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Research Papers

Multistable Morphing Mechanisms of Nonlinear Springs

[+] Author and Article Information
Chrysoula Aza

Department of Aerospace Engineering,
University of Bristol,
Bristol Composites Institute (ACCIS),
Bristol BS8 1TR, UK
e-mail: chrysoula.aza@bristol.ac.uk

Alberto Pirrera

Department of Aerospace Engineering,
University of Bristol,
Bristol Composites Institute (ACCIS),
Bristol BS8 1TR, UK
e-mail: alberto.pirrera@bristol.ac.uk

Mark Schenk

Department of Aerospace Engineering,
University of Bristol,
Bristol Composites Institute (ACCIS),
Bristol BS8 1TR, UK
e-mail: m.schenk@bristol.ac.uk

1Corresponding author.

Contributed by the Mechanisms and Robotics Committee of ASME for publication in the Journal of Mechanisms and Robotics. Manuscript received January 29, 2019; final manuscript received July 3, 2019; published online August 5, 2019. Assoc. Editor: Andrew P. Murray.

J. Mechanisms Robotics 11(5), (Aug 05, 2019) (14 pages) Paper No: JMR-19-1034; doi: 10.1115/1.4044210 History: Received January 29, 2019; Accepted July 08, 2019

Compliant mechanisms find use in numerous applications in both microscale and macroscale devices. Most of the current compliant mechanisms base their behavior on beam flexures. Their range of motion is thus limited by the stresses developed upon deflection. Conversely, the proposed mechanism relies on elastically nonlinear components to achieve large deformations. These nonlinear elements are composite morphing double-helical structures that are able to extend and coil like springs, yet, with nonlinear stiffness characteristics. A mechanism consisting of such structures, assembled in a simple truss configuration, is explored. A variety of behaviors is unveiled that could be exploited to expand the design space of current compliant mechanisms. The type of behavior is found to depend on the initial geometry of the structural assembly, the lay-up, and other characteristics specific of the composite components.

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Figures

Grahic Jump Location
Fig. 1

(a) Initially curved (radius Ri) composite strips are flattened to introduce prestress; (b) the strips are joined by rigid spokes to form a double-helix structure, which (c) can deform from a straight (light gray) to a twisted (dark) configuration (θ < 0)

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Fig. 2

Manufactured prototype of the double-helix structure in (a) straight, (b) stable twisted, (c) unstable twisted, and (d) fully coiled configuration

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Fig. 3

Load–displacement curves of double helices for different lay-ups of the form [β2/0/β2] or [β2/0/-β2], where β is the ply angle. The displacement Δl is normalized to the length L of the strips, with Δl/L = 0, representing the fully extended and Δl/L = 1 its fully coiled configuration. All double helices shown have two self-equilibrated configurations with no external force required to maintain the shape.

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Fig. 4

Schematic representation of the assembly of double helices in a truss-like configuration with both supports pinned [22]. The initial configuration is determined by the equilibrium length L0,i of the double helices and by the initial angle α0,i of the truss configuration.

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Fig. 5

Strain energy landscapes for a compliant mechanism of identical double helices assembled in a truss-like configuration. Results are for initial truss angles α0,1 = 35 deg and α0,1 = 70 deg, with composite strips of [05], [452/0/452], and [452/0/-452] lay-ups. The initial truss configurations are indicated with black lines. Points labeled 1–5 denote stable equilibria, while points A–H, J, K, and M identify positions of unstable equilibrium; points I–IV denote stable boundary equilibria. The positions of the truss apex under an applied vertical load (Ph = 0) and/or horizontal load (Pv = 0) are superimposed on the landscapes: red points indicate the equilibrium paths of the apex under the application of a vertical load, and blue points indicate the equilibrium paths of the apex under the application of a horizontal load (Color version online.)

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Fig. 6

Load–displacement curves of the assembled structure of identical double helices under the application of a vertical load at the apex. Results are for different initial truss angle (a) α0,1 = 35 deg and (b) α0,1 = 70 deg and of double helices with varying lay-ups. Dashed line represents sections of instability; points 1–5 are stable equilibrium points. Points A–H, J, K, and M are unstable equilibrium points. The load has been normalized with respect to the load value at the maximum peak (Pcrit) in each case. The displacement has been normalized with respect to the initial height of the truss structure.

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Fig. 7

Load–displacement curves of the assembled structure of identical double helices under the application of a horizontal load at the apex. Results are for different initial truss angle (a) α0,1 = 35 deg and (b) α0,1 = 70 deg and of double helices with varying lay-ups. Dashed line represents sections of instability. Points 1 and 2 are stable equilibrium points; points A–H, J, K, and M are unstable equilibrium points. The load has been normalized with respect to the load value at the maximum peak (Pcrit) in each case. The displacement has been normalized with respect to the initial width of the truss structure.

Grahic Jump Location
Fig. 8

Load–displacement curve (left) and deformation (right) of the assembled structure (with initial angle α0,1 = 35 deg) of two identical double helices (with a [452/0/452] lay-up) under combined loading (Ph = Pv) at the apex. Points 1 and 2 are stable equilibrium points; points J, K, and M are unstable equilibrium positions. Dashed lines (left) represent areas of instability. The load has been normalized with respect to the load value at the maximum peak (Pcrit). The displacement has been normalized with respect to the initial length L0.

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Fig. 9

Load–displacement curves (left) and deformation (right) of the assembled structure of double helices of different length under the application of (a) a vertical load and (b) a horizontal load at the apex. Points 1–4 are stable equilibrium points, and points A–E are unstable equilibrium points. The truss has initial angle α0,1 = 35 deg and double helices of [452/0/452] lay-up and lengths L1 = 95 mm and L2 = 71 mm. Dashed lines (left) represent areas of instability. The load has been normalized with respect to the load value at the maximum peak (Pcrit) for each case. The displacement has been normalized with respect to the initial height or width of the truss structure, respectively, for vertical or horizontal loading.

Grahic Jump Location
Fig. 10

(a) Load–displacement curve of a mechanism consisting of double helices compared with one of linear springs under a vertical load at the apex. Initial truss angle α0,1 = 70 deg is used for both assemblies. Points 1–4 are stable equilibrium points, and points A–F, J, K, and M are unstable equilibrium points. Dashed lines represent areas of instability. (b) Deformation of the assembled structures. (c) Axial force with respect to displacement of a linear spring (left) and of a double helix (right).

Grahic Jump Location
Fig. 11

(a) Load–displacement curves of a mechanism with different initial truss angles α0,1 consisting of double helices of [452/0/452] lay-up under the application of a vertical load at the apex. Points 1–4 are stable equilibrium points. Points A–F, J, K, and M are unstable equilibrium points. Dashed lines represent areas of instability. (b) Axial forces of double helices at selected equilibrium points for the different initial truss angles.

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Fig. 12

Load–displacement curves of the assembled structure of identical double helices for different symmetric lay-ups of the form of [β2/0/β2] for (a) β = 0 deg, 45 deg, 90 deg and (b) β = 30 deg, 45 deg, 60 deg under the application of a vertical load at the apex. Points 1, 2, and 5 are stable equilibrium points on the main paths; points 3 and 4 on the bifurcation paths. Points A, M, B, G, and H are unstable equilibrium points on the main paths; points C–F, J, and K on the bifurcation paths. Dashed lines represent areas of instability.

Grahic Jump Location
Fig. 13

(a) Load–displacement curve of a mechanism consisting of double helices compared with one of linear springs under a horizontal load at the apex. Initial truss angle α0,1 = 70 deg is used for both assemblies. Dashed lines represent areas of instability. (b) Deformation of the assembled structures of linear springs (left) and double helices (right). Points 1 and 2 are stable equilibrium positions; points J, K, and M are unstable equilibrium positions.

Grahic Jump Location
Fig. 14

Load–displacement curves of a mechanism of different initial truss angles α0,1 consisting of double helices of [05] lay-up, under the application of a horizontal load at the apex. Points 1 and 2 are stable equilibrium points; points J, K, and M are unstable equilibrium points. Dashed lines represent areas of instability.

Grahic Jump Location
Fig. 15

Load–displacement curves of the assembled structure of identical double helices for different symmetric lay-ups of the form of [β2/0/β2] for (a) β = 0 deg, 45 deg, 90 deg and (b) β = 30 deg, 45 deg, 60 deg under the application of a horizontal load at the apex. Points 1–5 are stable equilibrium points; points A, C, F–H, J, K, and M are unstable equilibrium positions. Dashed lines represent areas of instability.

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Fig. 16

Experimental set-up

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Fig. 17

(a) Load–displacement curves under the application of a vertical load at the apex of the truss of double helices. Dashed lines represent areas of instability for the analytical data. The truss has initial angle α0,1 = 70 deg and double helices of [90/45/0/45/90] lay-up, with dimensions L = 292 mm, R = 30 mm, Ri = 60 mm, and W = 10 mm. (b) Corresponding strain energy landscape with the positions of the truss apex under an applied vertical load (Ph = 0) at the end effector superimposed; individual (red) markers are used for the analytical and a continuous series of (green) markers for the experimental data. Points 1–4 are stable equilibrium points. Points A–F and M are unstable equilibrium points. (cf) Different configurations of the prototype during testing: (c) a configuration along the main path between points 1 and A with both helices twisted equally; (d) a configuration close to point M, with the helices collinear and twisted equally; (e) a configuration on the bifurcation path between points 1 and C, with the helices twisted to different extent; (f) a configuration close to point 3, with the helices collinear but one in a slightly twisted and the other in a coiled configuration (Color version online.)

Grahic Jump Location
Fig. 18

Load–displacement curves of double helices: (a) for different dimensions L, R, and Ri, keeping a constant ratio Ri/R = 2 and L ≈ 2πR, lay-up [452/0/452], W = 5 mm; (b) for different widths W, lay-up [452/0/452], L = 95 mm, R = 15 mm, Ri = 30 mm; (c) for different initial curvatures of the strips Ri, lay-up [452/0/452], L = 95 mm, R = 15 mm, W = 5 mm; (d) for different radius R, lay-up [452/0/452], L = 95 mm, Ri = 30 mm, W = 5 mm; (e) for different lengths L, lay-up [452/0/452], R = 15 mm, Ri = 30 mm, W = 5 mm. The displacement Δl is normalized to the length L of the strips in each case for comparison purposes, with Δl/L = 0 representing the fully extended and Δl/L = 1 its fully coiled configuration.

Grahic Jump Location
Fig. 19

Load–displacement curves of the assembled structure of identical double helices under the application of a vertical load at the apex, with different (a) dimensions L, R, and Ri, keeping a constant ratio Ri/R = 2 and L ≈ 2πR, lay-up [452/0/452], W = 5 mm; (b) radius R, lay-up [452/0/452], L = 95 mm, Ri = 30 mm, W = 5 mm; (c) initial curvatures of the strips Ri, lay-up [452/0/452], L = 95 mm, R = 15 mm, W = 5 mm; (d) widths W, lay-up [452/0/452], L = 95 mm, R = 15 mm, Ri = 30 mm. Points 1 and 2 are stable equilibrium points on the main path, and points 3 and 4 on the bifurcation path. Points A, M, and B are unstable equilibrium points on the main path, and points C–F on the bifurcation path. Dashed lines represent areas of instability.

Grahic Jump Location
Fig. 20

Load–displacement curves of the assembled structure of identical double helices under the application of a horizontal load at the apex with different (a) dimensions L, R, and Ri, keeping a constant ratio Ri/R = 2 and L ≈ 2πR, lay-up [05], W = 5 mm; (b) radius R, lay-up [05], L = 95 mm, Ri = 30 mm, W = 5 mm; (c) initial curvatures of the strips Ri, lay-up [05], L = 95 mm, R = 15 mm, W = 5 mm; (d) widths W, lay-up [05], L = 95 mm, R = 15 mm, Ri = 30 mm. Points 1 and 2 are stable equilibrium points; points J, K, and M are unstable equilibrium points. Dashed lines represent areas of instability.

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