Optimal complexity of deployable compressive structures
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The usual use of fractals involves selfsimilar geometrical objects to fill a space, where the selfsimilar iterations may continue ad infinitum. This is the first paper to propose the use of selfsimilar mechanical objects that fill an alloted space, while achieving an invariance property as the selfsimilar iterations continue (e.g. invariant strength). Moreover, for compressive loads, this paper shows how to achieve minimal mass and invariant strength from selfsimilar structures. The topology optimization procedure uses selfsimilar iteration until minimal mass is achieved, and this problem is completely solved, with global optimal solutions given in closed form. The optimal topology remains independent of the magnitude of the load. Mass is minimized subject to yield and/or buckling constraints. Formulas are also given to optimize the complexity of the structure, and the optimal complexity turns out to be finite. That is, a continuum is never the optimal structural for a compressive load under any constraints on the physical dimension (diameter). After each additional selfsimilar iteration, the number of bars and strings increase, but, for a certain choice of unit topology shown, the total mass of bars and strings decreases. For certain structures, the string mass monotonically increases with iteration, while the bar mass monotonically reduces, leading to minimal total mass in a finite number of iterations, and hence a finite optimal complexity for the structure. The number of iterations required to achieve minimal mass is given explicitly in closed form by a formula relating the chosen unit geometry and the material properties. It runs out that the optimal structures produced by our theory fall in the category of structures we call tensegrity. Hence our selfsimilar algorithms can generate tensegrity fractals. © 2009 The Franklin Institute.
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Skelton, R. E., & de Oliveira, M. C.
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