Statically Balanced Tensegrity Mechanisms By Schenk

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Statically Balanced Tensegrity Mechanisms By Schenk

  1. 1. Theory and Design of Statically Balanced Tensegrity Mechanisms Graduation Colloquium Mark Schenk Department of BioMechanical Engineering Faculty of Mechanical, Maritime and Materials Engineering Exam Committee prof. dr. ir. Peter Wieringa dr. ir. Just Herder dr. Simon Guest dr. ir. Arend Schwab
  2. 2. Contents <ul><li>Introduction </li></ul><ul><li>tensegrity structures </li></ul><ul><li>static balancing </li></ul><ul><li>MSc. thesis </li></ul><ul><li>objectives </li></ul><ul><li>Zero Stiffness Tensegrity Structures </li></ul><ul><li>(zero) stiffness </li></ul><ul><li>Prototype </li></ul><ul><li>Conclusions </li></ul>statically balanced tensegrity mechanisms
  3. 3. Tensegrity Structures – introduction & examples – <ul><li>Special class of bar framework </li></ul><ul><li>pin-jointed </li></ul><ul><li>prestressed </li></ul><ul><li>pure compression, pure tension </li></ul><ul><li>-> cables and bars </li></ul><ul><li>pretension required for stiffness </li></ul><ul><li>-> tens ile int egrity = tensegrity </li></ul><ul><li>-> delicate balance </li></ul>Needle Tower II , 1969 Kröller-Müller Museum
  4. 4. Tensegrity Structures – engineering applications – <ul><li>Engineering applications: </li></ul><ul><li>obvious architectural / design appeal </li></ul><ul><li>light-weight </li></ul><ul><li>stowable / deployable </li></ul><ul><li>biologically inspired </li></ul>
  5. 5. Static Balancing – introduction & applications – <ul><li>Static Balancing </li></ul><ul><li>= continuous equilibrium </li></ul><ul><li>= neutral stability </li></ul><ul><li>= zero stiffness (Herder 2001) </li></ul><ul><li>Energy-free / energy-efficient design </li></ul><ul><li>e.g. rehabilitation, robotics </li></ul>
  6. 6. Static Balancing – zero-free-length springs – <ul><li>Zero-free-length springs </li></ul><ul><li>tension is proportional to length </li></ul>normal pretensioned zero-free-length
  7. 7. Static Balancing – design methodology & limitations – <ul><li>Current design methodology </li></ul><ul><li>basic designs are extended </li></ul><ul><li>by means of modification rules </li></ul><ul><li>complex results possible </li></ul><ul><li>Limitations </li></ul><ul><li>ad hoc solutions, no real generic understanding </li></ul><ul><li>bottom-up vs. top-down </li></ul><ul><li>“ limited” to 2D </li></ul>
  8. 8. Statically Balanced Tensegrity Mechanisms – a combination of two fields – <ul><li>tensegrity structures + static balancing </li></ul>= zero stiffness tensegrity structures = statically balanced tensegrity mechanisms Previous examples: (Herder, 2001)
  9. 9. Statically Balanced Tensegrity Mechanisms – applications & relevance – <ul><li>Totally new class of structures / mechanisms </li></ul><ul><li>robotics, rehabilitation, deployable structures </li></ul><ul><li>Academic interest </li></ul><ul><li>new insights for static balancing -> generic theory? </li></ul><ul><li>tensegrities <-> 3D pin-jointed structures </li></ul>
  10. 10. Graduation project – title & objective – <ul><li>Title </li></ul><ul><li>“ Theory and Design of Statically Balanced Tensegrity Mechanisms.” </li></ul><ul><li>Objectives </li></ul><ul><li>fundamental understanding zero stiffness tensegrities </li></ul><ul><li>-> generic description of static balancing </li></ul><ul><li>-> develop design guidelines </li></ul><ul><li>build demonstration model </li></ul>
  11. 11. Graduation project – research approach – <ul><li>Theoretical research : zero stiffness </li></ul><ul><li>Stiffness of tensegrity structures </li></ul><ul><li>engineers </li></ul><ul><li>structural engineering </li></ul><ul><li>mathematicians </li></ul><ul><li>rigidity theory </li></ul>combination
  12. 12. Stiffness of Structures – tangent stiffness matrix – <ul><li>Stiffness of a structure </li></ul><ul><li>counting rules (e.g. Maxwell’s rule) </li></ul><ul><li>tensegrity is prestressed </li></ul><ul><li>tangent stiffness matrix </li></ul><ul><li>Tangent stiffness matrix </li></ul><ul><li>displacements d and forces F </li></ul><ul><li>formulations -> zero-free-length springs </li></ul>
  13. 13. Tangent Stiffness Matrix - zero stiffness - <ul><li>Zero stiffness </li></ul><ul><li>Two components: </li></ul><ul><li>both components are zero </li></ul><ul><li>components cancel out </li></ul>
  14. 14. Zero Stiffness Tensegrity – conventional structures – <ul><li>Conventional interpretation </li></ul><ul><li>internal mechanisms (undesirable) </li></ul><ul><li>stabilized by self-stress </li></ul>P δ L/2 L/2
  15. 15. Zero Stiffness Tensegrity – structures with zero-free-length springs – <ul><li>Zero stiffness tensegrity structures </li></ul><ul><li>affine transformations (scaling/shear) </li></ul><ul><li>preserve length of conventional members </li></ul>
  16. 16. Zero Stiffness Tensegrity - example structure - top side
  17. 17. Zero Stiffness Tensegrity – length-preserving affine & conic – <ul><li>Length-preserving affine transformation </li></ul><ul><li>iff all bar directions lie on conic </li></ul><ul><li>infinitesimal -> finite </li></ul><ul><li>only zero-stiffness modes </li></ul><ul><li>Number of zero-stiffness modes </li></ul><ul><li>no. of unique bar directions k on conic </li></ul><ul><li>k ≥ 6 : zero-stiffness modes = 1 </li></ul><ul><li>k < 6 : zero-stiffness modes = 6 - k </li></ul>
  18. 18. Zero Stiffness Babytoy <ul><li>Babytoy </li></ul><ul><li>6 bars </li></ul><ul><li>3 bar directions </li></ul><ul><li>direction lie on conic </li></ul><ul><li>k ≥ 6 : zero-stiffness modes = 1 </li></ul><ul><li>k < 6 : zero-stiffness modes = 6 - k </li></ul><ul><li>3 zero-stiffness modes </li></ul>
  19. 19. Prototype Structure - description - <ul><li>Prototype structure </li></ul><ul><li>demonstrate properties </li></ul><ul><li>from structure to mechanism </li></ul><ul><li>Prototype design </li></ul><ul><li>classic tensegrity structure </li></ul><ul><li>no zero-free-length springs </li></ul><ul><li>3 bars, 9 springs </li></ul><ul><li>-> 3 springs on each bar </li></ul>
  20. 20. Prototype Structure - final model - <ul><li>… but, there is a lot of friction </li></ul>
  21. 21. Conclusions <ul><li>Theory and Design of Statically Balanced Tensegrity Mechanisms </li></ul><ul><li>theory of statically balanced tensegrity mechanisms </li></ul><ul><ul><li>length-preserving affine transformations </li></ul></ul><ul><ul><li>conventional member directions on conic </li></ul></ul><ul><ul><li>finite mechanisms </li></ul></ul><ul><li>design of a statically balanced tensegrity mechanism </li></ul><ul><ul><li>working prototype </li></ul></ul>
  22. 22. Questions
  23. 23. Previous examples: (Herder, 2001) 0 @ ® ¯ ° ± ² µ ^ K b l a b l i e b 1 A

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