mathematics the first equations more than tree struts tetrahedron dissimilar elliptical

The first equations

The mathematical relation between struts and strings of the most simple tensegrity is shown on this page.

A few definitions before we start calculating.

s = the length of a strut

a = the length of a string at the bottom of the tensegrity.
The three strings in this simple tensegrity form an equilateral triangle.

b = the length of the strings at the top of the tensegrity. These also form an equilateral triangle.

c = the length of a string that connects the top of one strut with the bottom of another strut.

tensegrity 113

A schematic picture of the same tensegrity is shown here. The length of string a as well as the position of the equilateral triangle are fixed (see the red triangle). The length of the struts s and the strings b are chosen as well. The only variation is the angle between the blue and the red triangle, which is illustrated by showing each string b and strut s in five different positions.

The strings c are not shown in this picture, and precisely the length of this string c has to be found with mathematics.

There is in fact only one possible position for the blue triangle formed by the strings b, so the question is: what is the angle between the blue and the red triangle? Or in more detail: which position of the blue triangle results in the shortest distance between one angular point of the red triangle and one of the blue triangle? Because it is string c that connects these two angular points and the only possible way that c will be taut between these two points is when it is just as long as the shortest distance possible.

tensegrity 114 What is the position of b compared to a?

One of the five positions of b in the picture above is shown in this drawing as well as a few extra lines that can help understanding the equations below.

The equations include the parameter z, which is equal to the height of the tensegrity (and is not shown in the picture).

With the help of Pythagoras one can figure out that:

s2 = (a + x)2 + y2 + z2
c2 = x2 + y2 + z2

(1) and (2) together gives:

c2 = s2 - a2 - 2*a*x

A tensegrity can only be stable when c is as short as possible. From equation (3) one can see that the largest x results in the smallest c. So xmax will result in the shortest and actual c.

tensegrity 115

This picture shows the largest possible x (given a specific a, b and s) without any further calculations. For everybody that looks at this picture it will be clear that it is impossible to find a larger x, but if one wants to see the "mathematical proof" one can read page 44 and 45 of Bob Burkhardt's A Practical Guide to Tensegrity Design.

xmax can be described as:

xmax = rb - ½ a

in which rb is the radiant of the circle in which triangle b fits (see the horizontal dotted line)

For every equilateral triangle the rb : b ratio is:

rb = ½ b / sin(60)

Equation (3), (4) en (5) give:

s2 - c2 = a * b / sin(60)

Equation (6) gives the final relation among the four lengths a, b, c and s.

tensegrity 116


Different simple tensegrities (all with only three equal struts) can be made with equation (6). The ratio between the different strings and struts is fixed by this equation:

s2 - c2 = a * b / sin(60)

You can turn a 3D image of this simple tensegrity on page 3D images

Marcelo Pars