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
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Introduction History Stability Rigidity Tensegrities Stress Energy Tensegrities
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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Introduction History Stability Rigidity Tensegrities Stress Energy Tensegrities
History
1948 - Ken Snelson
Figure: Forest Devil, 1975, stainless steel, 34.5 x 68 x 51 inches
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Introduction History Stability Rigidity Tensegrities Stress Energy Tensegrities
Snelson
Figure: Needle Tower, 1968, aluminum & stainless steel, 60 x 20 x 20 feet
Matthew Thomas Tensegrities and Rigity
I d i Hi S bili Ri idi T i i S E T i i
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Introduction History Stability Rigidity Tensegrities Stress Energy Tensegrities
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.
Matthew Thomas Tensegrities and Rigity
I t d ti Hi t St bilit Ri idit T iti St E T iti
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Introduction History Stability Rigidity Tensegrities Stress Energy Tensegrities
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.
Matthew Thomas Tensegrities and Rigity
Introduction History Stability Rigidity Tensegrities Stress Energy Tensegrities
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Introduction History Stability Rigidity Tensegrities Stress Energy Tensegrities
Rigid But Not Super Stable
All elements are bars.
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Introduction History Stability Rigidity Tensegrities Stress Energy Tensegrities
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Introduction History Stability Rigidity Tensegrities Stress Energy Tensegrities
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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Introduction History Stability Rigidity Tensegrities Stress Energy Tensegrities
Stable Structures
Blue = CableRed = Strut
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Introduction History Stability Rigidity Tensegrities Stress Energy Tensegrities
Another Stable Structure
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Introduction History Stability Rigidity Tensegrities Stress Energy Tensegrities
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.
Matthew Thomas Tensegrities and Rigity
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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/
Matthew Thomas Tensegrities and Rigity
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