# Different Configurations of the Non-Minimal Prismatic Tensegrities
## 2 Theory
### The Elastic Energy Approach  
Examples of non-minimal prismatic tensegrities are presented in Figure 5a,b. According to Equation (1), angle \(\alpha\) is a function of \(t\); therefore, the coordinates of the lower base can be written as  
\[n_{i}=[r\cos(\frac{2\pi}{t}\{i-1\}),\quad r\sin(\frac{2\pi}{t}\{i-1\}),\quad 0] \quad for\quad i=1:t, \tag{4}\]  
while for the upper base as  
\[n_{j}=[-r\sin(\theta+\frac{\pi}{t}\{2j-1\}),\quad r\cos(\theta+\frac{\pi}{t} \{2j-1\}),\quad h]\quad for\quad j=t+1:2t. \tag{5}\]  
In order to define the lengths of bars, cross cables and additional members, three vectors are created from four nodes, i.e., \(n_{1}\) (for \(i=1\)), \(n_{\mathrm{t+1}}\) (\(j=t+1\)), \(n_{\mathrm{2t-1}}\) (\(j=2t-1\)) and \(n_{\mathrm{2t}}\) (\(j=2t\)). Here, one node of the lower base is used, i.e., \(n_{1}\), and three nodes of the upper base are used, i.e., \(n_{\mathrm{t+1}}\), \(n_{\mathrm{2t-1}}\) and \(n_{\mathrm{2t}}\). Therefore, the coordinates of these nodes are as follows:  
\[\begin{array}{llll}n_{1}=[r,\quad 0,\quad 0],\quad n_{t+1}=[-r\sin(\theta+ \frac{\pi}{t}),\quad r\cos(\theta+\frac{\pi}{t}),\quad h],\\ n_{2t-1}=[-r\sin(\theta-\frac{3\pi}{t}),\quad r\cos(\theta-\frac{3\pi}{t}), \quad h],\quad n_{2t}=[-r\sin(\theta-\frac{\pi}{t}),\quad r\cos(\theta-\frac{ \pi}{t}),\quad h],\\ \end{array} \tag{6}\]  
while the lengths of members are \(b=|n_{t+1}n_{1}|\), \(s=|n_{2t}n_{1}|\) and \(a=|n_{2t-1}n_{1}|\). After simplifications the lengths of the prisms are as follows:  
\[\begin{array}{llll}h=\sqrt{b^{2}-2r^{2}[1+\cos\frac{\pi}{t}\sin\theta+\sin \frac{\pi}{t}\cos\theta]},\quad s=\sqrt{b^{2}-4r^{2}\sin\frac{\pi}{t}\cos \theta}\\ a=\sqrt{b^{2}-4r^{2}\sin\frac{2\pi}{t}[\sin\frac{\pi}{t}\sin\theta+\cos\frac{ \pi}{t}\cos\theta]}].\end{array} \tag{7}\]  
Figure 5: Non-minimal: (**a**) pentex, (**b**) hexaplex and (**c**) drawing for lengths definition.  
The elastic energy of the t-bar prism is:  
\[U=\frac{t}{2}k_{s}[s-s_{n}]^{2}+\frac{t}{2}k_{b}[b-b_{n}]^{2}+\frac{t}{2}k_{a}[a-a _{n}]^{2}+tk_{l}[2-2\cos(2\pi/t)][r-r_{n}]^{2}, \tag{8}\]  
where the last component of the equation originates from the relation between the radius and the base cable length established from the law of cosines as \(l=r\sqrt{2-2\cos(2\pi/t)}\). Now, finding the extremum of the energy \(U\) and comparing it to zero produces a system of three nonlinear equations (due to the fact, that the system is defined by three variables):  
\[\begin{cases}\partial U/\partial\theta=0\\ \partial U/\partial b=0\end{cases}\rightarrow\left\{\begin{array}{l}tk_{s}[s- s_{n}]\partial s/\partial\theta+tk_{a}[a-a_{n}]\partial a/\partial\theta=0\\ tk_{s}[s-s_{n}]\partial s/\partial b+tk_{b}[b-b_{n}]+tk_{a}[a-a_{n}]\partial a/ \partial b=0\\ tk_{s}[s-s_{n}]\partial s/\partial r+tk_{a}[a-a_{n}]\partial a/\partial r+2 tk_ {l}[2-2\cos(2\pi/t)][r-r_{n}]=0\end{array}\right. \tag{9}\]  
where partial derivatives are given in the Appendix A. Critical points, the local maximum and the local minimum, or saddle point, are verified by analyzing the following Hessian:  
\[\mathbf{H}(\theta,b,r)=\begin{bmatrix}\partial^{2}U/\partial\theta^{2}& \partial^{2}U/\partial\theta\partial b&\partial^{2}U/\partial\theta\partial r \\ \partial^{2}U/\partial b\partial\theta&\partial^{2}U/\partial b^{2}&\partial^{ 2}U/\partial b\partial r\\ \partial^{2}U/\partial r\partial\theta&\partial^{2}U/\partial r\partial b& \partial^{2}U/\partial r^{2}\end{bmatrix}. \tag{10}\]  
The second derivatives of the system, as well as the second derivatives of the energy, are also gathered in the Appendix A. The member forces are:  
\[N_{s}=k_{s}(s-s_{n}),\quad N_{b}=k_{b}(b-b_{n}),\quad N_{a}=k_{a}(a-a_{n}), \quad N_{l}=k_{l}(l-l_{n}), \tag{11}\]  
where \(N_{s},N_{b},N_{a},N_{l}\in R\) are axial forces in cross cables, bars, additional members and horizontal cables, respectively.  
A trivial solution to Equation (9) is \(\theta=0\), \(r=r_{n}\) and \(b=b_{n}\), which means no energy is in the system and the prism remain unstressed and unstable.