magnets can be calculated as a combination of multipole interactions. For example, if two identical cylindrical magnets with radius ${R}_{{\rm{Mag}}}$ (in ${\rm{m}}$) and length $d$ (in ${\rm{m}}$) are placed end to end at a large distance $r\gg {R}_{{\rm{Mag}}},$ the force between them can be approximated by

$$\begin{eqnarray}&&{F}_{{\rm{Mag}}}(r)\approx \displaystyle \frac{\pi {B}_{r}^{2}{{R}_{{\rm{Mag}}}}^{4}}{4{\mu }_{0}}\left[\displaystyle \frac{1}{{r}^{2}}+\displaystyle \frac{1}{{(r+2d)}^{2}}-\displaystyle \frac{2}{{(r+d)}^{2}}\right].\end{eqnarray} \tag{ 10 }$$

On the other hand, based on the Amphère model [43], if two or more magnets are small enough or sufficiently distant such that their shapes and sizes are not important, we can model both magnets as magnetic dipoles with magnetic moments ${{\bf{m}}}_{1}$ and ${{\bf{m}}}_{2}.$ The force exerted by a dipole moment ${{\bf{m}}}_{1}$ on another dipole moment ${{\bf{m}}}_{2}$ can be determined as the force due to the magnetic field of ${{\bf{m}}}_{1}$ on ${{\bf{m}}}_{2},$ i.e.

$$\begin{eqnarray}&&\begin{array}{l}{{\bf{F}}}_{{\rm{Mag}}}({\boldsymbol{\delta }},{{\bf{m}}}_{1},{{\bf{m}}}_{2})=\displaystyle \frac{3{\mu }_{0}}{4\pi {\delta }^{5}}\left[\Space{0ex}{3.2ex}{0ex}({{\bf{m}}}_{1}\cdot {\boldsymbol{\delta }}){{\bf{m}}}_{2}+({{\bf{m}}}_{2}\cdot {\boldsymbol{\delta }}){{\bf{m}}}_{1}\right.\\ \left.\,+\,({{\bf{m}}}_{1}\cdot {{\bf{m}}}_{2}){\boldsymbol{\delta }}-\displaystyle \frac{5({{\bf{m}}}_{1}\cdot {\boldsymbol{\delta }})({{\bf{m}}}_{2}\cdot {\boldsymbol{\delta }})}{{\delta }^{2}}{\boldsymbol{\delta }}\right],\end{array}\end{eqnarray} \tag{ 11 }$$

where ${\boldsymbol{\delta }}$ is the distance-vector from the vector dipole moment ${{\bf{m}}}_{1}$ to ${{\bf{m}}}_{2},$ with $\delta =\parallel {\boldsymbol{\delta }}\parallel .$ Note that with vector calculation, this model includes both magnitudes and direction. For example, if two axially magnetized magnets are aligned along the z-axis, pointing in the z-direction, and separated by distance $r,$ the magnetic force can be simplified into

$$\begin{eqnarray}&&{F}_{{\rm{Mag}}}(\delta ,{m}_{1},{m}_{2})=\displaystyle \frac{3{\mu }_{0}{m}_{1}{m}_{2}}{2\pi {\delta }^{4}},\,{\rm{in}}\,z \mbox{-} \mathrm{direction}{\rm{.}}\end{eqnarray} \tag{ 12 }$$

Here, if the two magnets are identical, the magnetic dipole moment $m={B}_{r}V/{\mu }_{0}={B}_{r}Sd/{\mu }_{0},$ with $V$ denoting the volume (in ${{\rm{m}}}^{3}$) of the magnet; the distance between the two dipole moment is $\delta =r+d.$ Hence, equation (12) can be rewritten as

$$\begin{eqnarray}&&{F}_{{\rm{Mag}}}(z)=\displaystyle \frac{3{\left({B}_{r}Sd\right)}^{2}}{2\pi {\mu }_{0}{\delta }^{4}},\,{\rm{in}}\,z \mbox{-} \mathrm{direction}{\rm{.}}\end{eqnarray} \tag{ 13 }$$

To show the accuracy of these models, two NdFeB magnets (Grade N52, ${R}_{{\rm{Mag}}}=6.35\,{\rm{mm}},$ $d=12.7\,{\rm{mm}},$ axially magnetized, residual flux density ${B}_{r}=1.44\,{\rm{T}}$) are used for attraction and repulsion tests. Specifically, we 3D-printed two holders (Formlabs, photoreactive resin, standard black) for fixing the magnets; they are screwed to the universal testing machine with the magnets' dipoles aligned (figure 3(b)). Tension and compression tests are performed between 0 and 60 mm in a quasi-static way with speed 5.0 mm min⁻¹. Figure 3(c) shows the recorded force–distance relations for both the attractive-magnet and the repulsive-magnet layouts, which agree very well with the experimental calibration data provided by K&J Magnetics. Based on the magnet parameters, the relationship between the magnetic force and the separation distance corresponding to the multipole interaction model (i.e. equation (10)) and the magnetic dipole moment model (i.e. equation (13)) can be obtained, respectively, which are also plotted in figure 3(c). Comparing the theoretical curves with the experimental data, the multipole interaction model can hardly be used when the two magnets are getting close ($r\lt 10\,{\rm{mm}}$), but the magnetic dipole moment model agrees well with the experimental data in the whole range. Therefore, in what follows, the magnetic dipole moment model (equation (11)) will be adopted for calculating the magnetic potential energy.

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Using vector notations, the magnetic potential energy for two magnetic dipole moments ${{\bf{m}}}_{1}$ and ${{\bf{m}}}_{2}$ separated by a vector ${\boldsymbol{\delta }}$ (pointing from ${{\bf{m}}}_{1}$ to ${{\bf{m}}}_{2},$ $\delta =\parallel {\boldsymbol{\delta }}\parallel$) can be expressed as

$$\begin{eqnarray}&&\begin{array}{l}{{\rm{\Pi }}}_{{\rm{Mag}}}=-\displaystyle \int {{\bf{F}}}_{{\rm{Mag}}}({\boldsymbol{\delta }},{{\bf{m}}}_{1},{{\bf{m}}}_{2})\cdot {\rm{d}}{\boldsymbol{\delta }}\\ \,\,=\,-\,\displaystyle \frac{{\mu }_{0}}{4\pi {\delta }^{3}}\left[\displaystyle \frac{3({{\bf{m}}}_{1}\cdot {\boldsymbol{\delta }})({{\bf{m}}}_{2}\cdot {\boldsymbol{\delta }})}{{\delta }^{2}}-({{\bf{m}}}_{1}\cdot {{\bf{m}}}_{2})\right].\end{array}\end{eqnarray} \tag{ 14 }$$

The elastic potential energy of an origami structure may come from two aspects: the rotations of rigid facets with respect to elastic hinge-like creases and the elastic deformation of non-rigid facet panels.

• (a)  
  The SMO structure

For the rigid-foldable SMO structure, the elastic potential energy solely results from the hinge-like creases, because the facets remain rigid during folding and just rotate with respect to the creases. We assign ${k}_{{\rm{A}}}$ and ${k}_{{\rm{B}}}$ as the torsional spring stiffness per unit length for the creases in cell A and cell B, respectively, and ${k}_{{\rm{C}}}$ as the torsional spring stiffness per unit length for the connecting creases between the two cells. Hence, the torsional stiffness constants (${K}_{ki}$ and ${K}_{{\rm{C}}}$) corresponding to the dihedral angles (${\rho }_{ki}$ and ${\rho }_{{\rm{C}}}$) can be determined, where the subscript $k=A,\,{\rm{B}}$ and $i=1,\,2,3,\,4.$ In cell A, ${K}_{{\rm{A}}1}\,={K}_{{\rm{A}}3}={k}_{{\rm{A}}}b,$ ${K}_{{\rm{A}}2}={K}_{{\rm{A}}4}={k}_{{\rm{A}}}{a}_{{\rm{A}}};$ in cell B, ${K}_{{\rm{B}}1}={K}_{{\rm{B}}3}={k}_{{\rm{B}}}b,$ ${K}_{{\rm{B}}2}={K}_{{\rm{B}}4}={k}_{{\rm{B}}}{a}_{{\rm{B}}};$ for the connecting crease, ${K}_{{\rm{C}}}={k}_{{\rm{C}}}b.$ Then the total elastic potential energy of the SMO structure can be expressed as

$$\begin{eqnarray}&&\begin{array}{l}{{\rm{\Pi }}}_{E \mbox{-} M}=\displaystyle \frac{1}{2}\left[\displaystyle \sum _{i=1}^{4}{K}_{{\rm{A}}i}{\left({\rho }_{{\rm{A}}i}-{\rho }_{{\rm{A}}i}^{0}\right)}^{2}\right.\\ \,\,+\,\left.\displaystyle \sum _{i=1}^{4}{K}_{{\rm{B}}i}{\left({\rho }_{{\rm{B}}i}-{\rho }_{{\rm{B}}i}^{0}\right)}^{2}+4{K}_{{\rm{C}}}{\left({\rho }_{{\rm{C}}}-{\rho }_{{\rm{C}}}^{0}\right)}^{2}\right].\end{array}\end{eqnarray} \tag{ 15 }$$

Here, the superscript '0' denotes the dihedral angle corresponding to the stress-free stable configuration at ${\theta }_{{\rm{A}}}={\theta }_{{\rm{A}}}^{0},$ where no crease suffers to deformations.

• (a)  
  The Kresling-ori structure

The elastic potential energy of the non-rigid-foldable Kresling-ori structure results from both the hinge-like creases and the deformable facets. We assign ${k}_{{\rm{cre}}}$ as the torsional spring stiffness per unit length for the creases, and ${k}_{{\rm{bend}}}$ as the torsional spring stiffness per unit length for the virtual fold that approximate the panel bending. According to the experiment performed by Silverberg et al [32], for 120 lb paper, ${k}_{{\rm{bend}}}$ will be two orders of magnitude higher than ${k}_{{\rm{cre}}}.$ Noting that a relief cut is assumed along the fold $\overline{{\rm{AB}}},$ hence, only the elastic potential energies of the folds $\overline{{\rm{BC}}},$ $\overline{{\rm{VC}}},$ and $\overline{{\rm{AC}}}$ in the unit panel $\overline{{\rm{ABCD}}}$ need to be considered. Actually, they could also characterize the potential energy of the whole structure, because all the other folds are rotationally or reflectional symmetric to them. Thus, the total elastic potential energy of a Kresling-ori structure (with $n$ unit panels) can be obtained as

$$\begin{eqnarray}&&\begin{array}{l}{{\rm{\Pi }}}_{{\rm{E}} \mbox{-} {\rm{K}}}=\displaystyle \frac{n}{2}\left[2{k}_{{\rm{cre}}}{a}_{{\rm{K}}}{\left({\rho }_{{\rm{BC}},i}-{\rho }_{{\rm{BC}},0}\right)}^{2}\right.\\ \,\left.\,\,+\,{k}_{{\rm{cre}}}{l}_{{\rm{K}}}{\left({\rho }_{{\rm{AC}},i}-{\rho }_{{\rm{AC}},0}\right)}^{2}+2{k}_{{\rm{bend}}}{l}_{{\rm{v}}}{\left({\rho }_{{\rm{VC}},i}-{\rho }_{{\rm{VC}},0}\right)}^{2}\right].\end{array}\end{eqnarray} \tag{ 16 }$$

Here the subscript 'i' denotes the calculation step, $i=1,\ldots ,\,{\rm{End}};$ $i=0$ indicates the initial open configuration of the structure.

We first investigate the magnetic-elastic coupled Miura-ori structure based on Design I. Specifically, the two magnets are placed at the top vertex '2' and the bottom vertex '11' of the SMO structure (refer to figure 1(b)); they keep aligning along the z-axis and pointing in the z-direction during the complete folding process. Table 1 lists the geometry and physical parameters of the SMO structure and the used magnets (NdFeB, Grade N52). Three stress-free angles (${\theta }_{{\rm{A}}}^{0}=15^\circ ,\,30^\circ ,$ and $45^\circ$) of the SMO structure are studied; for each case, the total potential energy corresponding to the no-magnet, the repulsive-magnet, and the attractive-magnet layouts are examined. Substituting the parameters listed in table 1 into equations (14) and (15), the total potential energy of the magnetic-elastic coupled Miura-ori structure can be obtained via equation (8).

Table 1.  Parameters of the magnetic-elastic coupled Miura-ori structure.

Parameters	Values	Parameters	Values
${a}_{{\rm{A}}}$	38.1 mm	${k}_{{\rm{B}}}=40{k}_{{\rm{A}}}$	0.6 N rad−1
${a}_{{\rm{B}}}$	45.7 mm	${R}_{{\rm{Mag}}}$	6.35 mm
$b$	50.8 mm	$d$	12.7 mm
${\gamma }_{{\rm{A}}}$	${75}^{\circ }$	${B}_{r}$	1.44 T
${k}_{{\rm{A}}}={k}_{{\rm{C}}}$	0.015 N rad−1

Figure 4(a) displays the total potential energy of the structure with respect to the structure height ${H}_{{\rm{M}}}.$ When the stress-free angel ${\theta }_{{\rm{A}}}^{0}=15^\circ$ and there is no magnet, the structure is mono-stable, with its only stable configuration (i.e. the potential well) locating at ${H}_{{\rm{M}}}=17.58\,{\rm{mm}}.$ Adding repulsive or attractive magnet pairs, the structure remains mono-stable; however, its stable configuration can be significantly shifted to ${H}_{{\rm{M}}}=41.49\,{\rm{mm}}$ and ${H}_{{\rm{M}}}=10.68\,{\rm{mm}},$ respectively. When the stress-free angle ${\theta }_{{\rm{A}}}^{0}=30^\circ$ and there is no magnet, the structure already has two stable states locating at ${H}_{{\rm{M}}}=12.91\,{\rm{mm}}$ and 42.55 mm. Adding a repulsive or attractive magnet pair, the structure's stability profile is qualitatively changed to mono-stable. For the repulsive-magnet layout, the stable configuration at ${H}_{{\rm{M}}}=12.91\,{\rm{mm}}$ is no longer stable, and the stable configuration at ${H}_{{\rm{M}}}=42.55\,{\rm{mm}}$ is pushed to 47.89 mm; for the attractive-magnet layout, the stable configuration at ${H}_{{\rm{M}}}=42.55\,{\rm{mm}}$ is no longer stable, and the stable configuration at ${H}_{{\rm{M}}}=12.91\,{\rm{mm}}$ is shifted to 9.90 mm. When the stress-free angle ${\theta }_{{\rm{A}}}^{0}=45^\circ ,$ the structure is also bi-stable if there is no magnet; the two stable configurations are widely separated, situated at ${H}_{{\rm{M}}}=10.27\,{\rm{mm}}$ and 56.79 mm. Setting two repulsive magnets, the structure is switched to mono-stable, with its only stable configuration locating at ${H}_{{\rm{M}}}=57.80\,{\rm{mm}}.$ Setting two attractive magnets, the bi-stability is reserved; the two stable configurations experience small shifts from H_M = 10.27 to 9.13 mm, and from 56.79 to 54.74 mm.

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In addition to controlling the number and positions of the stable configurations, the depths of the potential wells can also be effectively tailored. For example, attractive magnets can effectively deepen the depth of the potential well at the nested-in configuration. This will also affect the structure's static and dynamic characteristics.

The structure's force–displacement relation along the height direction can be obtained via

$$\begin{eqnarray}&&{F}_{H-{\rm{M}}}=-\displaystyle \frac{\partial {{\rm{\Pi }}}_{{\rm{M}}}}{\partial {\theta }_{A}}{\left(\displaystyle \frac{{\rm{d}}{H}_{{\rm{M}}}}{{\rm{d}}{\theta }_{A}}\right)}^{-1}.\end{eqnarray} \tag{ 17 }$$

Figure 4(b) shows the constitutive relations corresponding to the no-magnet, the repulsive-magnet, and the attractive-magnet layouts, with the stress-free angle ${\theta }_{{\rm{A}}}^{0}=45^\circ .$ The magnet-induced changes of the stability profile also transform the static characteristics. With repulsive magnets, the constitutive relation experience qualitative changes that the stable nested-in configuration and the negative stiffness segment are removed. The attractive magnets do not change the stability profile qualitatively; but quantitatively, the deepened potential well significantly raises the critical force level for a snap-through transition from the nested-in to the bulged-out configurations. Specifically, the critical force increases by more than two times from ${\hat{F}}_{H \mbox{-} {\rm{M}} \mbox{-} {\rm{w}}/{\rm{o}}}=2.37\,{\rm{N}}$ to ${\hat{F}}_{H \mbox{-} {\rm{M}} \mbox{-} {\rm{A}}}=7.97\,{\rm{N}},$ where the hat denotes the critical force, the subscript 'M' denotes the Miura-ori structure, 'w/o' and 'A' denote the no-magnet and the attractive magnet layouts, respectively.

We then investigate the magnetic-elastic coupled Kresling-ori structure based on the design given in figure 2(f). Specifically, 8 magnets are embedded on the top-layer vertices, and 8 magnets are embedded in the bottom-layer side centers. Table 2 lists the geometric and physical parameters of the Kresling-ori structure and the used magnets (NdFeB, Grade N52). Substituting the geometric parameters into the constraint equations (6) and (7), the strain in the fold $\overline{{\rm{AB}}}$ (${\varepsilon }_{\overline{{\rm{AB}}}}$) and the dihedral angles (${\rho }_{{\rm{BC}}},$ ${\rho }_{{\rm{VC}}},$ and ${\rho }_{{\rm{AC}}}$) can be determined for each rotation angle $\alpha$ (figure 5(a)). Setting the open configuration as the stress-free state and based on the obtained dihedral angles, the total elastic potential energy can be determined via equation (16). Figure 5(b) displays the profiles of the total elastic potential energy as well as its constituents. It reveals that the elastic potential energy originating from the virtual folds (i.e. panel bending) account for the main parts, and the Kresling-ori structure is intrinsically bi-stable, with two potential wells (stable configurations) locating at ${H}_{{\rm{K}}}=54.3\,{\rm{mm}}$ and ${H}_{{\rm{K}}}\,=2.2\,{\rm{mm}}.$ We remark here that generally, owing to the contribution of panel bending, the elastic potential energy profile of the Kresling-ori structure is much higher than the Miura-ori structure (except when approaching the folding limits of Miura-ori). Therefore, multiple magnets (in this design example, 16 magnets) have to be integrated into the Kresling-ori structure to generate strong magnetic potential energy so that the inherent elastic potential energy profile can be significantly affected.

Table 2.  Parameters of the magnetic-elastic coupled Kresling-ori structure.

Parameters	Values	Parameters	Values
$n$	8	${k}_{{\rm{bend}}}$	0.8 N rad−1
${R}_{{\rm{K}}}$	30.00 mm	${R}_{{\rm{Mag}}}$	6.35 mm
$\lambda$	0.98	$d$	6.35 mm
${k}_{{\rm{cre}}}$	8 mN rad−1	${B}_{r}$	1.44 T

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To conveniently examine the magnetic potential energy, the magnets in the top layer are numbered from '1' to '8', and the magnets in the bottom layer are numbered from '9' to '16'. Note that the relative positions among the magnets in the same layer remain immutable during folding; while for the magnets in different layers, their relative positions would experience significant changes, which would, therefore, alter the magnetic potential energy. Generally, the net force applied on the top-layer magnet 'j' (${{\bf{F}}}_{{\rm{Mag}},j},$ j = 1, ..., 8) is the vector sum of the magnetic forces exerted by the bottom-layer magnets (with dipole moment ${{\bf{m}}}_{i},$ i = 9, ..., 16 ) on the top-layer magnet j (with dipole moment ${{\bf{m}}}_{j}$). Hence, based on equation (11), we have

$$\begin{eqnarray}&&\begin{array}{l}{{\bf{F}}}_{{\rm{Mag}},j}=\displaystyle \sum _{i=9}^{16}{{\bf{F}}}_{{\rm{Mag}}}\left({{\bf{m}}}_{i},{{\bf{m}}}_{j},{{\boldsymbol{\delta }}}_{ij}\right)\\ \,\,=\,\displaystyle \sum _{i=9}^{16}\displaystyle \frac{3{\mu }_{0}}{4\pi {\parallel {{\boldsymbol{\delta }}}_{ij}\parallel }^{5}}\left[\Space{0ex}{3.2ex}{0ex}({{\bf{m}}}_{i}\cdot {{\boldsymbol{\delta }}}_{ij}){{\bf{m}}}_{j}+({{\bf{m}}}_{j}\cdot {{\boldsymbol{\delta }}}_{ij}){{\bf{m}}}_{i}\right.\\ \left.\,\,+\,({{\bf{m}}}_{i}\cdot {{\bf{m}}}_{j}){{\boldsymbol{\delta }}}_{ij}-\displaystyle \frac{5({{\bf{m}}}_{i}\cdot {{\boldsymbol{\delta }}}_{ij})({{\bf{m}}}_{j}\cdot {{\boldsymbol{\delta }}}_{ij})}{{\parallel {{\boldsymbol{\delta }}}_{ij}\parallel }^{2}}{{\boldsymbol{\delta }}}_{ij}\right].\end{array}\end{eqnarray} \tag{ 18 }$$

Equation (18) is capable of capturing both the vertical and torsional components of the magnetic forces. Since all magnets are identical, aligned along the z-axis, and pointing in the z-direction, the corresponding magnetic dipole moment can be expressed as ${{\bf{m}}}_{i}=\pm m{(0,0,1)}^{T},\,(i=1,\ldots ,16).$ For the repulsive-magnet layout, the top and bottom layer magnets take the opposite signs; for the attractive-magnet layout, they take the same sign. To vividly illustrate the above concept, as an example, figure 6(a) shows the magnetic force components exerted by the bottom layer magnets '9' to '16' on the top layer magnet '1' at a certain folded configuration. With respect to folding, the magnet '1' will experience a spatial translation, whose trajectory is given in figures 6(a) and (b). Along this trajectory, the magnitude of the net force applied on magnet '1' increases significantly as the height decreases, and its direction also undergoes noteworthy changes, as discretely depicted in figure 6(b).

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Assuming zero magnetic potential energy at the open configuration and based on equation (18), the magnetic potential energy of the whole system can be expressed as

$$\begin{eqnarray}&&{{\rm{\Pi }}}_{{\rm{Mag}}}=-\displaystyle \sum _{j=1}^{8}\displaystyle \sum _{i=9}^{16}\displaystyle \int {{\bf{F}}}_{{\rm{Mag}}}\left({{\bf{m}}}_{i},{{\bf{m}}}_{j},{{\boldsymbol{\delta }}}_{ij}\right){\rm{d}}{{\bf{u}}}_{j},\end{eqnarray} \tag{ 19 }$$

where ${{\bf{u}}}_{j}$ is the translation trajectory of magnet $j\,(j=1,\ldots ,\,{\rm{8}})$ with respect to folding; it is a function of the structure height ${H}_{{\rm{K}}}$ or the rotation angle $\alpha .$ Figure 6(c) displays the total potential energy of the Kresling-ori structure versus its height. Attaching magnets to the bi-stable Kresling-ori structure, the stable configuration at ${H}_{{\rm{K}}}=54.3\,{\rm{mm}}$ is little affected due to the weak magnetic forces. However, the stable configuration at ${H}_{{\rm{K}}}=2.2\,{\rm{mm}}$ experiences great changes. When the top-layer and bottom-layer magnets are attractive, the stable configuration at ${H}_{{\rm{K}}}=2.2\,{\rm{mm}}$ is moved to ${H}_{{\rm{K}}}=6.1\,{\rm{mm}},$ and the corresponding potential well is significantly depended. When the magnets are repulsive, in addition to a large translation of the stable configuration from ${H}_{{\rm{K}}}=2.2\,{\rm{mm}}$ to ${H}_{{\rm{K}}}=16.6\,{\rm{mm}},$ the configuration at ${H}_{{\rm{K}}}=0\,{\rm{mm}}$ becomes stable, i.e. the stability profile of the structure undergoes a qualitative change from bi-stable to tri-stable. Similarly, based on ${F}_{H \mbox{-} {\rm{K}}}=-{\rm{d}}{{\rm{\Pi }}}_{{\rm{K}}}/{\rm{d}}{H}_{{\rm{K}}},$ the structure's constitutive force–displacement relations corresponding to the no-magnet, the repulsive-magnet, and the attractive-magnet layouts can be obtained (figure 6(d)).

The above two examples vividly demonstrate the effects of magnets in controlling origami structures' stability profile as well as the corresponding static characteristics. Quantitatively, the magnets could translate the potential well (i.e. the stable configuration), deepen or shallows the potential well (i.e. alter its stability degree). Qualitatively, the magnets could eliminate the original stable configuration or add additional stable configuration to the structure. These changes would thus fundamentally alter the structure's constitutive force–displacement relations. We point out here again that if replacing the permanent magnets with electromagnets, such tailoring could be achieved more easily by controlling the electric currents.

Figures 7(a) and (b) show the design and prototype of the magnetic-elastic coupled Miura-ori structure based on the parameters given in table 1. The origami facets are water jet cut individually from 0.25 mm thickness austenitic stainless-steel sheets. Note that austenitic stainless steels can be classed as paramagnetic with relative permeabilities approaching 1.0 (generally in the range between 1.003 and 1.05 in the fully annealed condition). Such low permeabilities enable austenitic steels to be used as 'non-magnetic' materials, with their effects on the magnetic field being negligible. To prevent possible contacts between the facets and the top/bottom magnets, part of the facets around the vertex are cut out. These facets are pasted to a 0.13 mm thickness adhesive-back plastic film (ultrahigh molecular weight polyethylene) to form individual Miura-ori cells. The two cells are then connected along the connecting creases by adhesive-back films to form an SMO structure. Through this construction, the steel facets are significantly stiffer than the plastic creases to ensure rigid-foldability. In addition, we paste 0.1 mm thickness pre-bent spring-steel stripes at the folds of the top cell to provide strong torsional stiffness (i.e. a high ${k}_{2}$), thus imparting bi-stability to the SMO structure. To install the SMO structure on the universal testing machine, 3D-printed connectors are screwed to rectangular connection plates, which further connect with the SMO cell at the top and bottom creases through adhesive-back films. To ensure free rotation of the SMO structure as well as effective load transmission during compression/tension tests, non-magnetic ball bearings are embedded into embedded into the connectors, and a non-magnetic titanium screw rod is inserted through the bearing, with one end fixing with the 3D-printed magnet holder, and the other end fixing with the universal testing machine. Here, NdFeB magnets (see parameters listed in table 1) are employed for proof-of-concept experiments.

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To experimentally verify the effects of magnets, displacement-controlled compression and tension tests are carried out on the prototypes with the no-magnet, the repulsive-magnet, and the attractive-magnet layouts, respectively. Figure 7(c) shows the measured force–displacement relations corresponding to the three layouts. For each layout, five complete quasi-static tension and compression tests (with speed 10 mm min⁻¹) are performed; for each test, we average the tension and compression test data to get one curve. Averaging the five curves yields the measured average (the curves) and the standard deviations (shaded bands). Figure 7(c) reveals that when there is no magnet, the prototype exhibits bi-stability. By integrating repulsive magnets, the original bi-stable profile is altered to mono-stable, where the stable bulged-out configuration remains stable, while the nested-in configuration is no longer stable. By integrating attractive magnets, although the prototype remains bi-stable, the stable nested-in configuration is significantly translated, and the critical force for the snap-through switch is remarkably lifted. Therefore, in terms of the qualitative characteristics, the experimental curves are in good agreement with the theoretical predictions displayed in figure 4(b).

It is worth pointing out that in the experiment, when the prototype is compressed toward its lower kinematic folding limit, the distance between the two attractive magnets (i.e. ${H}_{{\rm{M}}}$) becomes very small, and the magnetic force becomes extremely strong to possibly damage the prototype. In detail, experimental trials reveal that the strong magnetic attraction would deform the steel facets, tear the adhesive film, and let the two magnets stick together. To prevent such a scenario and to ensure safety, the compression tests are manually stopped when a force fall is detected. Such a procedure does not affect the identification of qualitative characteristics.

Figures 8(a) and (b) show the design and prototype of the magnetic-elastic coupled Kresling-ori structure based on the parameters given in table 2. The origami structure is made of 0.076-mm-thickness moisture-resistant polyester film. We use laser-based machining techniques to cut and pattern flat sheet. Specifically, the creases are perforated to some extent such that the bending stiffness of the creases is weakened; we also cut small holes at the vertices where multiple creases intersect to prevent stress concentration. Through folding, rolling, and gluing, a Kresling-ori structure can be obtained. To embed magnets and to connect with the universal testing machine, the Kresling-ori structure is connected with two acrylic plates at the top and bottom. Note that during folding, the top plate would rotate with respect to the bottom plate. To ensure free rotation, similarly, ball bearings are embedded into the 3D-printed connectors, which further connect with the top and bottom acrylic plates. A non-magnetic titanium screw rod is inserted through the bearing, with one end fixing with the plate, and the other end fixing with the universal testing machine. Here, NdFeB magnets (see parameters listed in table 2) are employed for proof-of-concept experiments.

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Displacement-controlled compression and tension tests are carried out on the prototype for the no-magnet, the attractive-magnet, and repulsive-magnet layouts, respectively. Based on the same testing and data processing procedures as we used in subsection 4.1, figure 8(c) shows the measured force–displacement relations of the magnetic-elastic coupled Kresling-ori prototype corresponding to the three layouts. It reveals that when there is no magnet, the prototype is bi-stable (figure 8(c), left). By integrating attractive magnets, the structure's constitutive relation evolves into a mono-stable profile (figure 8(c), middle), which, however, contradicts with the theoretical prediction (which is a bi-stable profile, shown in figure 6(d), dashed). Carefully observing the curves and the prototype's deformation during folding, the attractive magnets have little effects on the first half of the curve and the stable 'open' configuration; with the height decreasing, large discrepancies occur, which are induced by two reasons. First, in theoretical analysis, the origami panels are assumed to be rigid, which are strong enough to drag the magnets so that their configurations exactly follow the folding kinematics. Based on such assumptions, the overall reaction force of the prototype will experience an inflection point from declining to increasing, generating a stable 'close' configuration (see the dashed curve in figure 6(d) and the dashed curve in figure 8(c), middle). However, in experiments, when the prototype is compressed to a relatively small height, the attractive force between the top-layer and the bottom-layer magnets becomes so strong that the origami panels made of thin polyester films are no longer able to drag the magnets following the folding kinematics. Hence, when the prototype is further compressed, although the top-layer magnets will gradually decline, their lateral positions with respect to the bottom-layer magnets will be locked. In this process, the polyester-film panels will be seriously deformed; however, due to the increasingly strong magnetic attraction, the overall reaction force of the prototype keeps negative and declines fast, giving rise to a mono-stable profile. Second, theoretical analysis assumes zero-thickness of the facets and creases, but practically, when the prototype is compressed into small height, the film thickness cannot be ignored anymore. The films will be contacted and pressed seriously, which generate significant elastic force that would affect the overall profile. This explains why the reaction force starts to increase when the prototype is compressed to below $H=2.64\,{\rm{mm}}.$

By integrating repulsive magnets, the original bi-stable profile evolves into tri-stable (figure 8(c), right), where the stable 'open' configuration remains stable, while the stable 'closed' configuration is translated to a configuration with larger height. Keep compressing the prototype beyond the stable 'closed' configuration, the reaction force increases again, and then experiences a sudden drop, suggesting a snap-through transition of the configuration. Theoretically, such drop will continue until the test ends, and the final configuration would be an additional stable state brought by the repulsive magnets (see figure 6(d)). However, in experiments, the sudden force drop is followed by a fluctuating interval, and then the force rises again until the end. Such discrepancies between the theoretical and experimental results are also caused by the non-ignorable material thickness of the prototype. When the prototype is compressed into small height, the films will be contacted and pressed, which therefore interrupts the configuration snap-through and increases the reaction force undesirably. Particularly, at the final stage of the test, the film compressions become significant, which will increase the reaction force.

In short, based on proof-of-concept prototypes of Miura-ori and Kresling-ori structures, the effects of magnets on the stability profiles and the force–displacement constitutive relations are verified, which includes qualitative transitions (switches among mono-stable, bi-stable, and tri-stable) and quantitative evolutions (translations of the stable configurations, deepening or shallowing of the potential wells). Due to the deformation and inter-pressing of the polyether-film panels, non-negligible discrepancies are observed when testing the magnetic-elastic coupled Kresling-ori prototype; this triggers an interesting research problem that has not been tackled, that is, what is the effects of magnet-origami integration when the origami panels and creases are flexible to exhibit large deformations.

Figure 9(a) schematically displays the setup for dynamic study. Specifically, the magnetic-elastic coupled Miura-ori structure is horizontally suspended on a fixed frame with light strings. A lumped mass $m$ connects with the origami structure at one end, and harmonic base excitations are applied on the structure at the other end. Here, the lumped mass is assumed to be much heavier than the origami structure, such that the origami can be equivalently considered as a massless nonlinear spring with force–displacement relation $F(u)$ and a massless viscous damper with damping coefficient $c.$ Thus, the system in figure 9(a) can be simplified into a spring-lumped-mass model (figure 9(b)), with $x$ and $y$ denoting the absolute displacements of the lumped mass and the base, respectively, and $u=x-y$ denoting and the relative displacement between the lumped mass and the base. The origin of the relative displacement (i.e. $u=0$) is set at the unstable equilibrium of the SMO structure without magnet. Based on the example shown in figure 4(b) (with parameters listed in table 1, and ${\theta }_{{\rm{A}}}^{0}=45^\circ$), the $u=0$ configuration is set at ${H}_{{\rm{M}}}=27.20\,{\rm{mm}}.$ Hence, the equation of the motion of the equivalent model yields

$$\begin{eqnarray}&&m\ddot{u}+F(u)+c\dot{u}=-m\ddot{y},\end{eqnarray} \tag{ 20 }$$

where the dots over the variables denote time derivatives. The base is subjected to harmonic excitation $y=A\,\sin (\omega t)$ with amplitude $A$ and frequency $\omega .$

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Note that the constituent force–displacement relations of the magnetic-elastic coupled Miura-ori structure possess strong nonlinearity originating from the origami geometry and the magnetic effect. For the sake of convenience in numerical studies, the constitutive relations corresponding to the no-magnet, the repulsive-magnet, and the attractive-magnet layouts are fitted with polynomials up to 27th orders, respectively. Figure 9(c) displays the theoretical and the fitted curves of the three constitutive relations, which are in good agreement; particularly, the stable and unstable equilibria of the system are well captured. Note that as expected, some undesired small fluctuations exist, and when $u$ approaches to the folding limits, the fitted relations cannot fully reflect the kinematic constraints. However, such fittings would still be effective in demonstrating the effects of magnets on dynamics.

The following parameters are assigned for numerical analyses: $m=0.15\,{\rm{kg}},$ $c=0.4\,{\rm{kg}}\,{{\rm{s}}}^{-1},$ and $A=6.0\,{\rm{mm}}.$ Based on the fitted force–displacement relations corresponding to the no-magnet, the repulsive-magnet, and the attractive-magnet layouts, equation (20) is solved via the 4th-order Runge–Kutta method with zero initial condition, i.e. $({u}_{0},{\dot{u}}_{0})=(0,0),$ respectively. With a step of 0.1 Hz, a discrete frequency sweep is performed between 2 and 20 Hz. With a fixed displacement excitation amplitude $A,$ the higher the excitation frequency, the larger the input energy. At each frequency, the simulation time is long enough to let the system enter steady-state. The peak-to-peak value of the relative displacement ($u$) and the system's displacement transmissibility in terms of the root-mean-square (RMS) value (specifically, ${T}_{{\rm{d}} \mbox{-} {\rm{RMS}}}$) are examined to uncover the system's dominant dynamic behaviors. Specifically, ${T}_{{\rm{d}} \mbox{-} {\rm{RMS}}}$ is defined as

$$\begin{eqnarray}&&{T}_{{\rm{d}} \mbox{-} {\rm{RMS}}}=\displaystyle \frac{{x}_{{\rm{RMS}}}}{{y}_{{\rm{RMS}}}}=\displaystyle \frac{\sqrt{({x}_{1}^{2}+{x}_{2}^{2}+\ldots +{x}_{N}^{2})/N}}{\sqrt{({y}_{1}^{2}+{y}_{2}^{2}+\ldots +{y}_{N}^{2})/N}},\end{eqnarray} \tag{ 21 }$$

where ${x}_{i}$ and ${y}_{i}$ ($i=1,2,\,\ldots ,\,N$) are numerical sampling points in the steady-state displacement-time histories of the lumped mass and the base, respectively.

Figure 10 displays the dynamic responses of the magnetic-elastic coupled origami structure corresponding to the three layouts. When there is no magnet applied, the structure is inherently bi-stable (refer to figure 4). Under external excitations, three qualitatively different types of responses can be observed: intra-well (in) oscillations that surround the nested-in stable configuration, intra-well (out) oscillations that surround the bulged-out stable configuration, and large-amplitude inter-well oscillations that surround the two stable configurations (figure 10(a)). They present distinctly different displacement transmissibility levels. When the excitation frequency is relatively low, the structure could exhibit both intra-well (in) and intra-well (out) oscillations. Note that due to the low energy barrier between the nested-in and the bulged-out stable configurations (refer to figure 4), transitions from the intra-well (in) oscillations to the intra-well (out) oscillations occur at very low frequency (2.3 Hz). By increasing the excitation frequency, the input energy becomes large enough to maintain inter-well oscillations, which continue appearing above 8.6 Hz.

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When attractive magnets are applied, the structure remains bi-stable. Similarly, three types of responses with distinct displacement transmissibility levels are observed, namely, intra-well (in) oscillations, intra-well (out) oscillations, and inter-well oscillations (figure 10(b)). However, with the attractive magnets, the potential well at the nested-in configuration is significantly deepened, which raises the energy barrier from the nested-in to the bulged-out stable configurations, and correspondingly, increases the critical force for snapping through to the bulged-out configuration (refer to figure 4). Hence, when the excitation frequency is relatively low, the input energy is insufficient to overcome the potential energy barrier, and the structure sustains at intra-well (in) oscillations. By increasing the excitation frequency to 6.5 and 6.6 Hz, inter-well oscillation and a transition to the intra-well (out) oscillation are observed for the first time. When the excitation frequency is further increased to 11.4 Hz, the input energy becomes sufficient to maintain large-amplitude inter-well oscillations. Comparing to the no-magnet layout, the critical frequency for transiting from intra-well (in) oscillations to intra-well (out) oscillations and the critical frequency for exhibiting persistent inter-well oscillations are significantly increased.

When repulsive magnets are applied, the structure turns into mono-stable (refer to figure 4), which fundamentally alter the system's dynamic responses (figure 10(c)). Overall, the frequency-response diagram shows representative characteristics of hardening nonlinearity. In detail, unlike the bi-stable scenarios that the oscillations surround different stable equilibria, here, basically, the structure is able to output two types of responses (namely, high-amplitude responses and low-amplitude responses) that surround the same stable equilibrium but are different in the amplitude level and the displacement transmissibility level. Note that some complex dynamic behaviors (e.g. responses with subharmonic components) are also observed, which also surround the only stable equilibrium.

It's worth mentioning here that with only one pair of initial conditions, the system's dynamics cannot be fully captured; this is because the structure possesses strong global nonlinearity such that its responses are extremely sensitive to initial conditions [44]. Carrying out a comprehensive dynamic analysis of the structure by considering the nonlinearities originating from the origami geometry and the magnets is an interesting topic to be explored.

In short, by numerically examining the dynamic responses of the magnetic-elastic coupled Miura-ori structure, the effects of the attractive and repulsive magnets on system dynamics are exemplified. The attractive magnets could quantitatively change the critical frequency for transiting intra-well (in) oscillations to intra-well (out) oscillations and the critical frequency for exhibiting continuous inter-well oscillations; the repulsive magnets could qualitatively switch the structure's dominant dynamic characteristics from bi-stability to hardening nonlinearity. Such qualitative and quantitative changes on dynamics suggest the possibility and feasibility of using magnets to control origami structures' dynamic responses, and hence, to harness their dynamic functionalities.