Form finding f Tensegrity Structures Form‐finding of Tensegrity Structures テンセグリティ構造の形状決定 Jingyao ZHANG (張景耀) Jingyao ZHANG (張景耀) Dept. of Design & Architecture Nagoya City University, Japan January 13, 2015 Form-finding of Tensegrity Structures Jingyao ZHANG January 13, 2015
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Form finding f Tensegrity StructuresForm‐finding of Tensegrity Structuresテンセグリティ構造の形状決定
Jingyao ZHANG (張景耀)Jingyao ZHANG (張景耀)
Dept. of Design & Architecture
Nagoya City University, Japan
January 13, 2015
Form-finding of Tensegrity Structures Jingyao ZHANG
January 13, 2015
Contents 内容
Concept of Tensegrity Applications
概念応用 Applications
Stability Form-finding
Key Problems for Preliminary Design
応用安定性
形状決定
Intuition Approaches Analytical Approaches
直観的方法
解析的方法 Numerical Approaches
• Adaptive Force Density Methodi l i h d
数値的方法
適応軸力密度法動的緩和法• Dynamic Relaxation Method
• Non-linear Analysis Method• Optimization Method
動的緩和法非線形解析法最適化手法• Optimization Method 最適化手法
Form-finding of Tensegrity Structures Jingyao ZHANG
g ( p g) Intuition Approaches Analytical Approaches (using symmetry)y pp g y y Numerical Approaches
• Adaptive Force Density Method• Dynamic Relaxation Method• Non-linear Analysis Method
O ti i ti M th d• Optimization Method
Form-finding of Tensegrity Structures Jingyao ZHANG
Form-finding of Arch
Arch(compression)
CCatenary(tension)
Form-finding of Tensegrity Structures Jingyao ZHANG
Form-finding of Basilica and Expiatory Church of the Holy Family
Constructed from 1882UNESCO World Heritage Site in 2005Church of the Holy Family Site in 2005
Sand bag model to find the natural Sand bag model to find the natural shape (minimum bending moment)
Form-finding of Tensegrity Structures Jingyao ZHANG
Form-finding of Cable-netケーブルネットの形状決定
Frei Otto(1925~ )
Minimal area C Constant stress
Soap bubble Experiment Cable net
Form-finding of Tensegrity Structures Jingyao ZHANG
Soap-bubble Experiment Cable-net
West German Pavilion (1967) 1967 World Fair Expo in Montreal, Canada
Form-finding of Tensegrity Structures Jingyao ZHANG
Contents
Introduction to Tensegrity Applications Applications Stability Form-findingg
Intuition Approaches Analytical Approaches (using symmetry)y pp g y y Numerical Approaches
• Adaptive Force Density Methodp y• Dynamic Relaxation Method• Non-linear Analysis Method• Optimization Method
Form-finding of Tensegrity Structures Jingyao ZHANG
Polyhedral Symmetric Structures多面体対称
Tetrahedral
Hexahedral
P l h d l
Octahedral
Polyhedral
Icosahedral DodecahedralIcosahedral
Form-finding of Tensegrity Structures Jingyao ZHANG
Tensegrity Tower テンセグリティ・タワー
Needle Tower(18m, 1968)
N dl T II
by K. Snelson
Needle Tower II(30m, 1969)
y
Hirshhorn Museum & Sculpture Garden Washington D C
Kröller Müller Museum, Otterlo Holland
Form-finding of Tensegrity Structures Jingyao ZHANG
Garden, Washington, D.C. Otterlo, Holland
Self-equilibrium Equations 自己釣合い方程式
0EzEyEx Self-equilibrium Equations:
E: Force density matrixx,y,z: Nodal coordinates
y
/f li
q q q q q q + + + 0
/k k kq f lForce Density
1 2 3 1 2 3
1 4 5 5 4
q q q q q qq q q q q
E
+ + + = 0
2 5 6 6q q q qq q q
E
Sym.
Form-finding of Tensegrity Structures Jingyao ZHANG
3 4 6q q q y
Self-equilibrium & Super-stability Conditions
1=
h
x y z P0Ex 1
2, ,
1i i
i
x y z P
h
0Ex
h >= 4 Three-dimensional
E
G
EK E P.S.D. for super-stability
G
K EE
p y
E has four 0 eigenvaluesOther eigenvalues are positive S t bilit
Self-equilibrium
Form-finding of Tensegrity Structures Jingyao ZHANG
Other eigenvalues are positive Super-stability
Adaptive Force Density Method (AFDM)適応軸力密度法
Initial force densities
DefinitionDefinition
Force density matrix Set four zero eigenvalues
EigenvalueAnalysis
T i t
y Set four zero eigenvalues
1 O
Terminate if satisfied
Definition
EigenvalueAnalysis
*T T
* 1
n h
n h
E ΨΛΨ Ψ Ψ
O Definition
Update force densities Force density matrix
n
O
Least square solution
p Force density matrix
•J.Y. Zhang and M. Ohsaki
Form-finding of Tensegrity Structures Jingyao ZHANG
gForm-finding of tensegrity structures subjected to geometrical constraints, Int. J. Space Structures, Vol. 21, 4, pp. 183-195, 2006.
Tensegrity Tower by AFDM
Unit cell
SymmetryMerit: Guarantee on super-stability
Low computational costs
Ten-story tower
Symmetry about z-axis Specified
z-coordinates
Low computational costsGeometrical constraints
Demerit: Less accurate shape control
Form-finding of Tensegrity Structures Jingyao ZHANG
yDemerit: Less accurate shape control
Contents
Introduction to Tensegrity Applications Applications Stability Form-findingg
Intuition Approaches Analytical Approaches (using symmetry)y pp g y y Numerical Approaches
• Adaptive Force Density Method• Dynamic Relaxation Method• Non-linear Analysis Method• Optimization Method
Form-finding of Tensegrity Structures Jingyao ZHANG
Dynamic Relaxation Method (DRM) 動的緩和法
Initial configurationDs = pEquilibrium Equation
Out-of-balance
New configuration
Motion EquationD: Equilibrium matrixs: Axial force vectorp: Out-of-balance force Out-of-balanceg
Motion Equation Motion Equation
Reset velocity to zero if kinetic energy reaches its peak value;
Terminate if the peak value is small
Mx+Cx+Kx = p 2 / 2E Mx Terminate if the peak value is small.
M: Mass matrixC: DampingK: Stiffness matrix
Final configuration
K: Stiffness matrixx: Displacement vectorE: Kinetic energy Self-equilibrated
•J Y Zhang and M Ohsaki
Form-finding of Tensegrity Structures Jingyao ZHANG
•J.Y. Zhang and M. Ohsaki, Free-form Design of Tensegrity Structures by Dynamic Relaxation Method, Proc. IASS-IACM 2012, Sarajevo, Bosnia and Herzegovina, April 2012.
Tensegrity Tower by DRM
Initial
Kinetic energy
Merit: Close to designed shape
I iti l Fi lFinal
Merit: Close to designed shape
Demerit: Low convergence
Form-finding of Tensegrity Structures Jingyao ZHANG
Initial Final
Contents
Introduction to Tensegrity Applications Applications Stability Form-findingg
Intuition Approaches Analytical Approaches (using symmetry)y pp g y y Numerical Approaches
• Adaptive Force Density Method• Dynamic Relaxation Method• Non-linear Analysis Method• Optimization Method
Form-finding of Tensegrity Structures Jingyao ZHANG
Non-linear Analysis (NLA) 非線形解析法
D: Equilibrium matrixs: Prestress vectorf: Unbalanced force vectorU ba a ced o ce ectoK: Stiffness matrixd: Nodal coordinates
Ds f・ConfigurationP t
Sufficiently small
END
Initial
Ds fKd = fUpdate
coordinates
・Prestresses
1d = K f
coordinatesprestresses
Full-rank
Form-finding of Tensegrity Structures Jingyao ZHANG