Linear Form Finding Approach for Regular and Irregular Single Layer Prism Tensegrity

Abstract

In an irregular prism tensegrity, the number of force equilibrium equations is less than the number of unknown parameters of nodal coordinates and member force ratios. As a result, the form-finding process normally becomes nonlinear with additional conditions or needs to be carried out with the use of iterative procedures. For cases of irregular prism tensegrity which involves large number of members, it was found that previously proposed methods of form-finding are not practical. Moreover, there is a need for a form-finding approach which is able to cater to different requirements on final configuration. In this paper, the length relation condition is introduced to be used in combination with the force equilibrium equation. With the combined use of length relation and equilibrium conditions, a linear form-finding approach for irregular prism tensegrity was successfully formulated and developed. An easy-to-use interactive form-finding tool has been developed which can be used for form-finding of irregular prism tensegrities with large number of elements as well as under diverse specific requirements on their configurations.

Appendix

Process of derivation of Eq. (7) is shown in this appendix.

Equations (5) and (6) are first reproduced:

$$\left\{{\begin{array}{l} {\bar{F}_{{d_{1} }} + \bar{F}_{{d_{n} }} + \bar{F}_{{t_{1} }} \cos \gamma_{1} + \bar{F}_{{c_{n} }} \cos \delta_{n} = 0} \hfill \\ {\bar{F}_{{t_{1} }} \sin \gamma_{1} + \bar{F}_{{c_{n} }} \sin \delta_{n} = 0} \hfill \\ \end{array} } \right.$$

(5)

$$\left\{{\begin{array}{l} {\sum \bar{L}_{{\left( {x,y} \right)}} = \bar{L}_{{l_{n} }} + \bar{L}_{{t_{1} }} \cos \gamma_{1} + \bar{L}_{{c_{n} }} \cos \delta_{n} = \bar{0}} \hfill \\ {\sum \bar{L}_{\text{z}} = \bar{L}_{{t_{1} }} \sin \gamma_{1} + \bar{L}_{{c_{n} }} \sin \delta_{n} = \bar{0} } \hfill \\ \end{array} } \right.$$

(6)

From the second equation of Eqs. (5) and (6):

$$\bar{F}_{{t_{1} }} \sin \gamma_{1} + \bar{F}_{{c_{n} }} \sin \delta_{n} = \bar{0}$$

$$\left( {\bar{L}_{{t_{1} }} \sin \gamma_{1} + \bar{L}_{{c_{n} }} \sin \delta_{n} } \right)\left( { - {\text{k}}} \right) = \bar{0}$$

which can be combined to yield:

$$\left( {\bar{L}_{{t_{1} }} \sin \gamma_{1} + \bar{L}_{{c_{n} }} \sin \delta_{n} } \right)\left( { - {\text{k}}} \right) = \bar{F}_{{t_{1} }} \sin \gamma_{1} + \bar{F}_{{c_{n} }} \sin \delta_{n} = \bar{0}$$

Since ($\bar{L}_{{t_{1} }}$, $\bar{F}_{{t_{1} }}$) and ($\bar{L}_{{c_{n} }}$,$F_{{c_{n} }}$) are pairs of vectors with similar orientations,

$$\bar{L}_{{t_{1} }} \left( { - {\text{k}}} \right) = \bar{F}_{{t_{1} }}$$

$$\bar{L}_{{c_{n} }} \left( { - {\text{k}}} \right) = \bar{F}_{{c_{n} }}$$

From the first equation in Eq. (6),

$$\left( {\bar{L}_{{l_{n} }} + \bar{L}_{{t_{1} }} \cos \gamma_{1} + \bar{L}_{{c_{n} }} \cos \delta_{n} } \right)\left( { - {\text{k}}} \right) = \bar{0}$$

$$\bar{L}_{{l_{n} }} \left( { - {\text{k}}} \right) + \bar{L}_{{t_{1} }} \left( { - {\text{k}}} \right)\cos \gamma_{1} + \bar{L}_{{c_{n} }} \left( { - {\text{k}}} \right)\cos \delta_{n} = \bar{0}$$

which can be rewritten as,

$$\bar{L}_{{l_{n} }} \left( { - {\text{k}}} \right) + \bar{F}_{{t_{1} }} \cos \gamma_{1} + \bar{F}_{{c_{n} }} \cos \delta_{n} = \bar{0}$$

$$\bar{F}_{{t_{1} }} \cos \gamma_{1} + \bar{F}_{{c_{n} }} \cos \delta_{n} = \bar{L}_{{l_{n} }} \left( {\text{k}} \right)$$

Considering the first equation in Eqs. (5), (7) is finally obtained:

$$\bar{F}_{{d_{1} }} + \bar{F}_{{d_{n} }} + {\text{k}}\bar{L}_{{l_{n} }} = \bar{0}$$