Tensegrities and Rigidity by Thomas

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Introduction History Stability Rigidity Tensegrities Stress Energy Tensegrities

Tensegrities and Rigity

Matthew Thomas

March 12, 2008

Matthew Thomas Tensegrities and Rigity
http://find/http://goback/


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Definitions for Tensegrities

Pairs of points designated:

cables - constrained not to get further apart

struts - constrained not to get closer togetherFor cables, think of string.

For struts, think of springs.

Some people also include bars, which have fixed length.



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History

1948 - Ken Snelson

Figure: Forest Devil, 1975, stainless steel, 34.5 x 68 x 51 inches



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Snelson

Figure: Needle Tower, 1968, aluminum & stainless steel, 60 x 20 x 20 feet


I d i Hi S bili Ri idi T i i S E T i i


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History

B. Fuller came up with the name for the tensegrity, named for

tensional integrity structures.

Russian K. Loganson may have had similar ideas predating Snelson.


I t d ti Hi t St bilit Ri idit T iti St E T iti


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Stability

Super Stability of a tensegrity means that all other tensegritieswith the same underlying graph either violate one of the distanceconstraints or are congruent to the given tensegrity - these couldbe in a different dimension.

Rigidity of a tensegrity means that any continuous motion of thevertices which preserve the cable and strut conditions extends to

an isometry of the ambient space.




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Rigid But Not Super Stable

All elements are bars.




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More Definitions

Two configurations p and q are congruent if every distancebetween vertices of p is the same for the corresponding

distance for corresponding vertices of q.

A tensegrity structure with configuration p is rigid if everyother configuration q sufficiently close to p satisfying thecable, bar, and strut constraints is congruent to p.




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Stable Structures

Blue = CableRed = Strut




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Another Stable Structure




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Types of Rigidity

We might define rigidity in one of the following two ways:

Infinitesimal rigidity defined in terms of infinitesimaldisplacements, i.e. velocity vectors.

Static rigidity defined in terms of forces and loads on thestructure.

Infinitesimal rigidity can be thought of as elasticity, while staticrigidity can be thought of as dealing with forces. These turn out to

be the equivalent.




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y y g y g gy g

Infinitesimal Motions and Flexes

Let {i, j} denote the cable/bar/strut connecting pi and pj. Aninfinitesimal flex, p, of a tensegrity structure is a vector pi

assigned to each vertex pi of the tensegrity such that:(pi pj)(pi

pj) 0 when {i, j} is a cable.

(pi pj)(pi pj

) = 0 when {i, j} is a bar.

(pi pj)(pi pj

) 0 when {i, j} is a strut.




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An example




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A Simple Tensegrity

Green = StrutBlue = Cable




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Types of Tensegrities

Bar frameworks - all barsSpider Webs - all cablesCircle/Sphere Packing - all struts




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Circle Packing

Red = Strut

Boundaries are pinned




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Stresses

A stress is an assignmnet of real scalars to pairs, {i,j},wi,j = wj,i. If {i,j} is not a member of the tensegrity (i.e. pi

and pj are not connected by a strut, cable, or bar) we may saywi,j = 0.

A proper stress is one in which wi,j = wj,i 0 when {i,j} is acable and wi,j = wj,i 0 when {i,j} is a strut. There is nocondition for bars.




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Equilibrium Stress

A stress on a tensegrity is an equilibrium stress if for eachnon-pinned vertex (if non-pinned vertices are included),i

wi,j(pj pi) = 0.

Note that wi,j is a stress, and everything else is a vector.




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A Simple Example




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Stress Example

We consider our simple tensegrity again.




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Stress Example

To find the equilibrium stress (or self-stress), we first label out

vertices and set up our equation. We need to fix some stresses, sowe will choose 1 for the stress of each cable. (We could have left itvariable.)




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Stress Example

We can find the stress w3,2 = w2,3. Notice we should have

w1,3

01

+ w2,3

11

+ w4,3

10

=

00

.

w1,3 = 1 and w4,3 = 1, so w2,3 = 1.




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Other Examples

We may have modifications to this equation if we have pinnedvertices.

As an example, consider a spider web, where we might havewi,j > 0 i,j.




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Energy

If we begin with a stress for a graph G, we can define astress-energy for our configuration where q = (q1, q2, q3, . . . , qn)

by Ew(q) =i


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Critical Points ofEw

Proposition

A configuration p is a critical point for Ew p is in equilibriumwith respect to the stress w.

Sketch of Proof:Let p be a critical point of Ew. Let p be an arbitrary direction,

with pi = 0 ifpi is a fixed vertex.

Ew(p+ tp) =

i


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Proof Continued

Now, at t = 0,dEw

dt= 2

wi,j (pi pj) (p

i p

j) = 0. Now let

p be 0 except in one non-pinned coordinate.

2j

wi,j (xi xj) = 0 for the x-coordinate. Doing this at all

unpinned vertices, we get the equilibrium condition. Theequilibrium condition implies the critical point because we have abasis for all ps.




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Spider Webs

Theorem

Let G be a spiderweb graph (all cables). This means that allnon-pinned vertices are connected by a chain of cables to a pinnedvertex. If G(p) is in equilibrium stress with respect to a nonzeroproper equilibrium stress, then G(p) is globally rigid.




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Stress Matrices

The symmetric matrix ij is defined as

0 0 0 0...

...0 1 1 0

... ...0 1 1 0...

...0 0 0 0

Which has a 1 in the ii and jj slot, and -1 in the ij and ji slot.The stress matrix associated with stress w is =

i


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Example of Stress Matrix

Since the ij-th entry of is wi,j, the stress matrix for our simple

tensegrity is

1 1 1 11 1 1 11 1 1 1

1 1 1 1

coming from




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Stresses

Notice that this matrix is positive semi-definite. This can be shown

to always be true. This gives us a sense of how to determinerigidity in tensegrities.




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Returning to Tensegrities

Computing coordinates with set stress:http://www.math.cornell.edu/~mdt29/maple/

Computing coordinates with variable stress:http://www.math.cornell.edu/~mdt29/varmaple/

Images of Tensegrities with symmetries of symmetric groups:http://www.math.cornell.edu/~mdt29/

http://www.math.cornell.edu/~mdt29/maple/http://www.math.cornell.edu/~mdt29/maple/http://www.math.cornell.edu/~mdt29/maple/http://www.math.cornell.edu/~mdt29/varmaple/http://www.math.cornell.edu/~mdt29/varmaple/http://www.math.cornell.edu/~mdt29/varmaple/http://www.math.cornell.edu/~mdt29/http://www.math.cornell.edu/~mdt29/http://www.math.cornell.edu/~mdt29/http://www.math.cornell.edu/~mdt29/http://www.math.cornell.edu/~mdt29/varmaple/http://www.math.cornell.edu/~mdt29/maple/http://find/http://goback/

Tensegrities and Rigidity by Thomas

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