June 8th, 2014, Kyoto Workshop on “Geometric Structures with Symmetry and Periodicity” Organized by Ileana Streinu and Monique Teillaud Coordinated motions of repetitive structures from a mechanical point of view Hiro Tanaka, Dept. Mechanical Engineering, The University of Tokyo.

Outline 1. Research Background 2. Repetitive Structures & their Joint Rotations 3. Mechanisms of 8-bar-jointed Structures 4. Bi-stiffness Property of the Motion Structure 5. Summary 1

1. Research Background Deformability of repetitive structures # Gibson LJ & Ashby MF (1997) hexagonal cells A polar diagram showing Young’s modulus bending-dominated square cells stretching-dominated (tensile load) bending-dominated (shear load) buckling and bending (compressive load) triangular cells (truss) stretching-dominated # Deshpande VS et al. (2001) 2 # Wicks N, Guest SD (2004)

1. Research Background Mechanical property of two-dimensional repetitive structures Poisson’s ratio Examples of repetitive structures re-entrant hexagonal units: : conventional materials : auxetic materials # Dr. Borcea’s talk is coming soon Indentation resistance missing ribs square units: # Evans KE, et al. (2000) # Smith CW, et al. (2000) 3

1. Research Background Proposition of repetitive structures with nonconventional deformation 4

Outline 1. Research Background 2. Repetitive Structures & their Joint Rotations 3. Mechanisms of 8-bar-jointed Structures 4. Bi-stiffness Property of the Motion Structure 5. Summary 5

2. Repetitive Structures & their Joint Rotations Rotation energy of n -coordinate flexible joint ( n -bar joint) beam member : i -th rotational displacent symmetry of rotational stiffness: Hessian matrix of n -bar joint energy with respect to rotational displacements 6

2. Repetitive Structures & their Joint Rotations In case of 4-bar joint ： 4-bar joint four independent moments and rotational displacements: two types of rotational stiffness: (nearest-neighbor interaction) (2 nd nearest-neighbor interaction) Hessian matrix Standard eigenvalue problem Calculating a pair of eigenvalue and eigenvector on n -joint A class of circulant matrices 7

2. Repetitive Structures & their Joint Rotations Fundamental rotational modes of 4-bar joint (multiple root) co-rotation mode asymmetry-rotation mode counter-rotation mode (C 4 -invariance) (C 1 -invariance) (C 2 -invariance) 8

2. Repetitive Structures & their Joint Rotations Buckling problem of periodical square cells under equi-biaxial compression analytical model joint modeling • 2 x 2 cells, 4 x 4 cells,… • displacement control • periodically boundary condition 4x4 cells joint flexibility : to be a rigid joint : to be a pinned joint 9

2. Repetitive Structures & their Joint Rotations Overall Buckling mode vs. local rotational mode mode 1 mode 2 mode 4 mode 3 co-rotations co/counter-rotations asym. rotations counter-rotations co-rotation mode Asymmetry-rotation mode counter-rotation mode 10

2. Repetitive Structures & their Joint Rotations joint flexibility: µ =1 Buckling mode vs. compression ratio (4 x 4 cells) 11

2. Repetitive Structures & their Joint Rotations Localized deformation of mode 2 consists of the co/asymmetry rotations quadruple mode 12

2. Repetitive Structures & their Joint Rotations Repetitive framework with 4-bar joints and elbowed beam members ： 4-bar joint Poisson’s ratio where d 1,2 is x 1 - or x 2 -displacement joint flexibility : to be a rigid joint : to be a pinned joint 13

2. Repetitive Structures & their Joint Rotations Two types of the structures Model I Model II Three type of manufacturing aluminum joints 14 pinned joint flexible joint rigid joint

2. Repetitive Structures & their Joint Rotations Overview of biaxial tensile tester substrate(SS400) ： 1200x1200 mm 2 15

2. Repetitive Structures & their Joint Rotations Uniaxial tensile testing for Model I connected by pinned joints connected by rigid joints zero Poisson’s ratio expanded in the lateral direction because the moment was laterally transferred via the co-rotations of rigid joints: Negative Poisson’s ratio 16

2. Repetitive Structures & their Joint Rotations Measurements of uniaxial tensile testing Model I: Poisson ratio vs Nominal strain Model II: Poisson ratio vs Nominal strain 17

2. Repetitive Structures & their Joint Rotations Out-of-plane deformation of Model II Under the tensile load, the beam members (red) are compressed 18

2. Repetitive Structures & their Joint Rotations Novel 3D structure: applications of Model II 19

Outline 1. Research Background 2. Repetitive Structures & their Joint Rotations 3. Mechanisms of 8-bar-jointed Structures 4. Bi-stiffness Property of the Motion Structure 5. Summary 20

3. Mechanisms of 8-bar-jointed Structures proposed structural unit call the closed-loop unit with four pairs of rhombic elements (REs) as 4RE-linkage rhombic element unknown connection 21

3. Mechanisms of 8-bar-jointed Structures Motion of 4RE-linkage motion of a rhombic element closed-loop configuration: geometrical constraint conditions: Motion of 4RE-linkage 22

3. Mechanisms of 8-bar-jointed Structures Animations of the obtained motions 23

3. Mechanisms of 8-bar-jointed Structures initial configurations Repetitive-assembly motions (2 x 2 cells) 1 5 2 6 3 7 4 24

3. Mechanisms of 8-bar-jointed Structures Deployment mechanism of φ 3 counter rotation a 8-bar joint 25

3. Mechanisms of 8-bar-jointed Structures Manufactured model actuated by a single rotary operation side view actuator Minimum size: 320 × 320 mm 2 , Maximum size: 830 × 830 mm 2 26

3. Mechanisms of 8-bar-jointed Structures Combined assembly the structure built of the multiple rotational modes of 8-bar joints mode 3 mode 4 mode 5 27

Outline 1. Research Background 2. Repetitive Structures & their Joint Rotations 3. Mechanisms of 8-bar-jointed Structures 4. Bi-stiffness Property of the Motion Structure 5. Summary 28

4. Bi-stiffness Property of the Motion Structures Proposed structure Motion Structure with 8 rotational symmetry (MS-8) particular connections The adjacent bars painted in the same color, blue or red, are rigidly connected at all the pivot joints. 29

4. Bi-stiffness Property of the Motion Structures Geometrical configuration of MS-8 6-bar joint 8-bar joint 6-bar joint position of each grayish node 30

4. Bi-stiffness Property of the Motion Structures Mechanism of MS-8 The two patterns of square cells are tilted at 45˚ with respect to each other. Motion I Motion II 31

4. Bi-stiffness Property of the Motion Structures Which motions does MS-8 select? loading condition balance point of the square 32

4. Bi-stiffness Property of the Motion Structures Numerical analyses of the proposed structure with the cell-to-cell contacts for motion I stiff for motion II soft Bi-stiffness property: switching of the stiffness due to two motions 33

4. Bi-stiffness Property of the Motion Structures Simple periodic assembly with the minimum units of the MS-8s The internal forces are transferred on the each structural unit via the four vertices, so Motion II only occurs no matter how we may apply the vertical stress from a long distance. 34

4. Bi-stiffness Property of the Motion Structures Repetitive structure of MS-8s with inserting springs the inserted spring transfer the central force to the center re- entrant vertex on each side of a structural unit. 35

Outline 1. Research Background 2. Repetitive Structures & their Joint Rotations 3. Mechanisms of 8-bar-jointed Structures 4. Bi-stiffness Property of the Motion Structure 5. Summary 36

5. Summary Summary • Short wave-length buckling modes of square cells could be characterized by the fundamental rotational modes of 4-bar joints. • The repetitive structures with 4-bar joints and elbow beam member exhibited the wide ranges of Poisson’s ratio as joint flexibility changes. • Several kinematical motions of 8-bar jointed structure such as a deployment mechanism were presented. • We proposed the motion structure with eight rotational symmetry (MS-8) and revealed that it displayed the bi-stiffness property. 37

List of my papers 2. Repetitive Structures & their Joint Rotations Tanaka H. and Shibutani Y., J. Mech. Phys. Solids, Vol. 57 (2009) 3. Mechanisms of 8-bar-jointed Structures Tanaka H, et al., Int. J. Solids. Struct., Vol. 49 (2012) 4. Bi-stiffness Property of the Motion Structure Tanaka H, Proc. R Soc. A, Vol. 469 (2013) Thank you! 38

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