Deployable linear and spiral array structures based on a Kresling-inspired mechanism with integrated scissor arms

Abstract

Recent developments have shown that spatial structures devised from origami or low-dimensional rigid linkage mechanisms can be used to construct deployable arrays for antennas or satellites. Yet, some of these structures are limited to deployment in fixed planes or directions, or do not define straightforward processes for deployment. To surmount these limitations, this research introduces a reconfigurable single-degree-of-freedom spatial structure devised from a Kresling-inspired mechanism with integrated scissor arms. Analytical models are constructed to demonstrate compaction, deployment, and acoustic wave guiding capabilities of the proposed, modular structure. The influences of the geometric parameters on compaction, deployment, and scissor arm orientation are also explored, and reveal modular scissor arm behavior and large deployment-to-compaction area ratios. The acoustic wave guiding capabilities of the Kresling-inspired scissor structure are exemplified via a structure using spiral scissor arms, thereby proposing a novel concept for the construction of deployable wave guiding arrays. Experimental studies with model arrays complement the analytical findings of both the geometric reconfigurations and wave guiding functionality. Finally, out-of-plane configurations are depicted to demonstrate the three-dimensional shape change capabilities of the Kresling-inspired scissor structure. The results in this study encourage broader exploration of the interfaces between origami inspired structures and rigid linkage mechanisms.

Keywords

Origami-inspired mechanisms reconfigurable structures acoustic arrays deployable arrays deployable antennas

1. Introduction

Spatial structures inspired from origami have broad applications in the fields of science and engineering because they are complex three-dimensional (3D) structures that are highly portable, reconfigurable, and deployable. These applications include solar arrays (Cai et al., 2021; Karmakar and Mishra, 2022; Li et al., 2022), robotic devices (Chen et al., 2022; Fonseca and Savi, 2020; Robertson et al., 2021; Yang et al., 2021), antennas (Georgakopoulos et al., 2021; Ha et al., 2022; Huang et al., 2022; Zhang et al., 2020), medical devices (Kim et al., 2021, 2022; Li et al., 2019; Zhao et al., 2022), and sensor technology to monitor pollution (Matthew et al., 2022).

In addition, researchers have introduced origami-inspired spatial structures to the field of acoustic wave guiding. Traditionally, wave guiding systems have been constructed with digital signal processing (DSP) techniques to emulate physical positioning of the transducers (Bai et al., 2013; Balanis, 2016; Johnson and Dudgeon, 1993). Yet, DSP techniques can be computationally expensive due to limitations with real-time control adaptation and computational stability (Johnson and Dudgeon, 1993; Oppenheim et al., 1999). Alternatively, researchers have explored arrays that utilize nonuniformly-spaced transducers instead of DSP techniques to achieve optimal wave guiding (Prime and Doolan, 2013; Rafaely, 2005; Wang and Ratilal, 2017). Yet, these arrays have portability challenges due to their size and construction, nor are they reconfigurable to optimize transducer placement for varying frequencies. Recently, Zou and Harne (2017) have shown that tessellated star origami functions as a reconfigurable and portable wave guiding array where the tessellated star facets are assumed to be vibrating baffled pistons that radiate acoustic pressure.

To advance research efforts for origami-inspired acoustic wave guiding arrays, Srinivas and Harne (2020) have employed flasher-based origami to create reconfigurable arrays with acoustic wave guiding characteristics similar to those found in rigid, canonical spiral arrays studied by Prime and Doolan (2013). Alternatively, researchers have used origami-inspired spatial structures to create adaptive radio-frequency (RF) antennas. For instance, Alharbi et al. (2018) have constructed a dipole RF antenna from accordion-based origami. Liu et al. (2019) and Zhang et al. (2022) have also utilized Kresling origami to construct RF antennas that can be compacted for storage and transportation, as well as be reconfigured to optimize RF performance. While these structures demonstrate the efficacy of wave guiding arrays and RF antennas inspired from origami, their deployment configurations are limited to a single plane or path governed by origami tessellations that cannot create multi-planar and multi-directional configurations. In addition, rigid body linkages are not considered or implemented to facilitate deployment and compaction.

Spatial structures derived from rigid linkages have also been studied for low-dimensional deployable arrays. For example, Cao and Cheng (2022) devise a deployable mechanism from a composite of Sarrus linkages for synthetic aperture radar (SAR) satellites. Zijie et al. (2022) propose a triangular prism-shaped parallel mechanism to deploy a planar space antenna. Cao et al. (2021) design a double-ring truss deployable satellite mechanism antenna that uses a new rectangular pyramid linkage unit with a large deployment-to-compaction ratio. Tian et al. (2022) study a space deployable antenna with large aperture and a large deployment-to-compaction ratio using multifold rib unit mechanisms. While these examples demonstrate deployable structures that are modular with single-degree-of-freedom behavior for satellite antennas, they do not propose structural configurations advantageous for acoustic wave guiding.

Motivated by the shortcomings in the state-of-the-art, the goal of this research is to study a deployable spatial structure devised by a design method that integrates origami-inspired structures with rigid-link mechanisms to create reconfigurable structures with large shape changes. This design method is demonstrated by the use of a Kresling-inspired mechanism with integrated scissor arms to create the proposed spatial structure. The proposed spatial structure has large deployment-to-compaction ratios, can create deployment configurations that are multi-directional and multi-planar, and can be configured into rotationally symmetric linear and spiral scissor arm array designs necessary to tailor the propagation of acoustic waves. The proposed spatial structure is also designed to reconfigure with single-degree-of-freedom motion.

The Kresling origami pattern is used to create thin-walled structures that undergo large uniaxial transformations as they are twisted (Jianguo et al., 2016; Kidambi and Wang, 2020). This transformation is advantageous for spatial structures with large shape changes because multiple layers of Kresling origami can be stacked in the uniaxial direction, which increases the magnitude of uniaxial transformation. In addition, the outer radius of Kresling origami remains constant during transformation, which facilitates the integration of rigid mechanisms that are actuated in directions alternate to the uniaxial direction of the Kresling origami. The illustrations in Figure 1(a) show that the Kresling-inspired mechanism originates from Kresling origami by a conversion of the creases into rigid links. Figure 1(a) also shows the links that correspond to valley creases are removed from the mechanism because they are redundant to the links that correspond to the mountain creases. The Kresling-inspired mechanism is also designed to transform similar to the Kresling origami such that it compacts as the angle of twist $δ$ increases between the top and bottom layers, as it is shown in Figure 1(a). Furthermore, the Kresling-inspired mechanism is designed to reconfigure with single-degree-of-freedom motion because the top and bottom layers are constrained to remain parallel to each other, which means the angle of twist $δ$ becomes the only required input for reconfiguration.

Scissor arms are rigid body mechanisms that are extensible, can undergo large shape transformations, and are commonly used in the design of deployable spatial structures (Dinevari et al., 2021; Sarisayin et al., 2022). These inherent properties indicate that scissor arms are favorable candidates for integration with the Kresling-inspired mechanism to create a spatial structure with large shape transformation in multiple directions. Four-bar linkages are used as interfaces between the Kresling-inspired mechanism and scissor arm arrays so that both can deploy and compact simultaneously due to changes in the angle of twist $δ$ . The illustrations in Figures 1(b) depict the geometric parameters used to design the Kresling-inspired mechanism, the four-bar linkages, and the scissor arm arrays. The illustration in Figure 1(c) presents the geometric parameters used to define the spacing and attachment of the scissor arm arrays and four-bar linkages around the Kresling-inspired mechanism.

Two types of scissor arm array configurations are studied in this research for the proposed spatial structure. The first type is the linear scissor arm array configuration, as it is shown in Figure 2(a), which has inherent large deployment-to-compaction ratio capabilities. The second type is the spiral scissor arm array configuration, as it is also shown in Figure 2(a), and it is advantageous for acoustic wave guiding because of the non-uniform transducer spacing. The single-degree-of-freedom deployment and compaction of the proposed structure for both types of scissor arm array configurations are exemplified in Figure 2(b), where the angle of twist $δ$ from the Kresling-inspired mechanism facilitates the reconfiguration.

This research employs analytical and experimental efforts to explore the effects of the Kresling-inspired mechanism and scissor arm integration on the deployed and compacted configurations of the array. Once a mechanical model is constructed, we augment this analytical tool to study acoustic wave guiding behavior from such deployable arrays. Section 2 presents the analytical models used to examine compaction, deployment, and acoustic wave guiding behaviors of the proposed structure. Section 2 also illustrates the influences of geometric parameters on compaction, deployment, and scissor arm plane orientation. Section 3 provides case studies for the linear and spiral scissor arm array configurations with experimental models. Section 4 communicates the experimental efforts used to validate acoustic wave guiding observations from the analytical model. Section 4 also communicates the analytical efforts used to examine the influence of structural reconfiguration on acoustic wave guiding properties. Section 5 demonstrates the modular out-of-plane behavior of the Kresling-inspired scissor structure via examples that use linear and spiral scissor arm array configurations. Section 6 concludes the report with a summary of the new findings and closing remarks.

2. Model formulations

In this section, the geometric model for the deployable Kresling-inspired scissor structure is developed based on an integration of a Kresling-inspired mechanism and scissor arms with a four-bar linkage as the interface. Far field acoustic radiation is then analytically computed with Rayleigh’s integral for point sources. Parameters from the geometric model are utilized to demonstrate their influences on compaction, deployment, and scissor arm plane orientation.

2.1. Geometric modeling

Figure 1(b) shows the geometry of the Kresling-inspired mechanism. The Kresling-inspired mechanism is defined by $m$ sides, a radius $r_{0}$ , and $m$ diagonal links with a length $a_{0}$ . An angle of twist $δ$ is rotated in the clockwise direction and is the angle of the Kresling-inspired mechanism top layer with respect to the Kresling-inspired mechanism bottom layer. The Kresling-inspired mechanism height $h$ is defined by equation (1):

h = \sqrt{{a_{0}}^{2} + 2 {r_{0}}^{2} (\cos δ - 1)}

(1)

The coordinates for the nodes on the mechanism are defined by equations (2) and (3) for the bottom $A$ and top $B$ layers, respectively:

(\begin{array}{l} A_{i_{x}} \\ A_{i_{y}} \\ A_{i_{z}} \end{array}) = (\begin{array}{l} r_{0} \cos \frac{2 π i}{m} \\ r_{0} \sin \frac{2 π i}{m} \\ 0 \end{array}), \forall i \in {1 \dots m}

(2)

(\begin{array}{l} B_{i x} \\ B_{i y} \\ B_{i z} \end{array}) = (\begin{array}{l} r_{0} \cos (δ - \frac{2 π i}{m}) \\ - r_{0} \sin (δ - \frac{2 π i}{m}) \\ h \end{array}), \forall i \in {1 \dots m}

(3)

Figure 1(b) also shows the geometry of the four-bar linkage used as an interface. The four-bar link lengths are defined as $l_{1}$ and $l_{2}$ . Dimension $d$ represents the distance between nodes 1 and 3 on the four-bar linkage, which also coincide with the top and bottom nodes of the Kresling-inspired mechanism, respectively. Dimension $q$ represents the distance between nodes 2 and 4 on the four-bar linkage. The four-bar linkage plane has been defined to rotate by $δ / 2$ , and is represented by its normal unit vector $v'$ in equation (4):

(\begin{array}{l} v_{x}^{'} \\ v_{y}^{'} \\ v_{z}^{'} \end{array}) = (\begin{matrix} \cos (- δ / 2) & - \sin (- δ / 2) & 0 \\ \sin (- δ / 2) & \cos (- δ / 2) & 0 \\ 0 & 0 & 1 \end{matrix}) (\begin{array}{l} v_{x} \\ v_{y} \\ v_{z} \end{array})

(4)

where $v$ is the normal unit vector of the four-bar linkage plane when $δ = 0$ . The coordinates for nodes 2 and 4 on the four-bar linkage are determined by solving equations (5) to (7) and equations (8) to (10), respectively.

{l_{1}}^{2} = {(n_{2 x} - n_{1 x})}^{2} + {(n_{2 y} - n_{1 y})}^{2} + {(n_{2 z} - n_{1 z})}^{2}

(5)

{l_{2}}^{2} = {(n_{2 x} - n_{3 x})}^{2} + {(n_{2 y} - n_{3 y})}^{2} + {(n_{2 z} - n_{3 z})}^{2}

(6)

\begin{array}{l} {l_{2}}^{2} + {| v^{'} |}^{2} = {((v_{x^{'}} + n_{3 x}) - n_{2 x})}^{2} + {((v_{y^{'}} + n_{3 y}) - n_{2 y})}^{2} \\ + {((v_{z^{'}} + n_{3 z}) - n_{2 z})}^{2} \end{array}

(7)

{l_{1}}^{2} = (n_{4 x} - n_{3 x})^{2} + (n_{4 y} - n_{3 y})^{2} + (n_{4 z} - n_{3 z})^{2}

(8)

{l_{2}}^{2} = (n_{4 x} - n_{1 x})^{2} + (n_{4 y} - n_{1 y})^{2} + (n_{4 z} - n_{1 z})^{2}

(9)

\begin{matrix} {l_{1}}^{2} + {| v' |}^{2} = ((v_{x}' + n_{3 x}) - n_{4 x})^{2} + ((v_{y}' + n_{3 y}) - n_{4 y})^{2} \\ + ((v_{z}' + n_{3 z}) - n_{4 z})^{2} \end{matrix}

(10)

Figure 1(b) also presents the geometric parameters that define the scissor arm array, which are the scissor arm length $l_{s}$ and the intersection position fraction $α$ . It is important to recognize that the scissor arm array is linearly shaped when $α =$ 0.5, and spirally shaped when $α \neq$ 0.5, as it is shown in Figure 2(a). The number of scissor arm units that can be on each scissor arm array is defined by $n_{Rad}$ , as it is shown in Figure 2(a). Nodes 2 and 4 on the four-bar linkage are also used to define the first two nodes on the scissor arm unit that intersect with the four-bar linkage. The remaining nodes that also define the first scissor arm unit are nodes 5, 6, and 7, as it is shown in Figure 1(b). The coordinates for node 5 on the first scissor arm unit are determined by solving equations (11) to (13):

((1 - α) l_{s})^{2} = (n_{5 x} - n_{2 x})^{2} + (n_{5 y} - n_{2 y})^{2} + (n_{5 z} - n_{2 z})^{2}

(11)

(α l_{s})^{2} = (n_{5 x} - n_{4 x})^{2} + (n_{5 y} - n_{4 y})^{2} + (n_{5 z} - n_{4 z})^{2}

(12)

\begin{matrix} ((1 - α) l_{s})^{2} + {| v_{s} |}^{2} = (n_{5 x} - (v_{sx} + n_{2 x}))^{2} \\ + (n_{5 y} - (v_{sy} + n_{2 y}))^{2} + (n_{5 z} - (v_{sz} + n_{2 z}))^{2} \end{matrix}

(13)

where $v_{s}$ is the normal unit vector of the scissor arm plane. The coordinates for nodes 6 and 7 are then defined by equations (14) and (15), respectively:

\begin{matrix} (\begin{matrix} n_{6 x} \\ n_{6 y} \\ n_{6 z} \end{matrix}) = l_{s} (\begin{matrix} \frac{n_{5 x} - n_{4 x}}{| v_{54} |} + n_{4 x} \\ \frac{n_{5 y} - n_{4 y}}{| v_{54} |} + n_{4 y} \\ \frac{n_{5 z} - n_{4 z}}{| v_{54} |} + n_{4 z} \end{matrix}) \end{matrix}

(14)

\begin{matrix} (\begin{matrix} n_{7 x} \\ n_{7 y} \\ n_{7 z} \end{matrix}) = l_{s} (\begin{matrix} \frac{n_{5 x} - n_{2 x}}{| v_{52} |} + n_{2 x} \\ \frac{n_{5 y} - n_{2 y}}{| v_{52} |} + n_{2 y} \\ \frac{n_{5 z} - n_{2 z}}{| v_{52} |} + n_{2 z} \end{matrix}) \end{matrix}

(15)

where $v_{52}$ and $v_{54}$ are vectors from nodes 2 and 4 on the four-bar linkage to node 5 on the scissor arm unit, respectively. The coordinates for the remaining nodes on the array with $n_{Rad}$ scissor arm units are determined by solving equations (16) to (20):

\begin{matrix} ((1 - α) l_{s})^{2} = (n_{(i + 2) x} - n_{ix})^{2} + (n_{(i + 2) y} - n_{iy})^{2} \\ + (n_{(i + 2) z} - n_{iz})^{2} \end{matrix}

(16)

\begin{matrix} (α l_{s})^{2} = (n_{(i + 2) x} - n_{(i + 1) x})^{2} + (n_{(i + 2) y} - n_{(i + 1) y})^{2} \\ + (n_{(i + 2) z} - n_{(i + 1) z})^{2} \end{matrix}

(17)

\begin{matrix} ((1 - α) l_{s})^{2} = (n_{(i + 2) x} - (v_{sx} + n_{ix}))^{2} + (n_{(i + 2) y} - (v_{sy} + n_{iy}))^{2} \\ + (n_{(i + 2) z} - (v_{sz} + n_{iz}))^{2} \end{matrix}

(18)

\begin{matrix} (\begin{matrix} n_{(i + 3) x} \\ n_{(i + 3) y} \\ n_{(i + 3) z} \end{matrix}) = l_{s} (\begin{matrix} \frac{n_{(i + 2) x} - n_{(i + 1) x}}{| v_{(i + 2) (i + 1)} |} + n_{(i + 1) x} \\ \frac{n_{(i + 2) y} - n_{(i + 1) y}}{| v_{(i + 2) (i + 1)} |} + n_{(i + 1) y} \\ \frac{n_{(i + 2) z} - n_{(i + 1) z}}{| v_{(i + 2) (i + 1)} |} + n_{(i + 1) z} \end{matrix}) \end{matrix}

(19)

\begin{matrix} (\begin{matrix} n_{(i + 4) x} \\ n_{(i + 4) y} \\ n_{(i + 4) z} \end{matrix}) = l_{s} (\begin{matrix} \frac{n_{(i + 2) x} - n_{ix}}{| v_{(i + 2) i} |} + n_{ix} \\ \frac{n_{(i + 2) y} - n_{iy}}{| v_{(i + 2) i} |} + n_{iy} \\ \frac{n_{(i + 2) z} - n_{iz}}{| v_{(i + 2) i} |} + n_{iz} \end{matrix}) \end{matrix}

(20)

where $i = 3 n$ , and is an integer bounded by $6 \leq 3 n \leq (3 + 3 n_{Rad})$ and $n \geq$ 1. The variables $v_{(i + 2) i}$ and $v_{(i + 2) (i + 1)}$ are vectors from nodes $i$ and $i + 1$ to node $i + 2$ , respectively.

As it is shown in Figure 1(c), $f$ four-bar linkages and scissor arm arrays can be attached to the Kresling-inspired mechanism, where $f \leq m$ , and are equally spaced every $m / f$ nodes around the Kresling-inspired mechanism, where $m / f$ is an integer. The coordinates for the remaining nodes on the $f$ four-bar linkages and scissor arm arrays are defined by equation (21):

\begin{matrix} (\begin{matrix} n_{x}' \\ n_{y}' \\ n_{z}' \end{matrix}) = (\begin{matrix} \cos (\frac{- 2 π}{f} i) & - \sin (\frac{- 2 π}{f} i) & 0 \\ \sin (\frac{- 2 π}{f} i) & \cos (\frac{- 2 π}{f} i) & 0 \\ 0 & 0 & 1 \end{matrix}) (\begin{matrix} n_{x} \\ n_{y} \\ n_{z} \end{matrix}) \forall i \in {1 \dots f} \end{matrix}

(21)

where $n_{x}$ , $n_{y}$ , and $n_{z}$ are the coordinates for a given node on the first four-bar linkage and scissor arm array.

The coordinates defined by equations (1) to (21) are used to construct the geometric model of the Kresling-inspired scissor structure. Changes to $δ$ in equations (1) to (21) correspondingly alter the coordinates to achieve reconfiguration of the geometric model.

2.2. Acoustic modeling and analysis

For this research, the acoustic transducers are positioned at the intersection point of each scissor arm unit, as it is shown in Figure 2(a). Based on the principle of acoustic reciprocity, the transducers can be considered to be either acoustic sources or receivers. Results in this study assume the transducers are point sources without loss of generality.

Figure 2(a) shows that spherical coordinates are used to define the acoustic pressure $p$ at a far field point, where $R$ is the radial distance, $β$ is the elevation angle, $φ$ is the azimuth angle, and the origin for the acoustic field is located at the geometric center of the Kresling-inspired mechanism layer $A$ . The total acoustic pressure emitted by the scissor arm array sources at a far field point is determined from Rayleigh’s integral for point sources, that is, a Green’s function (Kinsler et al., 2000). Equation (22) is the total acoustic pressure amplitude at the field point.

p (R, β, φ) = j ρ_{0} ω a^{2} u_{0} \sum_{i = 1}^{M} [\frac{e^{- jk r_{i}}}{r_{i}}]

(22)

where $ρ_{0}$ is the fluid medium density, $ω$ is the angular frequency of the time-harmonic oscillation of each source, $a$ is the radius of the point source and is presumed to be significantly smaller than the acoustic wavelength, $u_{0}$ is the amplitude of the normal velocity of the point source, $M$ is the number of point sources, $k = ω / c_{0}$ is the wavenumber where $c_{0}$ is the sound speed, and $r_{i}$ is the radial distance from the center of the ith point source to the field point. Air is the fluid medium used in this study, where $ρ_{0} =$ 1.21 kg/m³ and $c_{0} =$ 343 m/s. After the total acoustic pressure at a field point is determined, the sound pressure level (SPL) is calculated using equation (23):

SPL = 20 \log_{10} [\frac{p_{rms} (R, β, φ)}{p_{ref}}]

(23)

where $p_{rms} (R, β, φ)$ is the root-mean-square value of $p (R, β, φ)$ , and $p_{ref} =$ 20 µPa is the reference acoustic pressure for air.

2.3. Influences of geometric parameters on compaction, deployment, and scissor arm plane orientation

The Kresling-inspired scissor structure can be reconfigured to configurations of maximum compaction and deployment when specific geometric requirements are met. Certain geometric parameters also affect the orientation of the scissor arm array plane on the Kresling-inspired mechanism. Figure 3 presents the requirements to achieve maximum compaction and deployment, as well as the geometric parameter influences on scissor arm array plane orientation. Although the illustrations in Figure 3 are only shown for a Kresling-inspired scissor structure with three scissor arm arrays, the geometric requirements exemplified are extensible to other configurations or assemblies.

As it is shown in Figure 3(a), the angle of twist $δ$ required to achieve a specific height $h$ decreases as the radius $r_{0}$ increases relative to the diagonal link length $a_{0}$ . For example, the insets in Figure 3(a) demonstrate that when $r_{0} / a_{0} =$ 0.5, the Kresling-inspired mechanism requires $δ =$ 120° for $h / a_{0} =$ 0.5 and $δ =$ 180° for $h / a_{0} =$ 0, whereas the mechanism only requires $δ =$ 16.6° and $δ =$ 19.19° to achieve the same respective heights when $r_{0} / a_{0} =$ 3. The results in Figure 3(a) also show a no twist region that occurs when $r_{0} / a_{0} <$ 0.5 and is bordered by $δ =$ 180°, which means the mechanism height $h$ will not fall below $h$ at $δ =$ 180°. The results shown in Figure 3(a) apply to the Kresling-inspired mechanism for any side number $m$ because $δ$ depends on $h$ , $a_{0}$ , and $r_{0}$ only, as it is shown in equation (1).

The illustrations in Figure 3(b) demonstrate that for the Kresling-inspired scissor structure to achieve maximum compaction, it is necessary that $d =$ 0 and the scissor arm length $l_{s} = q_{max}$ , where $q_{max}$ equals the sum of the four-bar link lengths $l_{1} + l_{2}$ . In order for $d$ to be equal to zero, it is necessary that $h =$ 0 and the angle $γ =$ 0°, where $γ$ is the x–y plane angle between the two nodes that coincide with the Kresling-inspired mechanism and the four-bar linkage. The angle $γ$ corresponds to the results shown in Figure 3(a) for the angle of twist $δ$ , such that $γ = δ_{max}$ when $h / a_{0} =$ 1, where $δ_{max}$ is equal to $δ$ at $h / a_{0} =$ 0. The angle of twist $δ_{max}$ required for maximum compaction is also defined by equation (24):

δ_{max} = \frac{N (360^{°})}{m}

(24)

where the four-bar linkage nodes 1 and 3 coincide with the Kresling-inspired mechanism every $N$ nodes, and $m$ is the side number of the Kresling-inspired mechanism. The combination of the results for $δ$ in Figure 3(a) and equation (24) provides a framework that facilitates the selection of geometric parameters that satisfy the requirements for maximum compaction of the Kresling-inspired scissor structure.

The illustrations in Figure 3(c) show that the Kresling-inspired scissor structure with linear scissor arm arrays can extend to maximum deployment when $l_{1} = l_{2}$ , and $d_{max} = l_{1} + l_{2}$ , where $d_{max}$ is the maximum length of dimension $d$ and is equal to $a_{0}$ . As it is shown in Figure 3(c), $d_{max}$ occurs when the four-bar linkage and the diagonal link are coincident.

The results in Figure 3(d) demonstrate that the scissor arm plane angle $θ$ is a function of the height $h$ and the ratio of the four-bar link lengths $l_{1} / l_{2}$ . This instance also uses the following geometric parameters: $m =$ 6, $a_{0} =$ 3 cm, $r_{0} / a_{0} =$ 0.5773, $d_{max} = a_{0}$ , and $l_{1} + l_{2} = d_{max}$ . The general trend in the results shows that as the ratio $l_{1} / l_{2}$ increases, so does the angle $θ$ . For example, the insets in Figure 3(d) show that as the ratio $l_{1} / l_{2}$ increases from 0.56 to 1, the angle $θ$ increases from 0.95° to 11.06°. Furthermore, the insets in Figure 3(d) show that when the four-bar link length ratios are $l_{1} / l_{2} =$ 1.5 and $l_{1} / l_{2} =$ 2.45, the scissor arm plane angles are $θ =$ 50.91° and $θ =$ 79.35°, respectively. The results in Figure 3(d) also show regions where the four-bar links are incompatible because of the selected link lengths. Due to these incompatibility regions, $l_{1} / l_{2} =$ 1 is the only ratio that meets the requirements for maximum compaction. Though the angle $θ$ has a small effect on the configurations for linear scissor arm arrays, it has a significant impact on the configurations of spiral scissor arm arrays, which will be discussed further in Section 5.

3. Linear and spiral scissor arm array configuration case studies

In this section, two case studies are presented for the linear and spiral scissor arm array configurations of the Kresling-inspired scissor structure. Analytical and experimental results are explored to demonstrate the capabilities of the Kresling-inspired scissor structure for deployment-to-compaction ratios and acoustic wave guiding.

3.1. Case study I: Linear scissor arm array configuration

In this case study, the linear scissor arm array configuration is explored for the Kresling-inspired scissor structure. As it is discussed in Section 2, maximum deployment configurations may be realized with linear scissor arm arrays on the Kresling-inspired scissor structure. The results for $δ$ in Figure 3(a) and equation (24) provide guidelines to facilitate geometric parameter selection for maximum compaction. The illustrations in Figure 4(a) utilize these geometric parameter relationships to demonstrate maximum compaction and deployment configurations in the Kresling-inspired scissor structure with linear scissor arm arrays and four-bar linkages coincident every $N =$ 2 nodes on the Kresling-inspired mechanism. Figure 4(a) shows that when the side number $m =$ 6, it is necessary that the ratio $r_{0} / a_{0} =$ 0.5773 for maximum compaction. Figure 4(a) also shows when $m =$ 10 and $m =$ 14, the ratio $r_{0} / a_{0}$ should increase to 0.8507 and 1.1525, respectively, to maintain capability for maximum compaction. The illustrations in Figure 4(a) indicate that the angle of twist $δ$ required for maximum compaction decreases from 120° to 51.42° as $m$ increases from 6 to 14, and as the ratio $r_{0} / a_{0}$ increases from 0.5773 to 1.1525. These results suggest that deployable structures that require a greater number of scissor arm arrays should be constructed with a larger radius $r_{0}$ to diagonal link length $a_{0}$ ratio for maximum compaction, and that the required angle of twist $δ$ will decrease. Figure 4(a) also shows that the length of each linear scissor arm array is independent of the geometric parameters, and can be increased with the inclusion of additional scissor arm units to each array. The flexibility in the linear scissor arm array length allows the Kresling-inspired scissor structure deployment-to-compaction ratio to be simply modified. This makes it an excellent candidate for applications that require large deployment-to-compaction scaling.

Figure 4(b) shows the construction of the experimental model used to validate the reconfiguration behavior of the analytical model from compaction to deployment. Revolute joints are integrated at every node in the experimental model with offset distances and an offset angle to account for structural thickness. The offset distances and angle are added to the analytical model for a more accurate comparison, as it is shown in Figure 4(b). The offset distances are 1.6 cm between the Kresling-inspired mechanism and four-bar linkage, and 3.87 cm between the four-bar linkage and scissor arm array, as it is shown in Fig. 4(b). The offset angle is 125.68°, and it represents the angle between node 1 on the four-bar linkage and the plane defined by nodes $A_{2}$ , $B_{2}$ , $B_{4}$ on the Kresling-inspired mechanism when $δ =$ 0°, as it is shown in Figure 4(b). The offset angle also represents the angle between to node 3 on the four-bar linkage and the plane defined by nodes $A_{2}$ , $B_{2}$ , $B_{4}$ , as it is shown in Figure 4(b). The parts for the experimental model are constructed with a 3D printer (FlashForge Creator Pro) using acrylonitrile butadiene styrene (ABS), and are fastened together with 3D printed snap fit joints. Three linear scissor arm arrays and four-bar linkages are attached at every $N =$ 2 nodes on the Kresling-inspired mechanism, with six scissor arm units on each array. The geometric parameter dimensions used to construct the experimental model are the following: $m =$ 6, $r_{0} =$ 5.08 cm, $a_{0} =$ 7.62 cm, $l_{1} =$ 3.81 cm, $l_{2} =$ 6.35 cm, $l_{s} =$ 9.14 cm, and $α =$ 0.5.

The analytical and experimental model results that are shown in Figure 4(c) demonstrate how the maximum radius $R_{max}$ of the Kresling-inspired scissor structure at the compacted configuration increases as the angle of twist $δ$ decreases. The results in Figure 4(c) show that $R_{max}$ increases to $2.18 R_{max}$ as $δ$ decreases from 87.27° to 61.31°, and then increases to $2.52 R_{max}$ as $δ$ decreases from 61.31° to 48.7°. This means that for a change in angle of twist $Δ δ =$ 38.57°, the deployed area is 6.35 times greater than the compacted area of the structure. These results suggest that only a small change in angle of twist $δ$ is required for significant changes in the deployment radius and area of the Kresling-Scissor structure.

3.2. Case study II: Spiral scissor arm array configuration

In this case study, the spiral scissor arm array configuration is explored for the Kresling-inspired scissor structure where maximum deployment is not a necessary characteristic for acoustic wave guiding applications. Figure 5(a) illustrates a geometric comparison of the Kresling-inspired scissor structure with seven spiral scissor arm arrays to a multi-spiral array that is advantageous for acoustic wave guiding applications (Prime and Doolan, 2013). The multi-spiral array shown in Figure 5(a) is a rigid structure that cannot be reconfigured to a compacted configuration. The geometric model for a single spiral on the multi-spiral array is defined by equations (25) to (28), which define the arc length of the spiral and the polar coordinates for the acoustic transducers.

l_{n} = (\frac{n - 1}{N_{m} - 1}) l_{max}, \forall n \in {1 \dots N_{m}}

(25)

l_{max} = \frac{r_{min} \sqrt{1 + \cot^{2} (η)}}{\cot (η)} (\frac{r_{max}}{r_{min}} - 1)

(26)

θ_{1, n} = \frac{1}{\cot (η)} \ln (1 + \frac{\cot (η) l_{n}}{r_{min} \sqrt{1 + \cot^{2} (η)}}), \forall n \in {1 \dots N_{m}}

(27)

r_{1, n} = r_{min} e^{\cot (η) θ_{n}}, \forall n \in {1 \dots N_{m}}

(28)

The number of transducers on the spiral is defined by $N_{m}$ , and are equally spaced by arc lengths $l_{n}$ , with $l_{max}$ as the max arc length of the spiral. The minimum and maximum spiral radii are defined by $r_{min}$ and $r_{max}$ , respectively, and the spiral angle is defined by $η$ . The polar coordinates for each transducer on the spiral are represented by $θ_{1, n}$ and $r_{1, n}$ . The polar coordinates for the transducers on the other spirals of the multi-spiral array are defined by equations (29) and (30):

θ_{i, n} = θ_{1, n} + \frac{i - 1}{N_{a}} 2 π, \forall i \in {1 \dots N_{a}}, \forall n \in {1 \dots N_{m}}

(29)

r_{i, n} = r_{1, n}, \forall i \in {1 \dots N_{a}}, \forall n \in {1 \dots N_{m}}

(30)

where $N_{a}$ is the number of spirals on the array. The geometric parameter dimensions used to design the multi-spiral array in Figure 5(a) are: $N_{m} =$ 9, $N_{a} =$ 7, $r_{min} =$ 5 cm, $r_{max} =$ 31.1 cm, and $η = 0.25 π$ rad. The Kresling-inspired scissor structure shown in Figure 5(a) has seven spiral scissor arm arrays with nine scissor arm units on each array. It is designed with the following geometric parameter dimensions: $m =$ 14, $a_{0} =$ 4.41 cm, $r_{0} =$ 5.08 cm, $l_{1} =$ 1.76 cm, $l_{2} =$ 2.65 cm, $l_{s} =$ 4.41 cm, and $α =$ 0.53. As it is shown in Figure 5(a), the Kresling-inspired scissor structure deploys to a maximum radius of $R_{max} =$ 33 cm at $δ =$ 30°, which is comparable to the size and shape of the multi-spiral array with a maximum radius of $R_{max} =$ 31.1 cm. Figure 5(a) also shows that the Kresling-inspired scissor structure can be compacted to a maximum radius of $R_{max} =$ 10.2 cm at $δ =$ 50°, which is three times less than that of the multi-spiral array. These results indicate that the Kresling-inspired scissor structure can be used to construct multi-spiral acoustic arrays that can be deployed for acoustic wave guiding and compacted for transportation and storage purposes.

The acoustic model results shown in Figure 5(b) complement the geometric comparison of the Kresling-inspired scissor structure and the multi-spiral array shown in Figure 5(a). Nine acoustic transducers are equally spaced on each array of the Kresling-inspired scissor structure, and on each spiral of the multi-spiral array. The results in Figure 5(b) show that the Kresling-inspired scissor structure and the multi-spiral array produce comparable acoustic wave guiding results, where the maximum sidelobe level (MSL) in decibels (dB) represents the difference between the main lobe SPL and the next highest peak SPL. Large negative MSL indicates a highly directive and effective wave guiding array. The results show that there is only a 2 dB difference between the MSL of the multi-spiral array and the Kresling-inspired scissor structure across the frequency spectrum considered, with the exception of 9 and 4 dB differences at 700 and 900 Hz, respectively. The inset in Figure 5(b) represents the acoustic beam pattern created by the Kresling-inspired structure at a driving frequency of 1900 Hz. The beam pattern at 1900 Hz has a MSL that is −8 dB and provides a visual that demonstrates how acoustic waves are guided to form a major lobe and minimize side lobes because of the structure geometry. The results shown in Figure 5(b) suggest that the Kresling-inspired scissor structure with spiral scissor arm arrays can produce acoustic wave guiding behavior comparable to multi-spiral arrays that are designed for acoustic wave guiding.

An experimental model is used to validate the reconfiguration behavior of the analytical model from compaction to deployment for the structure configuration with five spiral scissor arm arrays. Revolute joints are integrated at every node in the experimental model with offset distances and an offset angle to account for structural thickness. The offset distances are also incorporated into the analytical model for a more accurate comparison. The first offset distance is 1.75 cm, and it represents the distance between the nodes on the Kresling-inspired mechanism and nodes 1 and 3 on the four-bar linkages that originally coincide on the analytical model. The second offset distance is 3.87 cm, and is the distance between nodes 2 and 4 on the four-bar linkages and the scissor arm array nodes that originally coincide on the analytical model. The offset angle is 111.21°, and it defines the angles between the same plane that is depicted in Figure 4(b) and nodes 1 and 3 on the four-bar linkages. The parts used to construct the experimental model are fabricated with a 3D printer (FlashForge Creator Pro) using ABS, and are fastened together with 3D printed snap fit joints. The geometric parameter dimensions used for this experimental model are the same as the dimensions used for the experimental model in the linear scissor arm array case study, except for the following: $m =$ 10, $l_{1} =$ 3.05 cm, $l_{2} =$ 4.06 cm, and $l_{s} =$ 6.86 cm. The illustration in Figure 5(c) demonstrates how the intersection position fraction $α$ is incorporated into the experimental model to create spiral scissor arm arrays, where $α$ is now equal to 0.45 instead of 0.5 to produce the spiral effect. Five scissor arm arrays and four-bar linkages are attached every $N =$ 2 nodes on the Kresling-inspired mechanism, with six scissor arm units on each spiral scissor arm array.

Figure 5(d) shows the compaction to deployment results for the analytical and experimental models. The results show that the maximum radius $R_{max}$ of the Kresling-inspired scissor structure at the compacted configuration first increases to $1.48 R_{max}$ before it decreases to $1.19 R_{max}$ at the deployed configuration as $δ$ decreases from 85.67° to 68.07° and 53.74°, respectively. The decrease in $R_{max}$ as the Kresling-inspired scissor structure fully deploys is due to the spiral scissor arm expansion that occurs before the spiral scissor arm arrays wrap around the center axis of the structure. The experimental model results shown in Figure 5(d) indicate that the Kresling-inspired scissor structure can be used to physically realize reconfigurable acoustic wave guiding arrays that can be deployed and compacted with spiral scissor arm arrays.

4. Far field acoustic wave propagation from Kresling-inspired scissor structure arrays

A proof-of-concept prototype is fabricated to validate the analytical model predictions for far field acoustic wave guiding from the Kresling-inspired scissor structure with five spiral scissor arm arrays. Beam patterns of the acoustic wave guiding results from both experimental and analytical efforts are gathered and compared in this section. Analytical efforts from the Kresling-inspired scissor structure with seven spiral scissor arm arrays are also presented in this section to demonstrate the influence of structural reconfiguration on acoustic wave guiding.

4.1. Specimen design and fabrication

The experimental model with the five spiral scissor arm arrays shown in Figure 5(d) is also utilized as the structure for the proof-of-concept specimen that validates far field acoustic wave guiding. To replicate the acoustic transducers in the analytical model, 30 elliptical miniature loudspeakers (Parts Express, Springboro, OH) are bonded to the intersection point of each scissor arm unit. Acrylic sheets that are 3.18-mm-thick are laser cut and bonded to the experimental model to hold the five spiral scissor arm arrays at the prescribed spiral configuration of the analytical model, as it is shown in Figure 6(a). Measurements are taken for the deployed configuration of the experimental model at $δ =$ 53.74°. The miniature loudspeakers are driven in unison with an external amplifier and function generator.

4.2. Experimental setup

An anechoic chamber with dimensions 4.3 m × 6.1 m × 3.4 m is used for measurements of the far-field acoustic pressure generated by the proof-of-concept specimen, as it is shown in Figure 6(a). A microphone (PCB Piezotronics 130F20, Depew, NY) is used to measure the acoustic pressure from $β =$ 0° to $β =$ 45° at azimuth angles 0°, 24°, and 48° on the proof-of-concept specimen, as it is shown in Figure 6(b) and 6(c), and it is positioned $R =$ 3.7 m away from the specimen. A function generator is used to generate a tonal signal that is sent to an amplifier that drives the loudspeakers of the array. Measured acoustic pressure data are post-processed using MATLAB to determine the sound pressure level (SPL) along the effective microphone path.

4.3. Comparison between analytical and experimental results

The SPL results in Figure 6 present a comparison between far field beam patterns of the analytical and experimental models. The driving frequencies considered in the experiments are 500, 1000, and 2000 Hz. The transducer radius is $a =$ 1 cm. The normal velocity amplitude is $u_{0} =$ 1.5 cm/s for 500 and 2000 Hz, and $u_{0}$ = 5.5 cm/s for 1000 Hz. In Figure 6, the SPL beam patterns are displayed on the radial axes, and the corresponding elevation angles are displayed on the angular axes.

The results in Figure 6(d), (g) and (j), display uniform SPL across the elevation angle $β$ for azimuth angles 0°, 24°, and 48°. This monopole-like sound radiation is due to diminished potential constructive interference because the frequency wavelength (68.6 cm) is much larger than the radius of the experimental model (27.9 cm).

As the driving frequency is increased to 1000 Hz, the proof-of-concept specimen starts to exhibit the formation of major lobes at $β =$ 0° (broadside) for all three azimuth angles, as shown in Figs. 6(e), (h) and (k). These major lobes are comprised of the angular regions with high SPL before the first local SPL minimum, which occurs at $β =$ 35° for this instance.

The results in Figure 6(f), (i), and (l) show when the frequency is increased to 2000 Hz, the major lobe width narrows from a 60° angular width (for 1000 Hz) to a 30° angular width. This narrowing of the beam width for an increase in frequency corresponds to smaller wavelengths providing more potential constructive and destructive interference that project focused acoustic pressure at broadside and suppress acoustic pressure at off-axis locations. As it is shown in the SPL results in Figure 6, there is good qualitative and quantitative agreement for the experimental results and analytical predictions, which validates the analytical model formulation. These results suggest that the Kresling-inspired scissor structure with spiral scissor arm arrays can be realized as a deployable acoustic wave guiding array.

4.4. Influence of structural reconfiguration on acoustic wave guiding

The analytical model with seven spiral scissor arm arrays shown in Figure 5(a) is utilized to demonstrate the influence of structural reconfiguration on far field acoustic wave guiding. Figure 7 presents a comparison of the acoustic wave guiding results for the analytical model as it reconfigures from a compacted configuration at $δ =$ 50° to a deployed configuration at $δ =$ 30°. The beam patterns in Figure 7 are constructed from a modification of equation (22), such that the acoustic pressure amplitude is ignored and the amplitude of the beam pattern is defined by equation (31):

D (β, φ) = | \sum_{i = 1}^{M} [\frac{e^{- jk r_{i}}}{r_{i}}] |

(31)

The sound pressure level (SPL) results shown on the beam patterns in Figure 7 are calculated using equation (32):

SPL = 10 \log_{10} [D (β, φ)]

(32)

Driving frequencies of 1000 and 2000 Hz are used to create the beam pattern SPL results in Figure 7 at a far field point radial distance of $R =$ 20 m.

The results in Figure 7(d) to (f) show when $δ$ is decreased from 50° to 30°, the major lobe width narrows from a 180° angular width to a 92° angular width for a frequency of 1000 Hz. Similarly, the results in Figure 7(g) to (i) show when $δ$ is decreased from 50° to 30°, the major lobe width narrows from a 126.8° angular width to a 41° angular width for a frequency of 2000 Hz. The results in Figure 7(d) to (i) also show when $δ$ is decreased from 50° to 30°, the SPL drop in the major lobe increases from 2 to 22 dB (for 1000 Hz) and decreases from 25 to 18 dB (for 2000 Hz). These results indicate that the Kresling-inspired scissor structure with spiral scissor arm arrays can be realized as an acoustic wave guiding array that can be reconfigured to modulate acoustic wave guiding properties.

5. Out-of-plane structural configurations

The Kresling-inspired scissor structure proposed in this research has modular characteristics that it inherits from Kresling geometry. Moreover, the use of scissor arms affords the structure out-of-plane arm configurations. For instance, Figure 8(a) demonstrates that the Kresling-inspired scissor structure can be vertically stacked for an arbitrary number of layers because of the modular nature of Kresling geometry. In this example, Figure 8(a) illustrates that for three vertically stacked Kresling-inspired scissor structures the height at the deployed configuration is 4.54 times greater than the height at the compacted configuration. The length of the scissor arm arrays shown in Figure 8(a) can also vary with the inclusion or removal of additional scissor arm units. The schematics in Figure 8(a) also show that the rotation of the scissor arm arrays is affected by the rotation of adjacent layers. For example, point $b$ in Figure 8(a) is rotated 54° (scissor arm arrays are rotated by $δ / 2$ ) by the first and second layers before it is rotated an additional 27° by its own layer (layer 3).

Figure 8(b) demonstrates that the scissor arm arrays of the Kresling-inspired scissor structure can also be designed to deploy into 3D configurations. These spatial arrangements are controlled by the intersection position fraction $α$ , as well as the angle of twist $δ$ and the four-bar link lengths $l_{1}$ and $l_{2}$ , as it is discussed in Section 2 and shown Figure 3(d). For example, when $α =$ 0.47, $l_{1} = 0.5 a_{0}$ , $l_{2} = 0.5 a_{0}$ , $m =$ 12, and $l_{s} = r_{0} = a_{0}$ for a Kresling-inspired scissor structure with six scissor arm arrays that each have six scissor arm units, the scissor arm arrays emerge into a 3D spiral shape with a maximum height at $δ =$ 32° that is 18.02 times greater than the height at $δ =$ 59°. Furthermore, as the angle of twist $δ$ decreases from 40° to 32°, the radius of the structure in the x–y plane decreases from 2.62 to 2.10 times the radius $R_{xy}$ of the structure at the compacted configuration. The results in Figure 8(a) indicate that the linear scissor arm arrays can be vertically and radially modulated into any desired configuration, and the results in Figure 8(b) indicate that the spiral scissor arm arrays can be designed to create complex 3D configurations with the Kresling-inspired scissor structure. These characteristics suggest that the Kresling-inspired scissor structure is favorable for systems that use structures with embedded acoustic arrays, satellites, and antennas.

6. Conclusion

This research explores a deployable structure that is based on a Kresling-inspired mechanism with integrated scissor arms and is reconfigured with single-degree-of-freedom motion. The analytical results presented in this research demonstrate that the proposed structure is capable of being devised into shapes advantageous for extreme deployment-to-compaction ratio and directed acoustic wave applications. Experiments are conducted with proof-of-concept models to validate the analytical results for deployment and acoustic wave guiding. The results show that linear scissor arm configuration for the proposed structure is conducive to create arrays with favorable deployment-to-compaction ratios. The spiral scissor arm configuration may be utilized to create deployable arrays that are optimal for acoustic wave guiding. This research motivates further exploration of reconfigurable structures that are constructed from origami-inspired geometry and rigid link mechanisms.

Footnotes

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This project is supported by the National Science Foundation Faculty Early Career Development Award (Grant No. 2054970).

ORCID iD

Ryan L Harne

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