Optimization Methods for Force and Shape Design of Tensegrity Structures. J.Y. Zhang and M. Ohsaki Kyoto University, Japan

Transcription

1 Optimization Methods for Force and Shape Design of Tensegrity Structures J.Y. Zhang and M. Ohsaki Kyoto University, Japan

2 Contents Purpose Optimization techniques are effective for shape and force design of tensegrity structures. Concept & Applications Shape Design Energy Approach Direct Approach Minimum energy difference in cables and struts Minimum force deviation from target values Force Design Force & Stiffness Maximum stiffness Uniform forces Conclusions

3 Basics Tensegrity = Tension + Integrity (R.B. Fuller 1975) Assumptions Pin-jointed No external loads Struts (compression) + Cables (tension) No member failure Self-equilibrium Member forces Configuration Stiffness

4 Energy Approach? Cable Nets Cable nets: carry only tension.? Shape design problem: m 1 Minimize EA 2 Subject to? Variables: unstressed length, nodal locations? Stationary condition Equilibrium equation? Convex problem (Kanno & Ohsaki 2003) 2 0 i Li i 1 0 (1 i ) Li Bix di? Can specify member forces (or strain)

5 Energy Approach? Tensegrity Structures Tensegrity structures: tension + compression? Shape design problem: Minimize Subject to i Cable 1 1 EA L EA L i i i i i Strut 0 (1 i ) Li Bix di, i Cable 0 (1 i ) Li Bix di, i Strut? Variables: unstressed length, nodal locations? Stationary condition Equilibrium equation? Non-convex problem May not converge? Can specify member forces (or strain)? Unable to control member directions

6 Direct Approach? Objective & Idea? Direct assignment of member directions (Zhang et al. 2006) Fixed member Supported Free-standing Directed graph? Independent member directions? Difficulty for complicated structures

7 Direct Approach? Constraints? Two-step approach for form finding? Step 1: Find member directions (forces)? Step 2: Find nodal locations? Hard constraints (exactly satisfied) Hv = 0 Self-equilibrium equations Directions of fixed members Symmetry properties? Soft constraints (preferably satisfied) v cv Sv = 0 e.g., i j

8 Direct Approach? Member directions Step 1:? Minimize error from target values minimize subject to Hv = 0? Lagrangian? Stationary condition 1 T I 1 T II E( v) v v W v v ( Sv) W ( Sv) 2 2 L( v, ƒ) E( v) ƒ Hv I T II T I W S W S H v W v H O ƒ 0 T

9 Direct Approach? Nodal Coordinates Step 2:? Equilibrium equation w.r.t nodal coordinates FX 0? Rank of F < the number of unknown coordinates? Specify independent components of coordinates to obtain X

10 Direct Approach? Example 1 Top Side Perspective Hard constraints: Self-equilibrium equations Rotational symmetry Soft constraints: No 1 E ) T 1( v) ( vi vi ) ( vi vi 2 i K rank( F) 32 3n = 36 x, y, z, x ) (0, 0, 0,1.05) (

11 Direct Approach? Example 2 Soft constraint v 2v 7 19 Top Side Perspective Result v v 19 rank( F) 32 3n = 36 x, y, z, x ) (0, 0, 0,1.05) (

12 Force Design? Force Distribution Self-equilibrium Equation Ds 0 D? Equilibrium Matrix s? Member Force Configuration R: rank of D s f f f f i i m R m R fi? Force Mode Unknown Force Unknown Coefficient Force Deviation Stiffness Optimization Problem

13 Force Design? Stiffness Member Forces Configuration G E K K K E K M 0 stiffness Member Stiffness Q T G M K M M? mechanism Objective 1 Structure collapses in the weakest direction! Strengthen the Weakest Maximize the Minimum Eigenvalue of Q

14 Force Design? Force Deviation & Constraints Design Analysis Manufacture Construction Objective 2 Minimize deviation of member forces Constraints Positive force for cable Negative force for strut Given strain energy

15 Force Design? Formulation and Solution Objectives Maximize the Minimum Eigenvalue of Q Minimize deviation of member forces Constraints Positive force for cable Constraint Approach Negative force for strut Given strain energy Upper bound of force deviation

16 Force Design? Tensegrity Grid 38 Nodes 115 Members 8 Force Modes 1 Mechanism y x Top View Unit Cell Side View

17 Force Design? Pareto Optimality Maximum Stiffness Force Deviation Min. Force Deviation Pareto Optimality Target values: uniform distribution Maximum Stiffness

18 Conclusions? Optimization can be effectively used for shape and force design of tensegrity structures Shape Design Energy Difference Directed Graph Force Design? Specify member forces (strains)? Stationary condition satisfy self-equilibrium? Non-convex? Convergence problem? Direct assignment or force vectors? Member direction can be specified? Determine force components by optimization.? Shape and forces are controlled by modifying the target values and soft constraints.? Maximum stiffness? Minimum force deviation from target values? Pareto optimality to assist decision making