Rigidity And Tensegrity in Tensegrity structures

1. Rigidity and Tensegrity Robert Connelly Cornell University

2. Part I: The local theory Statics

3. What determines rigidity?

8. What model?

10. What sort of rigidity/stablility?

12. What energy function to choose?

17. An example of a stress matrix

20. Part II: The global theory Stress matrices applied again.

23. How do you tell when a given framework is globally rigid?

27. Part III: Bar frameworks More stress matrices using the generic philosophy.

38. Proof that the stress condition implies global rigidity The Tarski-Seidenberg theory of quantifier elimination implies that the situation below cannot happen, since the configuration p is generic. So a neighborhood of p can be mapped to a neighborhood of q by a diffeomorphism h, so that fh = f, and df p = df q dh p . This implies that any equilibrium stress for p is an equilibrium stress for q . If the rank of the associated stress matrix is maximal, this implies that p and q are affine images of each other, and ultimately that they are congruent.

49. An example of the Berg-Jordan Henneberg transformations

50. An example of the Berg-Jordan Henneberg transformations

51. An example of the Berg-Jordan Henneberg transformations

52. An example of the Berg-Jordan Henneberg transformations

53. An example of the Berg-Jordan Henneberg transformations

54. An example of the Berg-Jordan Henneberg transformations

55. An example of the Berg-Jordan Henneberg transformations

57. Part IV: Existence of realizations When does a graph have a realization with predermined edge lengths?