# TENSEGRITY TENSEGRITIES

## More than three struts

The picture shows a schematic drawing of a tensegrity with five struts. Each strut s and each tendon b is drawn in three different positions. The picture illustrates the following question: What position of the struts s reduces the distance c between the strut ends to a miminum?

The symbols in the equations have the following meaning:

s = the length of the struts

a = the length of the tendons at the bottom of the tensegrity.
The five tendons have the same size so automatically they form an equilateral pentagon.

b = the length of the tendons at the top of the tensegrity. They also form an equiletaral pentagon.

c = the length of the tendons that connect the top ends of the struts with the bottom ends of the struts. These tendons are not shown on this picture.

n = the number of struts This next picture shows one of the three strut positions from the picture above. In this picture the green tendon c is added as well as a few black lines that should simplify the following explanation.

One can use the same equation as in the former webpage:

 c2 = s2 - a2 - 2*a*x (3)

The equation implies the following: The larger the distance x the smaller the tendon c. The maximum x (resulting in the desired c) can be defined as:

 xmax = rb - ½ a (4)

So far no differences with the 3-strut tensegrity as described in the former webpage. The only difference is the length of radiant rb, of the circle in which the pentagon fits. The edge of the pentagon is b.

The relation between the radiant rb and edge b of an equillateral polygon is:

 rb = ½ b / sin(180/n) (7)

Equation (3), (4) and (7) result in:

 (s2 - c2) * sin(180/n) = a * b (8)

Equation (8) gives the relation among the four lengths a, b, c en s and the number of edges n (= the number of struts). But the top end of a strut does not have to be connected with the bottom end of the adjacent strut. It can also be connected with the bottom end of the next strut (the "second" strut). This is shown in this third picture.

In the picture the struts already have their optimum position, with the maximum length x. The length x is now defined by:

 s2 = (k + x)2 + y2 + z2 (9)
 c2 = x2 + y2 + z2 (10)

=>

 c2 = s2 - k2 - 2*k*x (11)

So, also in this case the longest c results in the shortest tendon c.

 xmax = rb - ½ k (12)

The length k in equation (11) and (12) is defined by:

 ½ k = ra * sin(180*2/n) (13)

The radiant ra can be defined the same way as rb:

 ra = ½ a / sin(180/n) (14)

Equations (11), (12), (13) en (14) give:

 ( s2 - c2 ) * sin2(180/n) = a * b * sin (180*2/n) (15)

Here, the value 2 in sin (180*2/n) is given by the fact that strut c connects the top end of one strut with the bottom end of the second consecutive strut.

Imagine v is the serial number that defines which struts are connected by tendon c (for instance v = 1 means one top end of a strut is connected with the first next strut bottom end. v = 2 means the top end of a strut is connected with the second strut bottom end, etc). The general equation is then as follows:

 ( s2 - c2 ) * sin2(180/n) = a * b * sin (180*v/n) (16) Summary

For simple one-layer tensegrities one can describe the relation among the lengths of struts and tendons and the number of struts as follows:

 ( s2 - c2 ) * sin2(180/n) = a * b * sin (180*v/n) (16)

in which:

s = the length of the struts

a = the length of the tendons at the bottom of the tensegrity.

b = the length of the tendons at the top of the tensegrity.

c = the length of the tendons that connect the top ends of the struts with the bottom ends of the struts.

n = the number of struts

v = a number that describes which two struts end (one top end and one bottom end) are connected.

Marcelo Pars