Coordinated motions of repetitive structures from a mechanical point - - PowerPoint PPT Presentation
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.
Workshop on “Geometric Structures with Symmetry and Periodicity” Organized by Ileana Streinu and Monique Teillaud
Hiro Tanaka,
June 8th, 2014, Kyoto
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square cells hexagonal cells triangular cells (truss) A polar diagram showing Young’s modulus Deformability of repetitive structures bending-dominated stretching-dominated (tensile load) bending-dominated (shear load) buckling and bending (compressive load) stretching-dominated
# Gibson LJ & Ashby MF (1997) # Deshpande VS et al. (2001) # Wicks N, Guest SD (2004)
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Mechanical property of two-dimensional repetitive structures Poisson’s ratio : conventional materials Indentation resistance : auxetic materials Examples of repetitive structures re-entrant hexagonal units: missing ribs square units:
# Dr. Borcea’s talk is coming soon
# Smith CW, et al. (2000) # Evans KE, et al. (2000)
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symmetry of rotational stiffness:
: i-th rotational displacent
Rotation energy of n-coordinate flexible joint (n-bar joint)
beam member
Hessian matrix of n-bar joint energy with respect to rotational displacements
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A class of circulant matrices Standard eigenvalue problem Calculating a pair of eigenvalue and eigenvector on n-joint In case of 4-bar joint
： 4-bar joint
four independent moments and rotational displacements: two types of rotational stiffness: (nearest-neighbor interaction) (2nd nearest-neighbor interaction)
Hessian matrix
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co-rotation mode (C4-invariance) asymmetry-rotation mode (C1-invariance) counter-rotation mode (C2-invariance) (multiple root)
Fundamental rotational modes of 4-bar joint
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analytical model
: to be a rigid joint : to be a pinned joint Buckling problem of periodical square cells under equi-biaxial compression joint flexibility joint modeling
4x4 cells
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Overall Buckling mode vs. local rotational mode mode 1 mode 2 mode 3 mode 4
co-rotations counter-rotations
co/counter-rotations co-rotation mode Asymmetry-rotation mode counter-rotation mode
joint flexibility: µ =1 Buckling mode vs. compression ratio (4 x 4 cells)
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Localized deformation of mode 2 consists of the co/asymmetry rotations quadruple mode
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Repetitive framework with 4-bar joints and elbowed beam members
： 4-bar joint
: to be a rigid joint : to be a pinned joint joint flexibility Poisson’s ratio where d1,2 is x1- or x2-displacement
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Model I Model II Two types of the structures
flexible joint rigid joint pinned joint
Three type of manufacturing aluminum joints
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substrate(SS400)：1200x1200 mm2 Overview of biaxial tensile tester
connected by rigid joints
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zero Poisson’s ratio
connected by pinned joints
Uniaxial tensile testing for Model I expanded in the lateral direction because the moment was laterally transferred via the co-rotations of rigid joints: Negative Poisson’s ratio
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Model I: Poisson ratio vs Nominal strain
Measurements of uniaxial tensile testing
Model II: Poisson ratio vs Nominal strain
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Out-of-plane deformation of Model II
Under the tensile load, the beam members (red) are compressed
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Novel 3D structure: applications of Model II
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proposed structural unit call the closed-loop unit with four pairs of rhombic elements (REs) as 4RE-linkage rhombic element unknown connection
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Motion of 4RE-linkage Motion of 4RE-linkage motion of a rhombic element closed-loop configuration: geometrical constraint conditions:
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Animations of the obtained motions
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Repetitive-assembly motions (2 x 2 cells)
initial configurations 5 6 7 2 3 4 1
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Deployment mechanism of φ3
a 8-bar joint counter rotation
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Manufactured model actuated by a single rotary operation
Minimum size: 320×320 mm2, Maximum size: 830×830 mm2 side view actuator
Combined assembly
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the structure built of the multiple rotational modes of 8-bar joints
mode 3 mode 4 mode 5
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Proposed structure
particular connections The adjacent bars painted in the same color, blue or red, are rigidly connected at all the pivot joints. Motion Structure with 8 rotational symmetry (MS-8)
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position of each grayish node
Geometrical configuration of MS-8
8-bar joint 6-bar joint 6-bar joint
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Motion I Motion II Mechanism of MS-8 The two patterns of square cells are tilted at 45˚ with respect to each other.
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Which motions does MS-8 select?
loading condition balance point of the square
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Numerical analyses of the proposed structure with the cell-to-cell contacts
for motion I for motion II stiff soft
Bi-stiffness property: switching of the stiffness due to two motions
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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.
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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.
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by the fundamental rotational modes of 4-bar joints.
exhibited the wide ranges of Poisson’s ratio as joint flexibility changes.
deployment mechanism were presented.
and revealed that it displayed the bi-stiffness property. Summary
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Tanaka H. and Shibutani Y., J. Mech. Phys. Solids, Vol. 57 (2009)
Tanaka H, et al., Int. J. Solids. Struct., Vol. 49 (2012)
Tanaka H, Proc. R Soc. A, Vol. 469 (2013)