This paper presents a new bistable collapsible compliant mechanism (BCCM) that is utilized in a lamina-emergent frustum. The mechanism is based on transforming a polygon spiral into spatial frustum shape using a mechanism composed of compliant links and joints that exhibits a bistable behavior. A number of mechanism types (graphs) were considered to implement the shape-morphing spiral, including 4-bar, 6-bar, and 8-bar chains. Our design requirements permitted the selection of a particular 8-bar chain as the basis for the BCCM. The bistable behavior was added to the mechanism by introducing a snap-through bistability as the mechanism morphs. A parametric CAD was used to perform the dimensional synthesis. The design was successfully prototyped. We anticipate that the mechanism may be useful in commercial small animal enclosures or as a frame for a solar still.

## Introduction

The objective of this study is to develop a design procedure for a shape-changing structure that morphs from a planar lamina to a frustum shape as an illustration of a general design strategy for shape-changing structures. The motivation for shape-changing structures occurs in many applications such as wing morphing for enhanced aerodynamic performance, complex deployable structures, and space-saving furniture [1,2]. Additionally, if shape-morphing designs have the ability to be manufactured on the microscale, they may provide useful functions, such as switches and relays [3]. Currently, shape-morphing structures often consist of a number of parts or mechanisms that may consist of links, springs, and joints, which can have high costs for manufacture, assembly, and maintenance. Compliant mechanisms offer advantages in these areas and can also be bistable [4,5]. Bistable compliant mechanisms have been used in many applications including a bistable mechanism for the rear trunk lid of a car, double toggle switching mechanisms, and multistable compliant Sarrus mechanisms [68].

In this paper, we describe the design of a bistable collapsible compliant mechanism (BCCM) that, when arrayed transforms from a planar lamina shape to a frustum shape and requires a small wrench, i.e., a torque with a parallel pulling or compressing force to switch between the frustum and lamina states. Compliant structures fabricated from a lamina (planar) sheet which have a motion out of the planar sheet are known as lamina-emergent mechanisms [9,10]. We are unaware of any other deployable mechanism with comparable motion to the lamina-emergent frustum described in this work. The original motivation for the design was a collapsible solar still. The solar still would be stored collapsed in a vehicle and assembled to purify water in an emergency. Other possible applications for this design could be a rapidly deployable tent or a collapsible animal enclosure or a bird cage.

### Background.

The majority of the background for this work is in the area of compliant mechanisms and its subdisciplines of bistability and shape-morphing. The advantages of compliant mechanisms over rigid-link mechanisms are considered in four important areas: the reduction of number of parts, the elimination of joint clearances, the integrated spring-like energy storage, and the potential reductions in cost [11]. Howell et al. developed the pseudo-rigid-body model concept to model the large deflection of compliant links by using rigid-body component analogies [12]. Therefore, a compliant mechanism can be analyzed using the rigid-link mechanism theory, especially, when the joints and pivots of the compliant mechanism are short and thin so that its bending stiffness is negligible [4]. Additionally, we found work on the type synthesis of mechanisms using a graph theory to be helpful. Shape-changing mechanisms have been successfully studied using the topological graph theory to synthesize morphing mechanisms [13,14]. A Graph theoretic approach was used to generate a creative design for a vehicle suspension by Liu and Chou [15]. Moreover, Feng and Liu used graph theory representations to evaluate lists of mechanisms for a deployable mechanism used as a reflector antenna [16]. The structure of the mechanism's connections can be defined by the graph representation [17].

The bistability of a mechanism means that it can be at a stable equilibrium in two different configurations [4,18,19]. These two configurations are local minima of the potential energy. The best way to understand bistability is by using the “ball-on-a-hill” analogy [4], which compares the strain energy in a compliant mechanism to the gravitational potential energy of a ball. In Fig. 1, a ball on an uneven surface is depicted. The ball is at equilibrium at positions A, B, C, and D. At positions A and C, the ball is at a minimum potential energy and will oscillate about that minimum if it is perturbed slightly from the positions shown.

On the other hand, at position B, the ball is at a maximum of potential energy, and if disturbed, it will not return to its original position but will move to one of the stable positions A or B. The equilibrium at D is described as neutral, because the potential energy curved is flat in the neighborhood of D.

The mechanism shown in Fig. 2 is a type of bistable mechanism called a snap-through mechanism. Snap-through mechanisms exhibit nonlinear potential energy curves and are bistable [20,21]. The snap-through mechanism in Fig. 2 consists of two (possibly nonlinear) springs (K1,K2) that are connected in series by revolute joints to ground points O1 and O2. In order to deduce the stable configurations of this mechanism, it is sufficient to know the distance between the ground pivots and the equilibrium lengths of the springs. If these three lengths satisfy the triangle inequality (i.e., the sum of any two of the lengths is greater than the length of the third), the mechanism will be bistable regardless of the stiffness characteristics of the springs. If the springs are sufficiently flexible, there will be a (possibly many) deflection path from the first equilibrium configuration to the second equilibrium configuration. The two springs (K1,K2) may be complicated functions yet the exact nature of their stiffness and motion does not have to be known to determine the second stable configuration, because that configuration is inherent in the symmetry which the second equilibrium configuration shares with the first. The importance of this snap-through mechanism is that it demonstrates that the snap-through bistability is a geometric condition and such mechanisms can be designed without motion, force, or stress analyses. To experimentally verify the bistability, it is sufficient to show that a mechanism has two configurations which it will hold while being perturbed (i.e., being shaken, turned upside down, etc.)

Alqasimi et al. demonstrated a shape-morphing space frame using a linear bistable compliant crank-slider mechanism [22] that is arranged in a specific pattern to produce a shape-changing structure [23]. Bistable shape-shifting surfaces can produce morphing structures with a line-of-sight integrity, i.e., effectiveness as a physical barrier [24]. Shape-changing structures and surfaces can add value to applications as in collapsible structures and folding geometries [2528].

### Overview.

The rest of the paper is organized as follows: first, we discuss the two configurations (lamina and frustum) and our strategy for morphing. Then, we discuss the type synthesis of a linkage to coordinate the motion, then the dimension synthesis of the mechanism and how to make it bistable. Finally, we show the prototype and results.

## Collapsible Compliant Mechanism

The frustum shape is decomposed into a number of submechanisms. In order to compress the frustum into its planar position, the straight sides of the frustum are bent into polygonal spirals, which allow these lengths to be effectively placed, as shown in Fig. 3. Here, a polygon spiral is used for the design, because its straight segments are compatible with rigid links in a mechanism. In this section, we discuss the design of the exterior of the polygonal spiral; in Sec. 4, we describe its interior, which is a mechanism for coordinating the bending of straightening in the spiral.

As shown in Fig. 4, the polygon spiral is designed in a pattern which can be adjusted and modified with a constant ratio, as
$ai+1ai=ri+1ri=r1R=cos(2πn)$
(1)

where R is the radius of the base, ri is the distance from the center of the circle to the ith spiral corner from the outer radius, and n is the number of sectors. The polygonal spiral has k segments.

The length of the spiral can be calculated as the summation of all side lengths (ai)
$L=a1+a2+⋯+ak$
(2)

$L=a1*(∑i=0k−1 cosi(2πn))i=0,1,2,3,….k$
(3)

$a1=R* sin(2πn)$
(4)

We have chosen the polygonal spiral terminate at 180 deg from its origin on the outer radius. This means that each spiral passes through half the sectors n, so the number of segments, k, is equal to n/2. The length of the spiral allows the calculation of the height of the frustum shape using the Pythagorean Theorem as shown in Fig. 5.

$H2=L2−(R−Rt)2$
(5)
where H is the height of the frustum, and t is the radius of the top frustum surface. The top radius, Rt, can be calculated by iterating Eq. (1) as
$RtR=(r1R)n2=cosn2(2πn)$
(6)
This ratio shows that the area of the top frustum surface is controlled by n, where the increase of n increases the radius, Rt. Solving for H in Eq. (5) using Eqs. (3), (4), and (6)
$H=R*{[sin(2πn)*∑i=0n2−1 cosi(2πn)]2−[1−cosn2(2πn)]2}12$
(7)

$β=tan−1(R−RtH)$
(8)
It can be clearly seen that the height equation depends on the number of sectors and the outer radius and the increase of sectors increases the height. However, the increase of sectors tends to increase the number of segments per spiral which requires a more complex interior design. Moreover, there is no large difference in the height of the frustum shape, because the ratio of the spiral, Eq. (1), is diminished with the increase in the number of sectors
$%height improvement=Hi+1−HiHi*100$
(9)
Table 1 shows the parameters needed for the frustum design. It can be seen that the percentage change in height diminishes with increasing sectors. Besides, as sectors increase, the sector width b, reduces as shown in Fig. 6.
$b=2R* sin(πn)$
(10)

As the sector width, b, becomes smaller, the design of the shape changing interior mechanism becomes more challenging.

## Design Criteria

The interior of the polygonal spiral is desired to be a one degree-of-freedom (1DOF) compliant mechanism, i.e., its pseudo-rigid-body model is a 1DoF mechanism, which can fit the two configurations. Every sector is designed as an independent mechanism and then coupled at the inner and outer circles (the top and the bottom of the frustum). Because the design is symmetric, the motion of the left side of each sector must correspond to its right side, i.e., the sides have equal lengths and rotations where adjacent sides in adjacent sectors are connected with hinges. The normal to the sectors are all parallel and vertical in the planar position, while in the frustum position, they bend to an angle $(2π/n)* tan β$ with respect to the adjacent sector. Our design has k = 4 segments, and without internal links, its minimum design is a 10-bar mechanism (see the dotted lines in Fig. 7). There are 230 possible 10-bar mechanism types [11] with 1DoF, which seemed like a poor place to begin the type synthesis. So, we considered subdividing the sector and coupling the resulting mechanisms.

The polygonal spiral has a glide-translational scaling symmetry, as shown in Fig. 8(b), where the dart-shaped quadrilateral labeled (1) has glide-translational scaling symmetry with quadrilateral (2), which has the same symmetry with (3), etc. As a result, each quadrilateral of the design (1, 2, 3, 4, etc.) can be designed independently. However, because each such design should have at least one-degree-of-freedom, having four independently designed quadrilaterals results in a minimum of four degrees-of-freedom per sector. Fortunately, a similar scaling applies in the frustum position (Fig. 8(a)), even though the glide-translation is different. To limit the degrees-of-freedom in each sector, we chose designs which couple the motion of the quadrilateral as shown in Fig. 8. We looked at 6-bar and 8-bar designs which couple two quadrilaterals with the objective of learning how to couple all four quadrilaterals together.

Using a 6-bar design as shown in Fig. 9(a), the coupling between sectors subdivisions is too rigid and the distance between the left and right sides does not compress. Figure 9(a) shows that the left and right sides rotate with respect to each other but are forced to maintain a constant radial distance. The 8-bar design shown in Fig. 9(b) is a more effective coupling, because it allows relative translation and enhanced collapsing of the sector sides. This shows a variety of possible motions of the mechanism and that it can have different radii circle between the bases of the quaternary links.

For planar mechanism, the following Kutzbach's equation is used to compute the mobility [11,29]
$M=3*(g−1)−2J1−J2$
(11)

where M is the degree-of-freedom, g is the number of links, J1 is the number of lower pairs, and J2 is the number of higher pairs.

Because the kinematic chain is designed for a compliant mechanism, all joints are modeled as revolute joints or lower pairs and there will not be any higher pairs. Substituting M = 1, g = 8, J2 = 0 into Eq. (11) gives J1 = 10. Based on this calculation, the 8-bar mechanism should have ten lower pairs for 1DoF. Many types of links can produce a mechanism but to reduce the complexity only binary (B, order = 2), ternary (T, order = 3), and quaternary (Q, order = 4) links and no multiple joints are considered. The total number of links can be [11]
$B+T+Q=g=8$
(12)

$2B+3T+4Q2=J1=10$
(13)

And the number of joints is found by Norton [11].

Thus, the maximum number of quaternary links will be two [30]. Possible 8-bar mechanisms are presented in three categories for Q = 0, 1, and 2 as shown in Fig. 10.

According to Norton, there are 16 types of 1DoF mechanisms, which are categorized as in Fig. 11 [11]. The kinematic chains in Fig. 11 are evaluated based on the following criteria:

1. (1)

The links in the outer loops should be an even number. This provides parallel links which are necessary for the spiral design (Sec. 1). This rules out designs c, g, h, and i.

2. (2)

The number of ternary and quaternary links has to be even to ensure the symmetry of the mechanism. Otherwise, it will increase the complexity of the design, which rules out all of category 2 designs j, k, l, m, and n.

3. (3)

The mechanism has to have a symmetric outer loop, which rules out designs e, f, and h.

4. (4)

The mechanism loops must be collapsible, thus, the ternary and quaternary links cannot be connected directly. Otherwise, this design will have a limited range of motion, as described for a six-bar in Fig. 11. This rules out designs b, d, and p. All sixteen mechanism configurations are evaluated based on these criteria. Only two mechanisms pass, which are shown in Fig. 12.

## Graph Theory

In this section, we used a graph theory to assist in the type synthesis of the interior of the polygonal spiral. The graphs of the two candidate mechanisms in Fig. 12 are shown in Fig. 13. In Fig. 13, links in the mechanism become vertexes in the graph and joints become edges [15,16].

Graph theory shows a symmetry in the connection between the links which allow us to extend the mechanism as in Fig. 14(a). This extension of the mechanism by adding similar segments is analogous to the process of polymerization in chemistry, in which single units (monomers) are combined to useful longer chains (polymers). Such polymerized mechanisms that do not change their mobility are useful as deployable mechanisms, such as those used in the aerospace industry. Polymerization provides a mechanism design that can increase the number of segments, and each segment can be scaled with respect to the previous one.

Still, in Fig. 14, we see that the polymerization of a mechanism may cause an increase in the mobility. By transferring the graph back to the mechanism state as in Fig.14(b) (mechanism 2), the result of Eq. (11) for (L = 14, J1 = 18, J2 =0) gives M = 3DoF. Thus, it cannot be used as a repeating pattern. On the other hand, mechanism 1 does not increase its mobility and passes all design criteria and is the best design for the collapsible compliant mechanism.

In designing the mechanism, a parametric CAD program is used to achieve the design goal in less time. Parametric CAD can provide clear visualization of the design approach. It allows kinematic chains and structure properties such as displacement to be straight analyzed and it allows design constraints to be specified, and link lengths are automatically adjusted to meet those specifications. In this study, the two positions of mechanism 1 are the main constraints of the design. Thus, we identify the initial and the final configurations of the design for synthesis of mechanism 1 using parametric CAD [31].

## Bistable Mechanism

The snap-through concept is used in the lamina-frustum mechanism to achieve the bistability. Figure 16 shows how the previous concept is applied on the lamina-frustum mechanism, where the nonlinear springs are the compliant links of the mechanism that accommodate the motion between the two stable configurations though their elastic behavior. Therefore, if a force is applied to the mechanism to any intermediate position, the mechanism will snap to one of the stable configuration.

### Design Prototype.

The design was prototyped using a laser cutting machine; the number of sectors, n, was chosen to be n = 8 due to limitations in cutting small sizes. Our design was chosen to have a base radius, R = 150 mm. All parameters of the design were derived using the relations given in Sec. 1 and using the parametric CAD and are shown in Table 2. These parameters can be used to represent the first and second positions of a sector, and all the primed (‘,”) links lengths are found by multiplying the unprimed values by the spiral ratio, once for the single primes, twice for the double primes as labeled in Fig. 17. In Fig. 17, the 8-bar mechanism from Sec. 3 is polymerized by a repeating pattern to satisfy the design parameters. Because the n = 8, the number of repeating patterns (P) can be calculated
$P=n2−1$
(14)

Initial design parameters are found to be as in Table 2 as follows:

• Number of sectors (n) = 8.

• Number of repeating patterns (P) = 3.

• Initial cylinder radius (R) = 150 mm.

Each sector creates an independent BCCM, and the assembly of eight sectors forms the frustum shape. The BCCMs were laser cut from a 1/8 in thick Polypropylene copolymer material (Fig. 18) in the position shown in Fig. 17(b). Each element is attached to the base by a torsion bar and two pins as shown in Fig. 19. The torsion bars are part of the ground link (l1G) and act as a hinge in the transition from the planar to the frustum position [32]. The top is designed to connect all sectors at link (l4). The top has enough flexibility, to accommodate the relative rotation of the sectors as the mechanism moves to the frustum position. These features, the base and the top, do not affect the mechanism's bistability. The top is made of polypropylene material laser cut from a 1/16 in thick sheet and the connections to each other

are flexible, so that each end can bend and twist independently, as shown in Fig. 20.

Figure 21 shows the lamina-emergent frustum in both stable positions. All eight BCCMs were attached to base and to top. A compressive force and 180 deg rotation were applied in the top to transition to the planar position from the frustum.

## Results and Discussion

This section will compare the results between the mathematical model and the prototype. The mathematical model predicts the height of the frustum, and the prototype height was measured experimentally. The prototyped BCCMs were fastened together to coordinate their out-of-plane movement and prevent the interference of the elements. The prototype loses a little height due to gravity causing the links to sag, additionally the polypropylene flexures tend to creep and exhibit some plastic deformation. Thus, the comparison between the two methods is done using the percentage error of the relative change between the height values as shown in Table 3. In this prototype, we verified the bistability experimentally by shaking the mechanism in its two stable positions, but did not optimize for fatigue life. Further, in our design of the mechanism, we tacitly assumed that each sector, which are planes in the stable configurations (see Figs. 17 and 18) would remain planes during their transition. This turned out not to be the case, resulting in torsional deformations in the small-length flexural pivots in the mechanism. This may have consequences for the fatigue life of the mechanism, though repeated experiments to date have not resulted in fatigue failures.

## Closure

This paper has presented a new bistable collapsible compliant mechanism (BCCM). Graph theory and kinematic chain design strategies were described. The collapsible bistable compliant mechanism of frustum-spiral planar shape was modeled geometrically and prototyped as a proof of concept.

## Funding Data

• National Science Foundation (CMMI-1053956).

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