design and analysis of 2d repetitive pattern
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Analysis and design of elastic material formed using 2D repetitive slit pattern
Taisuke Ohshima[1], Tomohiro Tachi[1], Hiroya Tanaka[2], Yasushi Yamaguchi[1] ![1]The University of Tokyo , [2] Keio University
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・Kerfing / Dukta® [1] ・Zigzag spring / Serpentine spring [2] ・Lamina Emergent Mechanisms(LEM) [3]
2D repetitive slit pattern
[3]An Introduction to Multilayer Lamina Emergent Mechanisms L. Delimont et.al
[2]from the web
[1] [2]
[3]
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Applications of 2D repetitive slit pattern
‘Spring’ stool by Carolien Laro
[1] US-Patent by Apple in 2013
Elastic buffer
Elastic hinges
[1]“Interlocking flexible segments formed from a rigid material” US 2013/0216740 A1
Bending
Kerf Pavilion @ MIT
Actuator or Deployable structure
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[2] LEM
[2]”Fundamental Components for Lamina Emergent Mechanisms"
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Research questions
High stiffness Processed flexible
Repetitive Pattern Material (RPM)
-3D printing -CNC cutting
・How does this pattern enable materials to be flexible ?
・How do we utilize this patten for designing flexibility ?
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fig from the web (*1)
(*1) http://www.pontrilasmerchants.co.uk/products/mdf.php
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Table of contents
1. Modeled relationship between pattern and resulting flexibility
2. Experiment to evaluate this model
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3. Dimensional analysis that explains characteristics of this pattern
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f (a,b,l,n,E,G) = stiffness of RPM
pattern parameter : (a,b,l,n)material parameter : (E,G)
E :Yoiung 's modulusG : shear modulus
Local beam
Stiffness function & pattern parameter
We define stiffness function f
RPM
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・We view RPM as 1D elastic rod
Concept of our model
MBIP = EBIPIBIPφ MBOP = EBOPIBOPφ MT = GT JTφTPs = Ksds
< Stiffness function in each deformation >
fBIP (a,b,l,n,E,G)Stiffness function in BIP-mode
Stretching Bending in plane Bending out of plane Twisting
S-mode BIP-mode BOP-mode T-mode
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φT =dθTdx
θT
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Stiffness functionLocal beamGlobal elastic rod
∝E n3a4bl
Ks = E12na3bl 3
∝E n3ab4
l
∝G na3.4b1.8
l
Overview of our contribution
S-mode
BOP-mode
BIP-mode
T-mode
Equation of deformation
Ps = Ksds
MBOP = EBOPIBOPφ
MT = GT JTφ
MBIP = EBIPIBIPφ
pattern parameter8
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Stiffness function in stretching (S-mode)
Global elastic rod Local beam
Ps = Ksds
fs (a,b,l,n) = E12na3bl 3
< Parameter >PS
P = PS
ds
Stiffness function
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Stiffness function in bending out of plane (BOP)Global elastic rod Local beam
∵ J is torsion constant
∵φBOP =θBOP
a + gMBOP = EBOPIBOPφBOP
MBOP
M = MBOP
Stiffness function
θBOP
< Parameter >
fBOP (a,b,l,n)= G(a + g)J(a,b) (pure torsion)
∝ Gna3.5b1.6
l
⎧⎨⎪
⎩⎪∵G is material parameter (shear modulus)
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Stiffness function
EBIPIBIP ∝E n3a4bl
Ks = E12na3bl 3
GT JT ∝E n3ab4
l
EBOPIBOP ∝Gna3.4b1.8
l
Overview of our contribution
S-mode
BOP-mode
BIP-mode
T-mode
Equation of deformation
Ps = Ksds
MBOP = EBOPIBOPφ
MT = GT JTφ
MBIP = EBIPIBIPφ
pattern parameter
(*1) using warping torsion model
Local beamGlobal elastic rod
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Dimensional Analysis
< Parameter >pattern parameter :(a,b,l,n)material parameter :(E,G)
a lb n
4 1 -1 3(3.7)
32 -1 3
3 -3 11
1-1.11.83.4
a = 4mm,b = 5mm,l = 50mm, n = 2,1≤ a ≤ 5, 4 ≤ b ≤ 8,40 ≤ b ≤ 80,1≤ n ≤ 8
⎧⎨⎪
⎩⎪
(3.2)
(1.7)
S-mode
T-mode
BOP-mode
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EBIPIBIP ∝E n3a4bl
Ks = E12na3bl 3
GT JT ∝E n3ab4
l
EBOPIBOP ∝Gna3.4b1.8
l
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Suitable pattern for elastic hinge
・S-mode has high sensitivity about “l”・BIP- and T-mode have high sensitivity about “n”
Decreasing “l” and increasing “n” realize compliant in BOP -mode but stiff in the other modes
Sensitive parameter
Elastic hinges
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Experiment result in BOP-mode
Physical testComputer simulation
・Used medium density fiber broad (MDF)
・Measured load and displacement with three-point bending
・Tested multiple samples by scaling pattern parameter.
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Laminated material (MDF)(*1)
fiber !(stiff)
glue!(compliant)
G ≠ E2(1+υ)
Shear modulus G of laminated materials (MDF)
E = 1261MPA Giso = 934 MPA (isotropic)
Glm = 126 MPA (laminated)
Measured shear modulus
Measured shear modulus
Measured G is ten times lower than isotropic G
G = E2(1+υ)
(*1) 構造用複合材料 影山和郎著
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Dimensional Analysis
< Parameter >pattern parameter :(a,b,l,n)material parameter :(E,G)
a lb n
4 1 -1 3(3.7)
32 -1 3
3 -3 11
1-1.11.83.4
a = 4mm,b = 5mm,l = 50mm, n = 2,1≤ a ≤ 5, 4 ≤ b ≤ 8,40 ≤ b ≤ 80,1≤ n ≤ 8
⎧⎨⎪
⎩⎪
(3.2)
(1.7)
S-mode
T-mode
BOP-mode
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EBIPIBIP ∝E n3a4bl
Ks = E12na3bl 3
GT JT ∝E n3ab4
l
EBOPIBOP ∝Gna3.4b1.8
l
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Experiment result in BOP-mode(1)
x: l (mm) y: stiffness = Load/Dsiplacement (N/mm)
Physical results Simulation results (Warping torsion)Simulation results (Pure torsion)
< Parameter >
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Experiment result in BOP-mode(2)
x: a (mm) y: stiffness = Load/Dsiplacement (N/mm)
< Parameter >
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Physical results Simulation results (Warping torsion)Simulation results (Pure torsion)
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Conclusion
・Proposed model explains local beam deformation determines stiffness of RPM
・Experiment result indicates this model is valid in BOP-mode
・Dimensional analysis explains how stiffness of RPM scales with changing pattern parameter
・We propose design guideline for elastic hinge with dimensional analysis and experiment
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Future work
・Implementing system to simulate and design elastic bending(hinge)
・Modeling buckling condition of local beam
・Utilizing this pattern for deployable structure
・Finishing experiment for the other deformation cases
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Thank You For Listening