Load–Displacement Characterization in Three Degrees-of-Freedom for General Lamina Emergent Torsion Arrays

Lamina emergent mechanisms (LEMs) are compliant mechanisms formed from a planar material and have some or all of their motion out of the plane [1–8]. They can be fabricated using two-dimensional processes and rely on the compliance of flexible members formed from the planar material to gain their motion [9]. Apart from common benefits of compliant mechanisms such as high precision, the absence of backlash and wear, and the limited amount of parts [9], LEMs are characterized by a low manufacturing cost, a relatively simple topology, and compactness in the initial state. Characterization of the load–displacement behavior of LEMs is an important step to appropriately design mechanisms with desirable behaviors. They have been used in a variety of applications, including microelectromechanical systems [1,10] and origami-based mechanisms [11,12]. A particularly useful LEM for origami-based mechanisms is the lamina emergent torsion (LET) joint. The LET joint achieves high rotational compliance while minimizing the required footprint of the joint, allowing for origami-like folding of panels and localized joints [13–15]. Recent studies have provided many folding and modeling techniques (e.g., elastic origami models [16], truss frameworks [17], topology optimization [18], and mechanical properties of paper folds [19]) and applications (e.g., kinetogami [20], deformable structures [21], and cylinders [22]) for origami-based mechanisms. Systems of panels and low-footprint joint designs are beneficial for applications where the function of the panels depends on the panel size. This feature can be used to create networks of monolithic panels and joints to obtain desired motions and functions (e.g., deployable devices [23–25], printed circuit boards [26], and actuation origami [27–29]).

LEMs have been placed in series and parallel to facilitate desired behaviors of systems, and work has been done to characterize the global bending stiffness in the desired degree-of-freedom (DOF) [30]. When LEMs are placed in series and parallel, they have been termed lamina emergent arrays (LEAs) or compliant arrays [31]. LEAs also have compliance in other DOFs, and motion in these other DOFs has generally been termed parasitic and undesirable. Specific topological changes of joints, known as surrogate folds for origami-based mechanisms, have been designed to reduce the parasitic motion [32] to behave more like the kinematics of smooth folding structures [33].

This paper describes the formulation of arrays of LET joints (LET arrays, a subset of LEAs) suitable for origami-based mechanisms in which panel-area conservation is desirable and develops a load–displacement relationship set of analytical equations to describe the arrays in three DOFs in the elastic region. The behavior predicted by the theoretical approach is then verified via the finite element analysis (FEA). An integrated simulation environment [34], in which a matlab parametric script guides the computer-aided design (CAD) geometry generations as well as the structural batch simulations, is used to automatically test several LET arrays for each DOF. The rest of the paper is organized as follows: Sec. 2 gives a description of LET array formulations to be studied in this paper, Sec. 3 describes the theoretical load-deflection laws of the LET arrays for each of the considered DOF, Sec. 4 provides a comparison between theoretical and FEA results, and Sec. 5 summarizes the work and presents concluding remarks.

Figure 1 shows the joint frame and motions considered in this paper. We will refer to the rotation about the x-axis (⁠$Rx(γ)$⁠) as folding (referring to the origami-based engineering nomenclature), translation along the y-axis (T_y(y)) as extension/compression (or for brevity, extension), and rotation about the z-axis (⁠$Rz(β)$⁠) as in-plane rotation. LET joints were introduced as joints well-suited for high rotation in the desired folding motion and where some compliance in other DOF were also observed [13]. A single LET joint was defined as a specific formulation of four compliant torsion segments in series and parallel. Two possible configurations were presented as the Outside LET and the Inside LET which had the same four torsion segments with slight topological differences resulting in different boundary conditions of the segments. The differing boundary conditions affected the off-axis motion, but had no significant effects on the desired folding motion.

Researchers, to understand these motions, have developed models describing the load-displacement relationships for folding, extension, and in-plane rotation for single joints. A summary of their work is listed in Table 1. The table also shows the gaps in the literature that are important for origami-based design and this paper fills these gaps. Specifically, we give general equations which account for the various boundary conditions for the three DOF for general LET arrays with any combination of series, parallel, and configuration. Figure 2 illustrates motivations for filling these research gaps. Shown in the figure is a system of panels and joints that uses LET arrays to fold a large area into a small volume in the form of an origami pattern [23]. On the left is a close-up of these panels and joints, in the middle is the unfolded system, and on the right are details of the folded configuration. In the side view of the folded configuration, a close-up of a LET array is shown with displacements in the y-axis and about the x and z axes. It is this type of behavior which this paper characterizes to better understand how such systems of panels and joint perform.

Standard LET joints assume four torsion segments: two in series and two in parallel. Other formulations are possible, which include odd numbers of torsion segments. To reduce confusion when discussing LET arrays made up of torsion segments, we will drop the term LET joints when describing the topology of the arrays. Instead, we will refer to LET array topologies using the following convention: SsPpc. S is the number of torsion segments in series, P is the number of torsion segments in parallel, and c is the configuration (whether the topology resembles an inside or outside LET joint, when applicable). For example, the designation 2s2pi is equivalent to an inside LET joint and 2s2po is the equivalent to an outside LET joint. A single torsion segment has the designation 1s1p and does not have a configuration c designation. Figure 3 shows possible formulations of LET arrays. The 2s2p location in the grid shows both configurations available to the 2s2p designation, with the outside LET on top and the inside LET on bottom. The outside/inside option is only available to designations with both even S and even P. The prototype shown in Fig. 2 indicates the LET array designations used in the left detail.

General LET arrays can be used to tailor stiffness values in different DOF while maintaining the ability to fold. For example, assuming the geometry of individual segments remain the same, LET arrays of the same S will have the same range of motion before failure in folding but will have different folding, extension, and in-plane rotation stiffness values, as is the case for the LET arrays indicated in Fig. 2. Note that, for example, LET arrays with a P = 2 formulation are equivalent to two separate LET arrays with a P = 1 formulation in terms of folding but are not necessarily equivalent in extension and in-plane rotation as the boundary conditions of the torsion segments can be different for each case. There are unlimited variations of formulating the compliant segments. This paper focuses on the topologies depicted in Fig. 3 for any S and P.

Load–displacement relationships (stiffness rates) for each of the three DOFs considered in this paper are given in this section. It is desirable in the design of systems of panels and joints to obtain equivalent stiffness rates, expressed as scalars, in each of the DOF for each joint. Throughout this section, fundamental dimensions are used and are labeled in Fig. 4. Torsion segments and bending segments are also indicated in the figure, about which will be discussed throughout the paper. l_T and w_T are the length and width dimensions of torsion segments, respectively. l_B and w_B are the length and width dimensions of bending segments, respectively. L and W are the length and width dimensions of overall array, respectively, and are functions of the dimensions of the segments in bending and torsion. Throughout these analyses, we assume isotropic material properties. These relationships may also be applicable to symmetric and balanced laminate composites that exhibit isotropic behavior, but it is likely that other considerations, such as for coupling behavior, would be required.

While LET arrays with varying dimensions are possible, we assume that for a particular LET array, the bending and torsion segments, separately, are of equal dimensions and therefore stiffness. This assumption allows for simplification of the predictive models. However, if LET arrays of varying dimensions are to be designed, the following guidelines are suggested: (1) to avoid unnecessary stress risers, ensure that dimensions of the torsion segments are the same along any particular row and (2) treat rows of identical geometry as unique LET arrays and solve the displacement of each LET array in series.

An outside LET torsion segment has a fixed-guided boundary condition, while inside LET torsion segment has a fixed-clamped condition. Here, we consider LET arrays that have a mixture of fixed-guided and fixed-clamped torsion segments, an example of which is shown in Fig. 6. The LET array has a 3s2p designation where four segments are fixed guided and two are fixed clamped. The differences between fixed guided and fixed clamped, in terms of loading conditions and mechanical responses, are discussed in Ref. [38].

To model the equivalent extension/compression spring stiffness K_y,eq of LET arrays, we describe individual segment stiffness rates along the rows first (adding in parallel) and then across the rows (adding in series). Unlike the uniform distribution of stiffness rates for LET arrays in folding, LET arrays in extension/compression have a nonuniform distribution due to the mixed boundary conditions. Thus, additional row types are introduced as follows (see Fig. 7): a boundary row is a torsion row that contains a fixed-clamped segment, and if exists, it does so only at the transition from panel to joint (maximum of two in an array). A regular row is a torsion row that does not have fixed-clamped segments along the left and right edges of the array, and if exists, it may repeat (no maximum, but dependent on S). A bending segment row is composed of only bending segments. Figure 7 shows a LET array, which has two boundary rows, two regular rows, and five bending segment rows (as another example, the array in Fig. 6 has one boundary and two regular rows).

Different spring models are used for in-plane rotation of LET arrays for the cases when P = 1 and P > 1. For P = 1, moment-loaded beams are added in series. For P > 1, a proposed analogous spring model is developed. These cases are discussed below.

To verify the analytical models, several LET arrays were tested using an integrated software framework [34], in which a matlab script manages the parametric study, a solidworks macro updates the geometry of the LET arrays, and ansys apdl provides the force/torque–deflection characteristics of each configuration via batch mode. A routine, overseen by matlab, was defined to automatically export the CAD files from solidworks and, subsequently, to perform the structural batch simulations of each candidate. In line with the proposed theoretical models, three DOFs were analyzed, namely $Rx(γ)$⁠, T_y(y), and $Rz(β)$⁠. For each of the DOF, a free mesh with second-order tetrahedral elements ansys solid187 was defined with a maximum element size of t. This element type is particularly suitable for bending-dominated problems, where the shear locking effect is undesired due to its effect on the bending stiffness. In all simulations, the nonlinear geometry (NLGEOM) option was turned on. As for the employed material, aluminum alloy 7075 (heat treated) is considered, owing to its high strength-to-modulus ratio [9]. The Young’s modulus and Poisson’s ratio are, respectively, E = 71.7 GPa and $ν=0.33$⁠. Concerning the boundary conditions, each LET array was fixed to the ground at one end and guided in a pure translation (along the y-axis) or rotation (about the x or z axes) on the other end. Since SOLID187 elements do not have rotational DOF, MPC184 elements were used to apply kinematic constraints (spider web of beams) between the solid model’s nodes and a master node (as shown in Fig. 9), onto which rotational displacement loads can be applied. Referring to Figs. 2 and 4, the tested array parameters were l_B = 2 mm; W_B = 4 mm; W_T = 2 mm; t = 1 mm; l_T = 10 mm, l_T = 20 mm, or l_T = 30 mm; S (ranging from 1 to 3); P (ranging from 1 to 5); and c (either i or o, where applicable).

One exception to the modified radius factors listed in Table 2 was the case when P = 2 for the R_z DOF. The spring model presented in Sec. 3.3.2 assumes that the columns remain vertical during loading with the global rotation occurring as the left and right columns extend in opposite directions. For P = 2, however, there are no other segments in parallel to enforce vertical displacement, and so the model overestimates the stiffness by artificially enforcing the vertical constraint. Therefore, a separate optimization was performed for the case when P = 2 for each of the l_T values. The optimal radius factors were as follows: (for l_T = 10 mm) $γfg=γfc=1.01$⁠, (for l_T = 20 mm) $γfg=0.53$ and $γfc=0.56$⁠, and (for l_T = 30 mm) $γfg=0.40$ and $γfc=0.49$⁠. These values are especially useful when considering the traditional outside and inside LET joints (which are equivalent to 2s2po and 2s2pi LET arrays, respectively).

It can be seen that for the T_y and R_z DOF, the force/torque–displacement relationships exhibit nonlinear behavior when S is low and P is high (see Figs. 13, 16, and 19) for both the FEA and modeled results. This is to be expected when considering the nonlinear terms in Eqs. (21) and (23) for these high-stiffness array formulations. It can also be seen that the errors are higher for these cases, especially for the R_z DOF. A possible explanation for this error in the R_z model is that for these stiff cases, the simplifying spring models do not account for compliance at the boundaries and at the interactions between the bending and torsion segments. This inaccuracy in model assumptions is less apparent in the less-stiff and linear regimes of the arrays. Other cases (higher S) show excellent agreement between the analytical and FEA results.

The comparison between the FEA and modeled results verified that the models can accurately characterize general LET arrays for elastic load–displacement behaviors. These models represent an important tool for designers of systems that incorporate LET arrays to enable design of behaviors of such systems. The equations used assume the material to remain in the elastic region throughout the motion and are particularly useful in exploring the design space of the formulation of the LET arrays. When finalizing a design, consideration of the elastic limits is required. Equations of maximum stress of LET joints [13,39] can be applied to determine the stresses of LET arrays.

In this paper, LET arrays were defined as torsion segments in series and in parallel, which can be used for origami-based applications and others. The load–displacement relationships for the arrays were characterized in three DOFs enabling the design of arrays that consider off-axis motion.¹ The proposed models were verified using FEA for several LET array configurations resulting from an automatic framework. By using the standard $γ=0.85$ value, conservative stiffness estimates are available. If more accurate stiffness estimated are desired, modified radius factors may be used. By using these modified values, load–displacement models had good agreement with the FEA results. Based on intuition and optimization results, shorter torsion segments may use higher $γ$ values and longer torsion segments may use lower $γ$ values. In some cases, the nonlinear behavior of the arrays were sources of error between the modeled and FEA results. Because the use of LET arrays is becoming more prevalent, the analytical models presented in this work enable their design. By using LET arrays, a designer can tailor the stiffness behavior for folding, extension, and in-plane rotation to realize folding and origami-based mechanisms. The provided stiffness values can be used for multibody dynamic or other system-modeling techniques in the design of such systems. By providing several plots of the force/torque–displacement relationships to show model verification, we have also showed the variability of the design space available to LET arrays.

The authors gratefully acknowledge critical discussions on in-plane rotation stiffness with Jared Butler, Brandon Sargent, Philip Stevenson, and Collin Ynchausti and the finite element modeling assistance from Kenny Seymour.

• NASA Space Technology Research Fellowship (Grant No. 80NSSC17K0145; Funder ID: 10.13039/100000104).