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Article

Bi-Stability in Flexure Beams: Introducing a Setup for Experimental Characterization

by
Alejandra Diaz-de-Leon
1,†,
Enrique Cuan-Urquizo
1,2,3,*,‡,
Armando Roman-Flores
1,2,*,‡ and
J. L. Gordillo
1,†
1
Tecnologico de Monterrey, School of Science and Engineering, Av. Eugenio Garza Sada 2501 Sur Monterrey, Monterrey 64849, Mexico
2
Tecnologico de Monterrey, Institute of Advanced Materials for Sustainable Manufacturing, Av. Eugenio Garza Sada 2501 Sur Monterrey, Monterrey 64849, Mexico
3
Laboratorio Nacional de Manufactura Aditiva y Digital (MADIT), Autopista al Aeropuerto, Km. 9.5, Calle Alianza Norte 100, Apodaca 66629, Mexico
*
Authors to whom correspondence should be addressed.
†
These authors are part of the Research Group on Robotics at Tecnologico de Monterrey, Mexico.
‡
These authors are part of the Metamaterials Lab Group at Tecnologico de Monterrey, Mexico.
Appl. Sci. 2022, 12(22), 11438; https://doi.org/10.3390/app122211438
Submission received: 24 October 2022 / Revised: 5 November 2022 / Accepted: 9 November 2022 / Published: 11 November 2022
(This article belongs to the Section Mechanical Engineering)
Browse Figures
Versions Notes

Abstract

:
Bi-stable mechanisms can remain in two positions without a power input or locking system. These mechanisms can achieve large displacements with low energy, requiring power only during switching. However, their design and analysis are challenging due to the non-linear behavior of their flexible members. Researchers commonly use experimental approaches to study and validate bi-stability, which demands efficient and affordable setups. This work presents a versatile and easy-to-fabricate setup to investigate the bi-stable behavior of flexure beams made of styrene plastic. The testing setup is made of laser-cut acrylic connectors and 3D-printed fixtures. The experiments demonstrate that all tested beams are bi-stable and highlight the impact of thickness on their behavior. The critical forces obtained range from 0.0277 N to 5.2724 N between the thinnest and the thickest samples. The distance traveled before snapping increments, with thickness, ranges from 18.78 mm to 32.6 mm. The samples were subjected to a cyclic compressive load and demonstrated a considerable decrement in the critical forces after the first load. Thicker flexural beams present a more significant deformation, causing fractures in some cases after the five loads. Regardless of the thickness, all samples traveled the same total linear displacement of 45 mm. The presented setup demonstrated consistency and rapidness in experimental bi-stability characterization, with the styrene plastic flexure beams proving to be suitable for studying bi-stability.

1. Introduction

Bi-stable mechanisms can stay in two positions without a power input or locking systems. These mechanisms can achieve large displacements with lower energy requirements and are versatile alternatives for various types of applications. Bi-stable mechanisms have been used in microdevices (MEMs) [1,2,3], aerospatiale applications [4], biomedical devices [5], robotic systems [6], and to add functionalities to autonomous vehicles [7,8]. The emerging area of soft robotics also harnesses bi-stable mechanisms due to their ability to move faster, amplify force outputs, overcome low actuation energy, and generate extreme compliance through their hyperplastic components [9]. Previous literature has introduced bi-stable elements that enable locomotion in soft robots [8,10,11]. Another use is in mechanical metamaterials with purposes such as deployable objects [12,13], shape-reconfigurable materials [14], or auxeticity (negative stiffness) [15,16,17].
When bi-stability is employed in compliant mechanisms, they gain advantages such as a decrease in the number of parts, reduced friction, a simplified manufacturing process, and a reduction of backlash, friction, and wear [18]. The deflected elements store and release strain energy to stay in two or more configurations, allowing the integration of motion and energy storage into a single component [19]. Compared with traditional rigid-body mechanisms that continuously consume energy, bi-stable compliant mechanisms only employ power during switching.
Regardless of their benefits and wide range of applications, bi-stable mechanisms are more difficult to design and analyze than traditional mechanisms, as they present high nonlinearity induced by the post-buckling behavior of their flexible beams [20,21]. Furthermore, they must maintain a low stress concentration to avoid fracture [22]. Researchers have proposed various approaches for investigating bi-stable mechanisms using analytical modeling, finite element analysis, and experiments. Some analytical approaches are pseudo-rigid body modeling (PRBM), which lacks accuracy at the end slope of the beam [23], and the elliptic integral solution (EIS) [24,25,26], which is more efficient but limited to simple geometries due to its high complexity. In contrast, non-linear finite element simulation is commonly used as a design evaluation method due to its practicality. However, the results do not strictly coincide with the experimental ones [11] because this type of solver assumes that the model does not have the imperfections found in real life [21].
As it is challenging to establish a method to determine the nonlinearities caused by the buckling behavior of the bi-stable mechanism, many authors opt to validate their analysis experimentally, specifically the force-distance relationships. To do so, the specimen must be placed in the measuring device through a setup and fixtures that adhere to the defined boundary conditions. As the bi-stable mechanism studied in this publication is based on the basic fixed-fixed beam topology [22], only setups for variations of this arrangement are considered in this section. For instance, to evaluate the force-displacement characteristic of the perching aerial vehicle presented by Zhang et al. [7], the gripper is fixed to the test plate and a hook is attached to the gauge. Both are joined with a string that pulls the specimen to measure tension [7]. Similarly, Chen et al. similarly attach the specimens of a multi-material, bi-stable structure to the testing machine with two customized 3D printed grippers [12]. These setups are restricted to the specific design and geometric characteristics of the samples. Other setups use glue to secure the bottom and top of the structures to the compression plate and base of the machine. This option is limited to specimens that require small forces to snap, either because of their small scale or because they are made of a material with low tensile strength [15,27]. A more sophisticated example uses steel fixtures designed to be bolted to the base of the machine and metal clamps that uses rods, springs, and ball bearings to allow the beam to slide back and forth horizontally [28]. Although this setup is robust, the fabrication of steel parts requires more sophisticated equipment and is expensive.
This manuscript presents an alternative to improve the consistency and quickness of experimental approaches for studying the bi-stability of flexure beams. This work has two objectives. First, the proposal of a versatile experimental setup that is easy to assemble and that allows the evaluation of beams of different lengths, thicknesses, initial angles, materials, and fabrication methods. It consists of a base with perforations in various places that allow modifying the testing area’s dimension. Parts can be fabricated with standard equipment and commercial materials. Second, it uses the setup to perform an experimental study of the behavior of bi-stable mechanisms (critical force, snapping distances, and deformation) made of different thicknesses of commercial styrene stripes.
The first part of this paper presents the design and construction of the bi-stable mechanism (Section 2.1), followed by a detailed description of the design and fabrication of the experimental setup in Section 2.2. Section 2.3 explains the implementation of the two experiments: one to evaluate the beams on cyclic loading and another to identify the maximum required force for the mechanism to snap-through (critical force). Section 3 presents and analyses the results of the experiments, including findings about the critical force (Section 3.2), the distances each mechanism travels to snap-through (Section 3.3) and the deformation of the buckling beams (Section 3.4). Finally, in Section 4, conclusions are drawn.

2. Materials and Methods

Each configuration of a bi-stable mechanism is called a stable equilibrium point or state, which commonly corresponds to different geometric configurations [29]. While transitioning from one state to the other, the system reaches an unstable equilibrium point, which evokes a phenomenon known as snap-through. Once passed, the mechanism does not require any external force and uses the stored energy to move to and remain in the second position [28]. This point is known as the snap-through point (SNP). The required input force to keep the system in equilibrium at a zero-stability position is called the critical force (CF). The maximum input force must be greater than the CF to switch from one stable position to the other [30]. A typical load-displacement curve of this behavior is shown in Figure 1a, where the different configurations (equilibrium positions) and snap-through point are indicated.

2.1. Bi-Stable Mechanism Design and Assembly

The design of the bi-stable mechanism studied in this work is based on the conventional fixed-fixed beam topology introduced by Jensen et al. [22]. As illustrated in Figure 1b, the presented design consists of a slender beam divided in the center by a slider. Each side of the beam is called a flexure beam and is attached to the slider through a pivot joint. The other end of each flexure beam is fixed. A base made of acrylic has perforations to allocate all the components with cap bolts (Figure 2a). The input force is applied at the central element, causing the slider to move downward and generate a linear motion.
The design of the components and the assembly strategy of the proposed bi-stable mechanism are inspired by the kit presented by Limaye et al. [31]. The proposed kit here consists of two elements: flexible slender beams and semi-rigid circular connectors with multiple slots. The elements can be press-fit by hand, allowing rapid assembly and testing of different compliant designs. From the original kit, we adapted the idea of having thin slots in the rigid components (sliders and connectors) to insert the individual flexure beams. The design of the slots locks the flexure beams in place. The fixed points at the ends of the flexure beam are referred to as anchors, and the component that receives the compression force to move along the slider is called the trigger. Figure 2b shows the components and layout. One end of the flexure beam is attached to the anchor, and the other is attached to the trigger. As shown in Figure 1b, the proposed design has two configurations. The length of the slider restrains the linear displacement, defining the form of the flexure beams in both initial and final configurations. To begin, the mechanism is set to the first configuration (first stable equilibrium position). When the force F is applied along the Y axis, the slider begins to move vertically until it overcomes the snap-through point and switches to the second configuration (second stable equilibrium position).

2.2. Test Setup and Component Specifications

To characterize the bi-stable behavior of the mechanism, a setup was designed, fabricated, and integrated into a TVT 6700 Perten Texture Analyzer (Perten Instruments, New South Wales, Australia). The main components of the setup are shown in the exploded view in Figure 3a. The actual setup is shown in Figure 3b.
The components of the bi-stable mechanism (flexure beams, slider-triggers, and anchors) are made of commercial materials that are easy to acquire and affordable. The flexure beams are made of different calibers of commercially available styrene plastic sheets. To create stripes, the sheets were laser cut in an STM-L1390 CO2 CNC laser cutter. The stripes have a total length of 55 mm and a height of 3 mm. On their part, the rigid components (the slider and the anchors) have holes for screws and slots for press-fit assembly of the flexure beams. The shape of the slot locks the flexure beam in place. These components were made of 3 mm cast acrylic (Acrylite®) and fabricated with the same laser cutter. The physical components are shown in Figure 4a.
The setup consists of a base with circular perforations in different places to position the components with M3 screws fixed with M3 cap bolts. As shown in Figure 2a, it also has a slender slot in the center to guide the slider as it moves. The setup is limited to an area of 146 mm by 83.5 mm. The distance of the linear displacement of the slider is 45 mm; this length is determined by the position of the screws in the trigger, which constrain the movement when meeting the slider ends. The base is also made of 3 mm acrylic cast fabricated with the same laser cutter. In turn, the two sides of the acrylic base are press-fit inserted into 3D printed fixtures bolted to the base of the machine. These fixtures were 3D printed with the blue eSun® 2.85 mm Polylactid Acid (PLA) filament in a Zortrax M300 FDM 3D printer.
To determine the range of operational actuation forces, different thicknesses of flexure beams were tested. The thickness was defined by the length of the calibers available on the market. Table 1 lists the five calibers used for this study, their nomenclature, and their thickness in millimeters. To join the flexure beams, the width of the slots in the anchors and triggers was adapted to meet each styrene thickness. The tailored designs for each thickness can be seen in Figure 4b. As shown in Figure 4c, one end of the styrene stripes is press-fit assembled to one slot of the anchor, and the other end to the trigger, shortening the 55 mm original length to an operative length of 53 mm.

2.3. Experiment Description

The objective of the experiments is to characterize the mechanical behavior of the bi-stable topology described in Section 2.1. Through the tests, the bi-stability of the flexure beams of 55 mm length and five different thicknesses is characterized along with their critical force CF required to snap (maximum input force), the snap-through point STP (linear displacement before it snaps-through), their deformations, and their performance while put under compression five times consecutively. To that end, two experiments were performed:
  • Performance by thickness. Three samples of each thickness were tested under compression to measure the maximum required force (CF), the distances at which these forces are reached, their snap-through points (STP) and their deformation.
  • Cycling loading. The same sample was subjected to compression five times to characterize its mechanical performance and determine an approximate behavior for real applications. In each test, the texture analyzer did the compression of the flexure beam, which was set back manually to the original position.
To begin the experiments, the machine was configured to move the compression plate 35 mm downward at a speed of 1 mm/s. A 672045 compression plate was used to drive the sliding trigger until the mechanism snapped. A Samsung A8 smartphone was placed in front of the sample to record the tests.
The software TexCalc 5® (v5.2.2.287, Perten Instruments, New South Wales, Australia) was used to obtain the data recorded from the machine. The measurements are plotted in a graph that shows the force in grams as a function of the displacement in millimeters, which represents the first part of the bi-stable curve. The software also generates CVS files with the load-displacement data, which was then processed in Matlab® (R2021a, MathWorks, Natick, MA, USA) using a median filter to reduce noise and generate the graphs shown in this paper.

3. Results and Discussion

3.1. Bi-Stability Validation and Total Linear Displacement

In this section, the results of the two experiments are presented and discussed. The tests demonstrated that all the samples under compression can trigger the bi-stability, regardless of their thickness. Figure 5a–e shows the bi-stable behavior in the load-displacement curves for the five thicknesses. The graphs represent the first part of the bi-stable curve (from state 1 to the snap-through point), as the second part (snap-back) cannot be measured by the texture analyzer. Each graph shows the curves for each of the three samples (a, b and c), and a thicker black curve represents the mean values of the three curves. All curves rise to their CF and then drop to zero (snap-through point), demonstrating bi-stability. The shaded gray areas correspond to the standard deviation caused by the variation among the three samples. Videos of the experiments can be found in the Supplementary Material section.
Although the critical force and the snap-through points vary within the same thickness, all samples exhibit linear displacement of the same total distance of 45 mm, which is the entire length of the slider (this observation can be seen in the videos of the experiments found in the Supplementary Material section). This indicates that the same linear displacement can be reached regardless of the thickness of the flexure beams and within a wide range of critical forces.

3.2. Influence of Flexure Beam Thickness on Bi-Stability Performance

3.2.1. Critical Forces as a Function of Thickness

The results from the “Performance by thickness” tests demonstrate that larger thicknesses require larger forces. Figure 5f shows the mean curves of the five tested thicknesses. The critical forces range from 0.0277 N (for a thinner flexure beam c10) to 5.2724 N (for a thicker flexure beam c30). The distance traveled before the critical force is reached (critical force point) also varies from 2.47 mm to 7.99 mm.
The variability of the critical forces also occurs within samples of the same thickness, due to fabrication and assembly imperfections inherent in experimental tests. Although the load-displacement curves converge near the end, they diverge at the beginning and around the highest part of the curve. The largest variation is presented in the caliber 10-flexure beams (0.25 mm thick), and variability decreases as thickness increases. When manually assembled and placed in the fixture, the components have millimetric differences that can be overlooked in thicker flexure beams but have a considerable affect the thinner ones. Also, in fine flexure beams, these irregularities are caused by the inability of the instrument to measure small forces accurately.
As shown in Figure 6a, the critical force increases gradually at the beginning of the curve that corresponds to calibers 10, 15, and 20 and then rises notably for calibers 30 and 40. The data exhibits a non-linear force-thickness relationship, as the deformation response is governed by buckling. The geometric parameters t (thickness) and l (length) have a non-linear effect on structures that deform by buckling (similarly to their participation in the bending of beams). In such slender structures, their bending and buckling properties and the thickness of their rectangular cross-sections follow a cubic scaling law.

3.2.2. Snap-Through Points as a Function of Thickness

From the experiments, we obtained STPs within a range of 18.78 mm to 32.6 mm. Figure 6b shows the mean of the snap-through points of the three tested samples of each caliber as a function of thickness. It demonstrates that larger thicknesses snap at longer distances, which indicates that thinner flexure beams snap-through faster, requiring less energy input to reach bi-stability. The STP-thickness relationship shown in Figure 6b is non-linear and tends to flatten at the end, suggesting that the STP of even thicker beams would uphold or even not be reached, making the flexure beams fail to snap as they gain thickness.
The snap-through points also present fluctuations within samples of the same thickness. As observed in the force-distance plots in Figure 5a–e, the curves converge at the end, indicating that samples of the same caliber travel a similar linear distance before snapping. However, snap-through points happen at slightly different distances in flexure beams of the same caliber, except for the c10 samples that present a more considerable range of variability. This behavior is better illustrated in the error bars in Figure 6b. These findings demonstrate that, regarding discrepancies in the accuracy of the setup because of manual assembly, the snap-through distances stay within a limited range, especially in larger widths, which can be suitable for concrete applications.
In accordance with the critical forces, the distances at which the mechanism snaps (snap-through points) also increase with thickness. Figure 6c compares the critical force to the snap-through points by standardizing their different values within a range between 0 and 1. The graph shows an increment in both CF and STP as the flexure beam becomes thicker but exhibits contrasting behavior: the CF curve increases sharply while the slope of the STP curve decreases gradually at the end of the curve. This behavior suggests that flexure beams with even larger thickness would continue to require greater input forces and would snap-through at a distance close to the limit exhibited in c40 or even fail to transit to the second stable equilibrium position.

3.3. Characterization of the Flexure Beams under Cyclic Loading

3.3.1. Critical Forces

To characterize the mechanical behavior of bi-stable flexure beams that would be used more than once in real-life applications, each sample was subjected to five loads under compression. The values of critical forces reached in each load are displayed in Figure 7. For samples c15, c20, c30, and c40, the first cycle has the highest critical force, and the following cycles show an input force reduction per load, indicating a loss of stress in the beam that suggests the elastic limit of the styrene stripe was exceeded. This behavior can be verified by the deformed shape of the flexure beams after the experiments (Figure 8), which will be discussed in Section 3.4.
For its part, the c10 flexure beam presents random variations in the results due to the equipment capabilities and affectations of the manual assembly. Small imperfections in fabrication and manual assembly have a higher influence on the thinnest beams. As the forces don’t necessarily decrease per load and the shape of the flexure beams remains almost straight after the tests (Figure 8a), it is concluded that the elastic limit of the c10 samples is never surpassed.

3.3.2. Snap-Through Point

In contrast to the critical force, which decreases greatly after the first load, the snap-through points remain consistent. Figure 9a–d shows the STP of each load per sample. The standard deviation confirms that the variations are never bigger than ±1.00 mm, which suggests that the snap-point is reached at a similar distance regardless of the cycle load. This aspect evokes predictable behavior that could provide the system with reliability for real-life purposes.

3.4. Flexure Beams Deformation

Besides the study of the hard data obtained by the texture analyzer, there is relevance in investigating the form of the flexure beams to validate the behavior observed in previous sections. The snapping sequence presented in Figure 1b shows that the system has three main buckling shapes as it moves along the slider. This section explores the deformation of the flexure beams at those points: the beginning, end, and snap-through moment, according to their thickness. The deformation of the flexure beams was analyzed by observing the specimens throughout the tests and through a thoughtful examination of the pictures and videos taken during the experiments.
Before being placed in the setup, the form of the flexure beams is completely straight, without deformations. Once allocated in the fixture, the 55 mm long flexure is compressed to fit a 50 mm width, acquiring the shape of an arch. As expected, under axial compression, the beam buckles to its initial form, which is the first state equilibrium position. The arch doesn’t reach its maximum height because it is restrained by the length of the slider. Therefore, the initial arches are slightly prestressed. This can be noticed in the starting point of the force-distance graphs in Figure 5, where the curves do not start at zero Newtons of force.
At the snap-through point, the system arrives at its unstable equilibrium position. Although buckling modes are difficult to predict, we observe that, at the STP, the arches of all sample beams remain symmetric about their centers, which means that both sides of the arch buckle in the same direction (upwards or downwards). In contrast, other studies found in the literature show both symmetrical and asymmetrical buckling modes [28,32,33]. Differences could be attributed to various sources. First, those samples in [28,32,33] consisted of a single beam, while those tested here are two independent beams loaded at the center. Regardless of the symmetry of the arches, the buckling direction in samples c20 and c30 varies. Figure 10c,d show these variations at the samples that buckle upwards or downwards indistinctively. Furthermore, Figure 10a,b,e show all c10, c15, and c40 samples buckled upwards.
As the beams are homogeneous and isotropic, there is no preference for buckling up or down. The diversity of buckling directions also seems irrelevant to the force-distance behavior of the beams. However, the change in thickness of the flexure beams affects the buckling beam’s non-linearity, thus the mechanical behavior. As observed in Figure 10a–c, thinner beams present smooth arches during snapping, but the curvature becomes less smooth as they increase thickness. c30 and c40 arches tend to bend toward the central slider, adding stress to the joints. Samples c30-1 and c40-1 even bend sharply near the joint, causing a partial fracture. This condition is exposed in Figure 8d,e.
Nevertheless, the thickness of the flexure beams impacts their deformation considerably. Figure 1b shows the form of a typical arch in its first stable equilibrium position which appears symmetrical and presents a smooth curvature. The thinner flexure beams c10, c15, and c20 seem to have such forms. In contrast, the arches of large calibers (c30 and c40) tend to be irregular and harder to set up. Although all samples are completely straight before being placed in the setup, most of them deform permanently after being compressed. Figure 8 shows the forms of the flexure beams after the cyclic loading experiments. Only c10s maintain their original straight form after the five cycles. While samples c15 and c20 bent slightly at the center, samples c30 and c40 show great deformations, demonstrating that their elastic limit was exceeded during the experiments. Figure 8f visualizes the different degrees of deformation among calibers, where only the c10 flexures remain straight and permanent deformation increases with width.
From the experiment results, it can be concluded that c10 flexures (0.25 mm) in this thickness-length ratio are unstable and not suitable for real-life applications. Even though they do not deform after five loads, they present inconsistent results regarding critical forces and snap-through points that are improbable to avoid since they come from small precision errors typical in physical devices. On their part, c30 and c40 samples that are 0.75 mm and 1.00 mm, respectively, require larger input forces and snap at longer distances, which would cause a demand for more energy input and storage. However, what makes c30 and c40 not convenient is their deformation grade during snapping. The cyclic loading experiments prove that flexure beams of 0.75 mm and 1.0 mm exceed their elastic limit fast and even present partial fractures in the stress points. c20 flexures (0.50 mm) demonstrate that they have mechanical properties that make them suitable for real applications: their deformations are slight enough to maintain their functionality. However, the critical force variations within the same thickness are large, and the distance traveled before snapping is long, which would require more energy input time. Finally, the deformation of c15 samples (0.38 mm) is minor, making it suitable for practical uses. The 24.553 mm linear displacement before snapping is also appropriate since it is close to half of the total length of the slider (45 mm), which indicates that the system would only require half the energy input time to achieve bi-stability.

4. Conclusions

The experimental setup introduced in this paper is shown to be easy to manufacture and implement. Its versatility facilitates the characterization of different samples, improving consistency and efficiency in experimental tests on bi-stable flexure beams. On its part, the use of stripes made of styrene plastic were not found in the literature for this type of application, but the results presented here demonstrate that styrene plastic is a proper material with suitable mechanical properties for bi-stable arches. In addition, from the experiment’s results, it is established that the thickness-to-length ratio is the most relevant feature of the studied topology. It determines the maximum input force required to snap, the snap-through point, and the deformation of the arches, which means that modifying thickness enables tunable mechanical characteristics in styrene flexure beams.
Both the experimental setup and the styrene flexure beams are useful for studying bi-stability and could be relevant for developing proof of concepts of bi-stable mechanisms in many fields. In addition, the topology studied in this paper has the potential to be integrated into functional devices, specifically those that require staying in two operational configurations, such as reconfigurable systems and deployable structures.

Supplementary Materials

The following supporting information can be downloaded at: https://www.mdpi.com/article/10.3390/app122211438/s1, Video S1 (Compilation of performance by thickness experiments) and Video S2 (Compilation of cyclic loading experiments).

Author Contributions

Conceptualization, A.D.-d.-L. and J.L.G.; methodology, A.D.-d.-L., E.C.-U. and A.R-F.; validation, A.D.-d.-L.; investigation, A.D.-d.-L.; writing—original draft, A.D.-d.-L.; visualization, A.D.-d.-L., E.C.-U. and A.R.-F.; supervision, E.C.-U., A.R.-F. and J.L.G.; formal analysis, A.D.-d.-L., E.C.-U. and A.R.-F.; writing—review and editing, E.C.-U., A.R.-F. and J.L.G. All authors have read and agreed to the published version of the manuscript.

Funding

The authors would like to acknowledge the financial support of Tecnologico de Monterrey in the production of this work and to the Consejo Nacional de Ciencia y Tecnologia (CONACYT) for the financial support to ADdL. PhD studies (CVU:342120).

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

Data is contained within the article.

Acknowledgments

Dra. Cristina Chuck (Tecnologico de Monterrey), CONACYT, Metamaterials Lab.

Conflicts of Interest

The authors declare no conflict of interest.

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Figure 1. (a) Typical bi-stable force-displacement graph. (b) Stable and unstable equilibrium positions of the bi-stable mechanism used in this study.
Figure 1. (a) Typical bi-stable force-displacement graph. (b) Stable and unstable equilibrium positions of the bi-stable mechanism used in this study.
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Figure 2. (a) Design and dimensions of the base of the setup. The layout of the flexure beams and components can be seen in thin lines. (b) Schematic of the components of the bi-stable mechanism.
Figure 2. (a) Design and dimensions of the base of the setup. The layout of the flexure beams and components can be seen in thin lines. (b) Schematic of the components of the bi-stable mechanism.
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Figure 3. (a) Enlarged 3D model of the setup on the texture analyzer. (b) Picture of the setup placed on the texture analyzer. All of the experiments were documented using a smartphone.
Figure 3. (a) Enlarged 3D model of the setup on the texture analyzer. (b) Picture of the setup placed on the texture analyzer. All of the experiments were documented using a smartphone.
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Figure 4. (a) Picture of actual components. The anchor is made of acrylic and the flexure beam of styrene plastic. (b) Schematics of anchor components. The width of their slots gets larger according to the caliber of the flexure beam. (c) The flexure beams are press-fit assembled to the anchor and trigger.
Figure 4. (a) Picture of actual components. The anchor is made of acrylic and the flexure beam of styrene plastic. (b) Schematics of anchor components. The width of their slots gets larger according to the caliber of the flexure beam. (c) The flexure beams are press-fit assembled to the anchor and trigger.
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Figure 5. (a–e) Force-Displacement graph per thickness (c10 to c40), which represents the first part of the bi-stable curve. Three samples of each thickness were tested and are plotted in the graphs as a,b, c. The thicker black line plots the mean of the three sets of data. The standard deviation appears in light gray. (f) Force-Displacement graph comparing the mean curves of the five tested thicknesses. As the thickness increases, so do the critical force and the snap-through point.
Figure 5. (a–e) Force-Displacement graph per thickness (c10 to c40), which represents the first part of the bi-stable curve. Three samples of each thickness were tested and are plotted in the graphs as a,b, c. The thicker black line plots the mean of the three sets of data. The standard deviation appears in light gray. (f) Force-Displacement graph comparing the mean curves of the five tested thicknesses. As the thickness increases, so do the critical force and the snap-through point.
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Figure 6. (a) The force-thickness graph exhibits a non-linear relationship (b) The snap-through point-thickness plot demonstrates that thicker flexure beams snap at longer distances. (c) Plot that exhibits the opposite growth of the critical force and snap-through point curves as a function of thickness.
Figure 6. (a) The force-thickness graph exhibits a non-linear relationship (b) The snap-through point-thickness plot demonstrates that thicker flexure beams snap at longer distances. (c) Plot that exhibits the opposite growth of the critical force and snap-through point curves as a function of thickness.
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Figure 7. (a) The c10 flexure beam presents variations in the CF due to the equipment capabilities and affectations of the manual assembly (b–e) The plots show a decrement in the critical force per load.
Figure 7. (a) The c10 flexure beam presents variations in the CF due to the equipment capabilities and affectations of the manual assembly (b–e) The plots show a decrement in the critical force per load.
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Figure 8. Image of the left side flexure beams after being subjected to compression five times. (a) c10 flexure beams (0.25 mm) (b) c15 flexure beams (0.38 mm) (c) c20 flexure beams (0.50 mm) (d) c30 flexure (0.75 mm) (e) c40 flexure beams (1.00 mm) (f) Compilation of samples of each thickness.
Figure 8. Image of the left side flexure beams after being subjected to compression five times. (a) c10 flexure beams (0.25 mm) (b) c15 flexure beams (0.38 mm) (c) c20 flexure beams (0.50 mm) (d) c30 flexure (0.75 mm) (e) c40 flexure beams (1.00 mm) (f) Compilation of samples of each thickness.
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Figure 9. (a–d) Snap-through points in relation to the load cycle graphs. The straightness of the curves suggests that the mechanisms travel a similar linear distance before snapping-through, regardless of the load number.
Figure 9. (a–d) Snap-through points in relation to the load cycle graphs. The straightness of the curves suggests that the mechanisms travel a similar linear distance before snapping-through, regardless of the load number.
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Figure 10. Image of samples used for the “Performance by thickness” experiments, at their snap-through points. (a) Samples of c10 flexure beams (0.25 mm) (b) Samples of c15 flexure beams (0.38 mm) (c) Samples of c20 flexure beams (0.50 mm) (d) Samples of c30 flexure beams (0.50 mm) (e) Samples of c40 flexure beams (1.00 mm).
Figure 10. Image of samples used for the “Performance by thickness” experiments, at their snap-through points. (a) Samples of c10 flexure beams (0.25 mm) (b) Samples of c15 flexure beams (0.38 mm) (c) Samples of c20 flexure beams (0.50 mm) (d) Samples of c30 flexure beams (0.50 mm) (e) Samples of c40 flexure beams (1.00 mm).
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Table
Table 1. Nomenclature, caliber, and thickness of the five styrene plastic stripes used in this study for making the flexure beams.
Table 1. Nomenclature, caliber, and thickness of the five styrene plastic stripes used in this study for making the flexure beams.
NomenclatureCaliberThickness
C10Caliber 100.25 mm
C15Caliber 150.38 mm
C20Caliber 200.50 mm
C30Caliber 300.75 mm
C40Caliber 401.00 mm
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Diaz-de-Leon, A.; Cuan-Urquizo, E.; Roman-Flores, A.; Gordillo, J.L. Bi-Stability in Flexure Beams: Introducing a Setup for Experimental Characterization. Appl. Sci. 2022, 12, 11438. https://doi.org/10.3390/app122211438

AMA Style

Diaz-de-Leon A, Cuan-Urquizo E, Roman-Flores A, Gordillo JL. Bi-Stability in Flexure Beams: Introducing a Setup for Experimental Characterization. Applied Sciences. 2022; 12(22):11438. https://doi.org/10.3390/app122211438

Chicago/Turabian Style

Diaz-de-Leon, Alejandra, Enrique Cuan-Urquizo, Armando Roman-Flores, and J. L. Gordillo. 2022. "Bi-Stability in Flexure Beams: Introducing a Setup for Experimental Characterization" Applied Sciences 12, no. 22: 11438. https://doi.org/10.3390/app122211438

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