Megascaling Rule: Conservation of Tensile Cross-Section across Fractal Dimensions

Leonardo noted that tree structure usually follows a simple conservation rule, that equivalent cross-sectional area is maintained at every branching scale. Thus tree-trunk cross-section equals branch cross-section equals twig cross-section. This is attributed to the tree's need for optimal strength and extension with minimal mass.

This rule approximates optimal megscaling of soft kites. Starting with the main rope(s) of a kite as the the thickest "trunk," a continual branching progression of scaling jumps occurs all the way down to the thread or membrane thickness. The diffuse kinetic energy of the wind "fans-in" by degrees to the main ropes to match the concentrated kinetic energy of the anchor, resulting in maximal static force handling. A branching factor of 2 is the natural minimum, with the ideal number of branching steps determined by the ratio of the wind load on a twig to the total load on the trunk.

This is how Mothra loadpaths are generally sized, with each major loadpath branching as smoothly as practical. It is rigger's art to choose COTS loadpath ropes to form a nearly ideal sequence, and especially to design the junctions to maintain full strength, avoiding knots or heavy expensive shackles with splices and soft-shackle methods.

Unlike trees, with severe scaling limits imposed by cantilever tower structure, branching tensile networks seem to mostly avoid normal scaling laws. Two factors in particular help; that the cross-sectional strength of a rope grows at the square of bluff-body drag, and that the energy of the airmass processed by a wing grows at the square of membrane area.

These advantages and the ready methods help predict soft-kite megascaling feasibility, even though more study remains to find the technological limits. Many subtleties to the fractal branching system are noted in trees- Leaves fail first to unload wind peaks. The progression of flexible compliance and statistical load averaging ensures that the trunk rarely fails before its branches.

http://phys.org/news/2012-01-leonardo-da-vinci-tree.html

• CoolIP*  and CC BY NC SA        ~Dave Santos         1Jan2013           AirborneWindEnergy/message/8319

Comment and development of this topic will be occurring here.       
All, send notes, links, drawings, papers, videos, plans, safety-critical findings, and photographs!

• Terms and aspects:   
  • v
• Related links and concepts:
  •  
• Commentary is welcome:
  • v