# An Energetic Approach for Comparing Monomaterial
Compliant Bistable Mechanisms

Kéliane Megret, Justin Dirrenberger, Jocelyn Monnoyer, Cyrille Sollogoub, Benjamin Goislard de Monsabert, Laure Fernandez, Stéphane Delalande, Stephane Viollet

► To cite this version:

Kéliane Megret, Justin Dirrenberger, Jocelyn Monnoyer, Cyrille Sollogoub, Benjamin Goislard de Monsabert, et al.. An Energetic Approach for Comparing Monomaterial Compliant Bistable Mechanisms. Journal of Mechanical Design, 2025, 147 (12), pp.123301. 10.1115/1.4068549. hal-05070812

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# An Energetic Approach for Comparing Monomaterial Compliant Bistable Mechanisms

Shape-morphing systems offer multiple functionalities in a single part by leveraging compliance and bistability principles. Compliant mechanisms, composed of flexible structural elements, derive motion from deflection rather than from traditional joints. Structures exhibiting two stable equilibria are termed bistable. These bistable mechanisms can replace complex rigid body assemblies, enhancing modularity. However, their application is challenging due to the lack of comprehensive methods for comparing mechanical behaviors qualitatively and quantitatively. This research introduces a novel methodology for investigating the energetic properties and actuation symmetry of bistable mechanisms. Utilizing systematic review and meta-analysis, the study categorizes a dataset of articles into two classes, providing a robust reference for studying bistable mechanisms. The analysis focuses on how critical parameters such as motion type and shape affect behavior and actuation symmetry, using load-displacement curves and related energy metrics. The findings present a new method to identify key parameters and offer valuable design guidelines for developing compliant, single-part, and sustainable mechanisms.
[DOI: 10.1115/1.4068549]

Keywords: compliant mechanisms, design of engineered materials system, mechanism synthesis

### Kéliane Megret<fnref n="1" />

Stellantis France,
75000 Paris, France;
Laboratoire PIMM, Arts et Métiers Institute of
Technology, Cnam, CNRS, 151 boulevard de
l'Hôpital,
75013 Paris, France;
Institut des Sciences du Mouvement,
Aix-Marseille Université,
CEDEX 09, 13284 Marseille, France
e-mail: kéliane.megret@ensam.eu

### Justin Dirrenberger

Laboratoire PIMM,
Institut universitaire de France (IUF), Arts et
Métiers Institute of Technology, CNRS, Cnam,
151 boulevard de l'Hôpital,
75013 Paris, France
e-mail: Justin.DIRRENBERGER@ensam.eu

### Jocelyn Monnoyer

Stellantis France,
75000 Pari, France
e-mail: jocelyn.monnoyer@stellantis.com

### Cyrille Sollogoub

Laboratoire PIMM,
Arts et Métiers Institute of Technology,
CNRS, Cnam,
151 boulevard de l'Hôpital,
75013 Paris, France
e-mail: cyrille.sollogoub@lecnam.net

### Benjamin Goislard de Monsabert

Aix-Marseille Université, CNRS, ISM,
CEDEX 09, 13284 Marseille,
Bouches-du-Rhône, France
e-mail: benjamin.goislard-de-monsabert@univ-amu.fr

### Laure Fernandez

Aix-Marseille Université, CNRS, ISM,
CEDEX 09, 13284 Marseille,
Bouches-du-Rhône, France
e-mail: laure.fernandez@univ-amu.fr

<footnote n="1">¹Corresponding author.</footnote>

<footnote>Contributed by the Mechanisms and Robotics Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received October 8, 2024; final manuscript received March 13, 2025; published online May 15, 2025. Assoc. Editor: Shikui Chen.</footnote>
Stéphane Delalande

Laboratoire PIMM,
Arts et Métiers Institute of Technology,
CNRS, Cnam,
151 boulevard de l'Hôpital,
75013 Paris, France
e-mail: stephane.delalande2@lecnam.net

Stéphane Viollet

Aix-Marseille Université, CNRS, ISM,
CEDEX 09, 13284 Marseille,
Bouches-du-Rhône, France
e-mail: stephane.viollet@cnrs.fr

# 1 Introduction

Shape-morphing systems have garnered significant interest in
various engineering domains over the past decade [1]. They offer
advantages over traditional designs in terms of weight, part count,

fabrication simplicity, cost, and energy efficiency [2]. Morphing can
be achieved through mechanical metamaterials [3,4], auxetics [5],
lattice structures [6], origami [7,8], or kirigami [9]. Recent advances
in additive manufacturing have made shape-morphing systems par-
ticularly appealing for achieving compliance and bistability in a

<figure><img src="image_3.png" /><figcaption>Fig. 1 Mechanical response to a loading cycle of a monostable A₁ and a bistable B₁ and C₁ system. Potential energy and load–displacement curves associated with a monostable structure (A₂ and A₃), an asymmetric (B₂ and B₃) and a symmetric (C₂ and C₃) bistable structure. A₂, B₂, and C₂ are inspired from Ref. [14].</figcaption></figure>
single part. Compliant mechanisms, composed of flexible structural elements like beams, derive motion from deflection rather than from classical movable joints [10,11]. Bistable structures exhibit two stable equilibria, maintaining these positions without additional external energy. Mechanisms with a single stable position are monostable, while those with more than two stable positions are multistable. Single-part bistable mechanisms offer modularity for spaces requiring various interactions [12,13], such as in vehicles. However, implementing these innovative mechanisms for specific applications remains challenging due to the lack of quantitative and qualitative methods for comparing their mechanical efficiency.

The behavior of monostable and bistable mechanisms is illustrated in Fig. 1. Multistable structures may rely on buckling to snap from one position to another. In that case, it requires that the applied external load surpasses the Euler critical force [15], the threshold at which structures begin to buckle. Upon reaching this force, flexible beams buckle, initiating large deformation. When the unstable position is reached, a rapid snap-through transition occurs. Figure 1 depicts a loading cycle for a monostable ($A_1$) and bistable structure ($B_1$ and $C_1$). The monostable mechanism returns to its original configuration, while the bistable mechanism maintains its current equilibrium configuration. Bistable mechanisms based on shape-morphing do not require additional mechanical joints [16], making them useful for designing and producing nonlinear and tunable behavior. Bistable mechanisms have been studied using three approaches: analytical models, numerical simulations via the finite element method, and experimental investigations. Analytical models have been proposed [11,17–20]. Numerical simulations have been developed by Liu et al. [21], and experiments have been conducted [22,23]. Each study assesses the mechanical properties of bistable mechanisms. Typically, a potential energy–displacement curve is plotted, but more commonly, a load–displacement curve is drawn, which is more convenient for experiments. To ensure bistability, the potential energy curve must exhibit two wells, leading to two sign changes in the load–displacement curve: the first corresponding to the unstable equilibrium and the second to the second stable state. Figure 1 illustrates these curves, with cell $B_2$ showing the energy curve and cell $B_3$ showing the force curve. Stable positions are indicated by numbers 1 and 2 and squares (■). If only one energy well is present, the system is monostable, as shown in $A_2$ in Fig. 1.

During the transition from one stable position to another, three phases occur. Initially, energy increases due to the external load until the system reaches an unstable equilibrium, denoted as $U_{in}$. Next, a snap-through event occurs, and the mechanism releases energy to reach the second stable equilibrium, denoted as $U_{out}$. When the bistable mechanism moves from position 2 to position 1, the energy barrier $U_{out}$ is encountered again. These energies contribute to creating a map that shows the distribution of applied and released energy based on the structure's properties, known as an energy landscape.

Bistable systems exhibit different behaviors when actuated: asymmetric [24] or symmetric [25]. A system is considered asymmetric when one direction of actuation requires less energy than the other, facilitating snapping in that direction. This results in an asymmetry in the potential energy, as shown in Fig. 1, cell $B_2$, where the energy wells have different depths, $U_{in}$ is greater than $U_{out}$. This asymmetry is also reflected in the force–displacement curve by comparing the positive and negative areas, as illustrated in cell $B_3$ of Fig. 1. Conversely, symmetric behavior occurs when the mechanism requires the same amount of external energy to snap from one position to another, regardless of the direction of actuation. In this case, the energy wells have the same depth, $U_{in}$ equals $U_{out}$, and the positive and negative areas in the load–displacement curve are equal, as shown in cells $C_2$ and $C_3$ of Fig. 1. Asymmetry is a crucial parameter in mechanism design as it influences the onset of hysteresis during activation. For example, in an emergency stop button, it should be easy to push but hard to pull. Similarly, in a circuit breaker, the force required

to reset the breaker must be significantly higher than the force needed to open the circuit.

Numerous designs for bistable mechanisms have been proposed. The diversity of mechanisms and the versatility of materials expand their application in robotics [26–35], aerospace [36–40], biomechanics [41–43], sensors [44–46], micro-electro-mechanical systems [47–55], energy harvesting [56–59], and energy absorption [60–63]. Moreover, various classifications have been developed to review the state-of-the-art. For example, Opdahl [64], followed by Jensen [65], categorized them based on mechanical configurations such as sliders and cranks. In the field of energy harvesters, Harne and Wang [66] highlighted the use of magnetic bistable and mechanical systems for efficient energy conversion while Emam and Inman [67] reviewed composite laminate potential. Research on composite structures includes Zhang et al. [68] who classified mechanisms by their driving methods and Lemos et al. [69] who provided a comprehensive analysis of bistable composite structures for space applications. Cao et al. [1] highlighted the versatility of bistable mechanisms in various fields by grouping them by applications. Finally, Chi et al. [70] and Zhang et al. [71] sorted bistable mechanisms by structural forms, from 1D beams to 3D dome shells. However, to our knowledge, there has been no comprehensive overview or systematic comparative methodology addressing the advantages and disadvantages of these bistable shape-morphing mechanisms in the literature.

Despite the rapid growth of the literature, there is no consensus or standard procedure for comparing and predicting the mechanical behavior of bistable mechanisms. This work aims to create and analyze a dataset based on the existing literature, proposing a systematic methodology for comparing monomaterial bistable mechanisms using a mechanical energy landscape. Through this methodology, we identified the impact of design parameters on the mechanical behavior of these mechanisms.

In this study, we conducted a systematic review to create a dataset and map the energy landscape of existing mechanisms. We proposed a two-level classification system. The first level classifies mechanisms based on their global motion: translation or rotation. The second level describes the design type of the mechanisms. We categorized mechanisms into classes to determine whether factors like shape influence the energy landscape or symmetry of behavior.

The article is organized into five sections. Section 2 outlines the methods used to collect and process data from the reviewed articles. Section 3 presents the results of this analysis. Section 4 provides a discussion of the study. Finally, Sec. 5 offers the conclusions.

## 2 Materials and Methods

This section describes the methodology used in this article. Section 2.1 details the collection of articles and the creation of a dataset. Section 2.2 introduces a new classification of bistable mechanisms. The procedure for computing energy and comparing the classes is provided in Sec. 2.3. Finally, Sec. 2.4 outlines the statistical analysis procedure.

### 2.1 Creation of the Dataset.
To conduct the study, the first step was to gather a sufficient number of articles to create a dataset. Our selection method combined a systematic review with a meta-analysis. The systematic review involved collecting available research using a clearly defined method to answer a specific question. The meta-analysis is the statistical process that analyzes and combines the results [72]. In this case, the research question was: “Is there a way to compare the behavior of monomaterial bistable mechanisms, particularly based on the load-displacement curve?”

To begin, records were identified through three online databases: Web of Science, Science Direct, and Wiley, using the keywords *bistable mechanism* and *bistable mechanisms*. After removing duplicates, we identified 559 potential articles. As presented in Sec. 1, the focus is on the load–displacement curve, and the 559 articles were screened. We only studied classical monomaterial bistable
<figure><img src="image_4.png" /><figcaption>Fig. 2 Flow diagram of our systematic review analysis. Adapted from the PRISMA 2020 flow diagram [74].</figcaption></figure>

mechanisms with bar-linkages, excluding bistable origami and bistable composite structures. Additionally, we excluded any articles that did not include a load-displacement curve or those that only presented an analytical model. This process resulted in the selection of 107 full-text articles. We needed to fully characterize the properties of the mechanisms, including the energies, geometry of the compliant beams, and the constitutive material (Young's modulus), to compare them effectively. Articles missing any of these properties were excluded, leaving 47 full-text articles that met all criteria. Some articles studied more than one mechanism, leading to the inclusion of several figures from the same article. For example, Yang [73] created a mechanism with eight bistables, each with varying parameters, generating eight figures. In total, 13 additional figures were added. Ultimately, 60 figures were included in the quantitative analysis.

A flow diagram of this process, based on the PRISMA 2020 flow diagram [74], summarizes the creation of the dataset and is provided in Fig. 2.

Finally, other recent studies were deliberately omitted in this article because they did not allow for the calculation of load-displacement curve due to a lack of data [26].

## 2.2 Classification of the Mechanisms.
During our systematic review, we found that classifications of mechanisms typically focused solely on their global motion, identifying the primary types as translation and rotation. However, this traditional criterion may not be sufficiently refined to effectively distinguish between different mechanisms. To address this, we introduced two criteria for classifying mechanisms: the traditional criterion, which considers translation and rotation, and a newly developed, more detailed criterion based on the shape of the mechanisms. This novel classification includes six distinct classes, named as follows:

* Translation straight beam mechanism (TrST)

* Straight reinforced beam mechanism (TrSR)

* Curved beam mechanism (TrCB) (inspired by Qiu et al. [75])

* Rotation straight beam mechanism (RoST)

* Safety pin mechanism (RoSP)

* Young mechanism (YM) (as defined by Jensen et al. [76])

Mechanisms in the TrST, TrSR, and TrCB classes exhibit global motion characterized by translation, whereas RoST and RoSP mechanisms are defined by rotational motion. Within the YM class, there is a diversity of mechanisms that can exhibit either translational or rotational motion.

Mechanisms based on compliant beams are often fixed to two rigid frames: one grounded and the other serving as the effector. The two stable positions of each class are represented by dashed lines in Fig. 3. The constitutive material property chosen for this study is Young's modulus *E*, expressed in MPa. The selected geometric properties are the in-plane thickness *d*, the apex height *h*, the half-span of the beam *s*, and the out-of-plane thickness *t*. All dimensions are given in mm. The number of compliant beams holding the effector is represented by *n*<sub>b</sub>. All the data were stored in a table and used to compute the energies from the load-displacement curve. Table 1 shows the first five mechanisms of the TrST class extracted from the dataset. The full dataset is available in the Supplemental Material, Sec. A, on the ASME Digital Collection.

Figure 4 illustrates the distribution of mechanisms in each class within the dataset. Figure 5 presents a statistical analysis of the dimensions and Young's modulus. Each dimension of interest is displayed with a violin plot, which combines a visualization of the data distribution of the points on the left and a box plot in the center [78]. The dataset primarily represents the mesoscale, with an average span of a few centimeters. The bistable mechanisms are mainly fabricated from polymers, though metals [79] are also used. The primary advantage of using polymers is the ability to employ 3D printing for rapid prototyping.

In this section, we have detailed the data curation process for the systematic review analysis. A dataset was generated, including geometric and material properties and a load-displacement curve for each item.
<figure><img src="image_5.png" /><figcaption>Fig. 3 Classification of mechanisms in the dataset, sorted by their shape. TrST, TrSR, and TrCB mechanisms have mostly a translation movement. RoST and RoSP mechanisms move in rotation. YM mechanisms can have both types of motion depending on their configuration. Mechanisms are composed of compliant links and rigid parts. The deformed shape resulting from the force *F* or moment *M* is shown in dashed lines.</figcaption></figure>

<table><caption>Table 1 Extract of the dataset, only representing the geometric and material properties for the first five mechanisms of the TrST class</caption><thead><tr><th>Author</th><th>Date</th><th>s (mm)</th><th>d (mm)</th><th>t (mm)</th><th>h (mm)</th><th>E (MPa)</th><th>Trans./Rot.</th><th>Cat.</th><th>n<sub>b</sub></th></tr></thead><tbody><tr><td>Calmé et al. [30]</td><td>2023</td><td>11.5</td><td>1.5</td><td>0.25</td><td>2</td><td>1944</td><td>T</td><td>TrST</td><td>4</td></tr><tr><td>Chen and Fulein [19]</td><td>2014</td><td>31.514</td><td>6</td><td>1.2</td><td>5.557</td><td>1420</td><td>T</td><td>TrST</td><td>4</td></tr><tr><td>Correa et al. [77]</td><td>2015</td><td>95</td><td>50</td><td>5</td><td>20</td><td>1582</td><td>T</td><td>TrST</td><td>4</td></tr><tr><td>Fulei and Chen [20]</td><td>2016</td><td>69.678</td><td>12.55</td><td>1.5</td><td>6.709</td><td>1379</td><td>T</td><td>TrST</td><td>2</td></tr><tr><td>Holst et al. [18]</td><td>2011</td><td>69.678</td><td>12.55</td><td>1,5</td><td>6.709</td><td>1379</td><td>T</td><td>TrST</td><td>6</td></tr></tbody></table>

Note: The full dataset is provided in the Supplemental Material, Sec. A.

<figure><img src="image_6.png" /><figcaption>Fig. 4 Repartition of the mechanism classes in the dataset. TrCB and TrST mechanisms are the most represented.</figcaption></figure>

**2.3 Data Processing.** In Sec. 1, Fig. 1 introduces two types of energies ($U_{in}$ and $U_{out}$) involved in the displacement of the bistable mechanism between its stable positions (cells $B_2$ and $C_2$). To compare the mechanisms and the classes presented in Sec. 2.2, a criterion was required. We proposed using the energies defined in Sec. 2.1 as this criterion. The detailed procedure is depicted in Fig. 6.

For the 47 full-text articles, we first collected the geometric and material properties, as mentioned in Sec. 2.2. The load-displacement curves were then manually digitized using *GRAPH GRABBER*

software. This manual method was necessary due to the wide variation in figure orientation and data presentation across articles. The number of points digitized ranged from 24 to 76, with an emphasis on inflection points. Each figure's resolution was controlled, setting a minimum of 200 dpi to ensure accurate digitization.

Next, the load-displacement curves were normalized to compare energies, as detailed in Table 2. The chosen normalization was class-dependent, relying on geometric and material properties. The classes TrST, TrSR, and TrCB, which are translational mechanisms, used a classical normalization factor. Parameters for the
<figure><img src="image_7.png" /><figcaption>Fig. 5 Repartition of the geometry and material properties of the reviewed mechanisms. The latter are mostly fabricated in polymers and measure a few centimeters. At the center of each plot, a boxplot is shown, and the filled part corresponds to the data distribution visualization [78].</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Fig. 6 Energy data collection procedure for an article. Usually, an article provides one force-displacement curve and is not normalized. If it is normalized, we must renormalize according to Eqs. (1) and (2), compute the energies, and then add the point to the figure.</figcaption></figure>

<table><caption>Table 2 Normalization formulas of force *F*, displacement *u*, moment *M*, and angle *A* depending on the classes</caption><thead><tr><th>Classes TrST, TrSR, and TrCB</th><th>Classes RoST and RoSP</th><th>Class YM</th></tr></thead><tbody><tr><td>{ f = <sup>L<sup>3</sup></sup>&frasl;<sub>Elhn<sub>b</sub></sub> Fu<sub>nor</sub> = <sup>1</sup>&frasl;<sub>h</sub> (1) }</td><td>{ m = <sup>s</sup>&frasl;<sub>Eln<sub>b</sub></sub> Ma = <sup>&pi;</sup>&frasl;<sub>180</sub>A (2) }</td><td>None</td></tr></tbody></table>

Note: *E* is the Young modulus and *I* is the second moment of inertia. YM is not normalized due to the complexity of the mechanisms' shapes. *L* and *I* are defined as $L = \sqrt{h^2 + s^2}$ and $I = dt^3/12$.

classes RoST and RoSP were provided by Wang et al. [80]. The YM class was not normalized due to the complexity of the shapes. In some cases, curves were already normalized [81] and were renormalized according to our chosen method. After normalization, the energies $U_{in}$ and $U_{out}$ were calculated, where $U_{in}$ (resp. $U_{out}$) is the integral of the load-displacement curve above (resp. below) zero. A trapezoidal rule was used for the integration. If an article contained additional load-displacement curves and associated parameters, the above procedure was repeated, and the resulting normalized data were included in the dataset, as demonstrated by Yang [73].

We presented the procedure for calculating the energies related to each class of mechanism. The energies, representing a synthesis of the mechanisms, were compared using several methods. Trends in mechanical behavior, such as symmetry, were revealed by plotting the energies for each classification criterion. Specifically, for the shape-based class, ellipses were drawn to highlight potential differences. These trends were confirmed through statistical analysis, which is described in Sec. 2.4.

**2.4 Statistical Analysis.** The statistical analysis is based on the energy ratio $U_{out}/U_{in}$ to highlight the potential effects of classes on energetic performance and to reveal any differences. Since the groups are of different sizes, we used a nonparametric test. The groups are independent. To evaluate the effect of the type of motion, a Mann-Whitney test [82] was conducted because there are only two groups. For the shape-based criterion, which has more than two classes, a Kruskal-Wallis test [83] was used to compare the classes. If a significant difference was found between any of them, a Nemenyi post hoc test [84] was applied to identify which classes differed by making a family-pairwise comparison. For all tests, a classical level of probability $p < 0.05$ was considered significant. If the $p$-value is below this threshold, it indicates that shape or type of motion significantly affects the energy landscape and symmetry. The analysis was conducted using R software, version 4.3.1 [85].

## 3 Results

The energy landscape calculated according to the procedure described in Sec. 2.3 is presented in this section. The energy landscape of the mechanisms and their symmetric behavior were assessed using two methods. First, we plotted the released energy $U_{out}$ versus the applied energy $U_{in}$. Then, we plotted the energy ratio $U_{out}/U_{in}$ against the applied energy. The analysis was conducted using two sorting criteria: type of motion and shape. Section 3.1 presents the results for the first criterion, while Sec. 3.2 presents the results for the second criterion.

**3.1 Energies Sorted by the Type of Motion.** Figure 7 shows the computed energies, sorted by the type of motion. We observed that the circles are much closer to the ideal symmetric line than the crosses, indicating that rotational mechanisms tend to be more symmetric and that the type of motion appears to influence mechanical behavior. A statistical analysis is needed to confirm that the
<figure><img src="image_9.png" /><figcaption>Fig. 7 Normalized released energy plotted versus normalized applied energy during a snapping from the first to the second stable position. A distinction between the translation (cross) and rota- tion (circle) movement is made. The dashed line represents an ideal symmetric mechanism, where Uin = Uout [11,14,18–20,22–25,30,44–46,50,51,62,73,75,77,79,81,86–111].</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>Fig. 8 Statistical analysis of (a) the released to the applied energy sorted by the type of motion. The medians are significantly different, which means that rotational mechanisms have a more symmetric behavior than translational mechanisms (Mann-Whitney test, *p-value = 6.02 × 10⁻⁵ < 0.05). (b) The ratio of stored to applied energy sorted by classes. Significant difference is noted with a star (*) (Kruskal-Wallis test + Nemenyi post hoc test, *p-value = 1.49 × 10⁻³).</figcaption></figure>

type of motion of the mechanisms impacts the energy levels and thus the symmetry of their mechanical response. The tests used are described in Sec. 2.4.

Figure 8(a) presents the results of the analysis of the energies of the mechanisms sorted by the type of motion. The energy ratio of rotational mechanisms is significantly higher than that of translational mechanisms (median = 0.860, [Q1 – Q3] = [0.660 – 0.971] versus 0.175 [0.106–0.322], W = 23, p < 0.05). This significant difference is noted in Fig. 8(a) with a horizontal bar highlighting

the two groups and a star (*) above the bar. In conclusion, the type of motion influences the behavior of bistable mechanisms. Readers can refer to the Supplemental Material, Sec. B, to view the same energy map where each article is represented by a specific color.

**3.2 Energies Sorted by Shape.** In Fig. 7, we showed that translational and rotational mechanisms can be clearly identified and separated. While trends are easily discernible for the type of
<figure><img src="image_11.png" /><figcaption>Fig. 9 To assess the symmetric behavior, the ratio of the normalized released energy and the normalized applied energy against the applied energy is plotted. A distinction between the translation (cross) and rotation (circle) movement is made. The dashed line represents an ideal symmetric mechanism, where $U_{in} = U_{out}$. A log scale is used to represent the range of properties. The center of the ellipses is calculated with the mean of the energy coordinates [11,14,18–20,22–25,30,44–46,50,51,62,73,75,77,79,81,86–111].</figcaption></figure>

<table><caption>Table 3 Dimensions of the ellipses presented in Fig. 9</caption><thead><tr><th>Class</th><th>x<sub>c</sub></th><th>y<sub>c</sub></th><th>l<sub>MA</sub></th><th>l<sub>ma</sub></th><th>φ (deg)</th></tr></thead><tbody><tr><td>TrST</td><td>3.92</td><td>0.125</td><td>7.81</td><td>1.16</td><td>-1.98</td></tr><tr><td>TrSR</td><td>76.0</td><td>0.302</td><td>3.41</td><td>1.14</td><td>17.12</td></tr><tr><td>TrCB</td><td>68.8</td><td>0.159</td><td>7.87</td><td>2.18</td><td>3.49</td></tr><tr><td>YM</td><td>0.616</td><td>0.750</td><td>11.7</td><td>0.455</td><td>-0.082</td></tr></tbody></table>

Note: For each class, $x_c$ and $y_c$ are the coordinates of the center, $\phi$ is the orientation of the ellipse. The major axis length $l_{MA}$ and the minor axis length $l_{ma}$ are given in a number of decades. A length equal to 2 represents a stretching over two decades.

motion, they are less apparent for shape-based classes. Therefore, a new representation of the energy landscape is needed. Using the energy ratio described in Sec. 2.4, we plotted a new energy map in Fig. 9. Each article in the dataset is represented, with colors indicating the shape of the bistable mechanism. The RoST, RoSP, and YM classes appear to be more symmetric than the other classes.

For each class, the computed mechanical energy points were enclosed by a fitted ellipse [112]. The coordinates of the points were averaged to determine the center of the ellipse. The orientation and radii of the ellipses are provided in Table 3, with minor and major axes given in a number of decades. External points (three for the TrST class and one for the TrCB class) were too distant from the others to be included within the ellipse. The limited number of mechanisms in the RoSP and RoST classes hindered their ellipse creation. The ellipses for YM and TrST are not rotated or only slightly rotated, indicating that mechanisms within these classes exhibit similar behavior across a wide range of

applied energy: mostly symmetric for YM and mostly asymmetric for TrST. The TrSR and TrCB ellipses are tilted, suggesting that these classes contain a wider range of mechanisms in terms of mechanical behavior and indicating a relationship between the energies. Notably, most translational mechanisms have a small energy ratio, indicating they are strongly asymmetric.

The results of the statistical analysis of the mechanisms sorted by shape are presented in Fig. 8(b). The Kruskal–Wallis test confirms that the shape of the bistable mechanism significantly influences the energy ratio ($\chi^2 = 19.59, p = 1.49 \times 10^{-3} < 0.05$). The Nemenyi post hoc test shows that the energy ratio $U_{out}/U_{in}$ of the YM class is significantly higher than that of the TrST class (median = 0.768, $[Q1 - Q3] = [0.646 - 0.951]$ versus 0.128, $[0.101-0.204], p = 0.007$), and that YM energy ratio is significantly higher than that of the TrCB class (median = 0.768, $[Q1 - Q3] = [0.646 - 0.951]$ versus 0.165, $[0.088-0.372], p = 0.044$). These significant differences are indicated in Fig. 8(b) with a horizontal bar highlighting the classes and a star (*) above the bar. Only the significant *p*-value is provided here for simplicity, but readers can refer to the Supplemental Material, Sec. D, for the *p*-values of all pairwise comparisons. Additionally, readers can consult the new energy map where each article is represented by a specific color in the Supplemental Material, Sec. C.

For both criteria, we demonstrated that the size of the mechanism does not influence the energy levels and symmetric behavior (Kruskal–Wallis, $\chi^2 = 2.04, p = 0.360 > 0.05$). The detailed demonstration is available in the Supplemental Material, Sec. E.

Through a systematic review analysis, we were able to map the energy landscape of multiple bistable mechanisms and assess the symmetry or asymmetry of motion-based and shape-based categories. We showed that global motion and shape are key parameters that influence the mechanical behavior of bistable systems.
# 4 Discussion

The goal of the study was to predict and evaluate the mechanical behavior of bistable mechanisms, particularly the symmetry of actuation. Our main hypothesis was that the global motion and shape of mechanisms are crucial factors that influence both energy levels and symmetry. To investigate this, we collected a large number of articles on bistable mechanisms using a meta-analysis method and computed the applied and released energy for each mechanism.

## 4.1 Symmetry and Mechanical Behavior.
By conducting a systematic review analysis, we proposed a new classification based on types of motion and shapes. Our study first revealed an imbalance between translational and rotational bistable mechanisms, with the latter being much less studied. We then showed that rotational mechanisms exhibit better symmetric behavior than translational ones. This difference in mechanical behavior can be explained by geometrical constraints. First, the angle between compliant links and grounded rigid parts remains constant when the mechanism deforms, hindering the mechanism from deforming symmetrically. Shan et al. [81] demonstrated that the higher this angle, the more asymmetric the mechanism, particularly for mechanisms with translational motion. Second, modulating beam thickness reduces bending stresses and promotes symmetric behavior [75,102]. Third, Vangbo [113], Qiu et al. [75], and Kim and Han [114] showed that constraining the second buckling mode enhances symmetry. In addition, Yang [73] designed a fully symmetric bistable mechanism from an asymmetric snapping beam by precompressing it and introducing prestresses. Dunning et al. also showed a drastic evolution of the energy ratio with varying preload- ing cases [115].

## 4.2 Limitations, Guidelines, and Use Cases.
Some limitations emerged during our study. First, we only considered the global motion to classify the mechanisms. However, distinguishing between rotation and translation can be challenging. For instance, Tran and Wang [105] proposed a mechanism with a shuttle at its center. The global motion, defined as the motion of the shuttle, is translational, so it was categorized as a translational mechanism. However, the compliant beams fixed to a grounded rigid frame and linked to the shuttle exhibit rotational movement. In our study, we chose to consider only the global motion. Additionally, the limited number of mechanisms in the rotation straight beam (RoST) and safety pin (RoSP) classes prevented a detailed analysis. With a larger number of mechanisms, other significant differences might emerge.

Despite these technical limitations in the classification, our study shows that rotational mechanisms should be favored when symmetric behavior is required, such as for regular on/off buttons. Conversely, translational mechanisms are preferred for asymmetric use cases, like emergency push buttons. The classification based on shape also provided guidelines for designing mechanisms according to desired symmetry properties. For symmetric applications, the YM is recommended. For asymmetric applications, the translation straight beam (TrST) shape is advised, as this class has the lowest energy ratio. The other classes exhibit a wider range of energy ratio values, making it difficult to provide specific recommendations for their use. However, the TrCB class warrants thorough analysis due to its two distinct intraclass groups.

## 4.3 Exploring the Design Space.
In our dataset, some articles contain multiple figures, as mentioned in Sec. 2.1. For quasi-identical mechanisms (having the same shape), some parameters varied. The modification of a single parameter in the design of the mechanism drastically affects the energy ratio and thus the mechanical behavior. For example, Yang [73] modified Young's modulus and the thickness of his mechanism to obtain a deterministic deformation sequence. The calculated energy ratio varied significantly, ranging from 0.0396 to 0.6995. The mechanism with an

energy ratio of 0.04 is highly asymmetric, while the other is symmetric. By using the meta-analysis method and computing the energy landscape, we observed this evolution of symmetry and the influence of a single parameter. This approach allows exploration of the design space and tuning of the mechanism's behavior for specific use cases. Similarly, further studies could be conducted to investigate the effect of other parameters on a single bistable structure, such as temperature [116], material, pressure, aging, or exposure to rays (UV, IR, X).

The range of possibilities for bistable mechanisms spans from the nanometer scale [42] to several meters in size [117]. We have shown that size does not interfere with the mechanism's properties in terms of energy. Figure 9 highlights that a few translational mechanisms are symmetric at smaller scales. Therefore, there is potential for further studies to design small mechanisms featuring symmetric energy. Conversely, rotational mechanisms are generally symmetric and require low amounts of energy to activate. Hence, there is an opportunity for additional exploration to create asymmetric and high-energy-demanding rotational bistable mechanisms.

## 4.4 Implication for Future Studies

### 4.4.1 Bistable Origami Structures.
In this study, the focus was solely on classical monomaterial bistable mechanisms with bar linkages. We identified key parameters, such as type of motion and shape, and demonstrated their influence on the symmetry of actuation. Historically, in the late 1990s and early 2000s, bistable structures garnered significant interest. Classical bistable mechanisms with bar linkages were extensively studied by Howell [11] and his colleagues, including Ref. [76]. Concurrently, the ancient art of origami, the art of folding paper, inspired researchers and revealed bistable properties. Guest and Pellegrino [118] twisted the Yoshimura pattern [119] and created a multistable structure with cylindrical triangles. Another pattern, created by Kresling [120], led to the bistable Kresling tower [121]. Although bistable origami structures were not studied here, they certainly warrant further investigation. Moreover, the novel methodology based on an energetic approach and systematic review proposed here can be applied to origami-based bistable structures.

### 4.4.2 The Role of Interactions.
We established links between design, motion, mechanical behavior in terms of symmetry, and use cases. Our findings revealed that the type of motion and shape are key parameters, and surprisingly, all the parameters were interdependent. These parameters are influenced by interactions with mechanisms, which were not considered here. Such interactions should be integrated into the initial stage of mechanism conception.

At the macroscale, bistable mechanisms represent new opportunities for human-machine interface [104], but their specific behavior could lead to new design constraints, integrating biomechanics [122] while grasping or handling [123,124] these new interfaces. These constraints will affect the geometry, shape, and consequently, the energy landscape. For instance, the energetic aspect of bistable mechanisms could be coupled with human action by considering the impact of the hand effector during manipulation [125]. The use of coupled mechanical impedance between mechanism and human action could potentially enhance the design of manipulated mechanisms. Despite these additional considerations, this study can assist in selecting a mechanism during the design process.

# 5 Conclusion

Bistable mechanisms have garnered increasing interest over the past decades, leading to the creation of numerous mechanisms. This study aimed to review the energetic properties and mechanical behavior, specifically the symmetry of actuation, of monomaterial bistable mechanisms. These properties were investigated through a systematic review analysis. A dataset was created from articles
and then sorted using two criteria: type of motion and shape. We proposed a new classification of mechanisms and an associated terminology. The energies involved during the actuation of the mechanisms were computed by integrating the load-displacement curve.

This methodology led to the creation of a criterion to assess the symmetry of actuation: the energy ratio, defined by the normalized released energy divided by the normalized external applied energy. Using this criterion, an energy landscape was drawn, providing a comprehensive mapping of bistable mechanisms and distinguishing between different classes. The proposed meta-analysis ensures both quantitative and qualitative comparisons of monomaterial bistable mechanisms.

We first demonstrated the influence of the type of motion on energy levels and symmetry of actuation. Rotational and translational mechanisms differ significantly, with rotational mechanisms being more symmetric than translational ones. This means rotational bistable mechanisms are likely to behave consistently regardless of the direction of activation. We then investigated the influence of the shape of the mechanisms. YM significantly differ from both TrSTs and translation TrCBs; YM are symmetric, while TrST and TrCB are asymmetric.

Overall, the results of this work provided a method to study the effect of key parameters of bistable mechanisms on their mechanical properties. This study ultimately revealed design guidelines for the conception of compliant, partless, and recyclable bistable mechanisms that can replace complex assembled rigid-body mechanisms.

## Acknowledgment

The authors thank Dr. Julien Serres for his help in statistical analysis and William Wilmot for his thoughtful comments on software development.

This work was conducted in the framework of the OpenLab Stellantis AMU "Automotive Motion Lab" and OpenLab Stellantis Arts et Métiers "Material and Process." This work was supported by CNRS, Aix-Marseille University, and le Cnam.

Part of the study was funded through a research collaboration between the Stellantis company, the Aix-Marseille university, and le Cnam. The authors will not receive any financial benefits from the results of this study.

K.M. was supported by a CIFRE doctoral fellowship from the ANRT and Stellantis (agreement #2021/0871).

## Conflict of Interest

There are no conflicts of interest.

## Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

## Author Contribution Statement

Kéliane Megret: conceptualization, data curation, formal analysis, investigation, methodology, software, vizualization, and writing—original draft. Justin Dirrenberger: conceptualization, funding acquisition, investigation, project administration, supervision, and writing—review and editing. Jocelyn Monnoyer: conceptualization, funding acquisition, project administration, supervision, and writing—review and editing. Cyrille Sollogoub: supervision, validation, and writing—review and editing. Benjamin Goislard de Monsabert: supervision, validation, and writing—review and editing. Laure Fernandez: supervision, validation, and writing—review and editing. Stéphane Delalande: supervision,

validation, and writing—review and editing. Stéphane Viollet: conceptualization, funding acquisition, project administration, supervision, and writing—review and editing.

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