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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.


  1. 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.

  2. 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

  3. 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)

  4. 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

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

  6. 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

  7. 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

  8. 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

  9. 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

  10. 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

  11. 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

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

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

  14. 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

  15. 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

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

  17. 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

  18. 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

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

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

  21. 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

  22. 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

  23. 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

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

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

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

  27. 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

  28. 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

  29. 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

  30. 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

  31. 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

  32. 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

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

  34. 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

  35. 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

  36. 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

  37. 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

  38. 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

  39. 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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