## 2 Methods of Analysis
### Simplest Two-Element Cable Structure  
The simplest two-dimensional structure is considered to explain in detail the process for determining the stiffness of the structures and the process of dynamic stability analysis. Despite the fact that it is not a tensegrity structure, its behavior fully reflects the behavior of tensegrity structures. This is because the structure is characterized by two inherent features of tensegrity systems, i.e., the self-stress state and an infinitesimal mechanism. For such a simple structure, it is possible to determine the stiffness and impact of the initial prestress level \(S\) on the unstable regions in explicit form. This approach makes it easier to understand the behavior of Geiger domes. Step 1--Compute the elongation matrix  
The truss, consisting of two elements \((n=2)\) with length \(l\) and stiffness \(EA\), characterized by two degrees of freedom \((m=2)\), is considered (Figure 3a). A force \(P\) applied to the 2 nd node in the vertical direction is taken into account (Figure 3b). The displacement vector (4)\({}_{1}\), vector of force (4)\({}_{2}\), and elongation matrix are as follows:  
\[\mathbf{q}= \begin{bmatrix}q_{1}\\ q_{2}\end{bmatrix} ,\ \ \mathbf{P}= \begin{bmatrix}0\\ P\end{bmatrix} ,\ \ \mathbf{B}= \begin{bmatrix}1&0\\ -1&0\end{bmatrix} . \tag{16}\]  
Step 2--Identification of immanent features of tensegrity structures  
Spectral analysis of matrix \(\mathbf{BB}^{T}\) leads to obtain two eigenvalues, one of which is zero \((\mu_{2}=0)\). The zero eigenvalue is related to the existence of one self-stress state, considered as an eigenvector related to the zero eigenvalue:  
\[\mathbf{BB}^{T}= \begin{bmatrix}1&-1\\ -1&1\end{bmatrix} \overset{(\mathbf{BB}^{T}-\mu\mathbf{I})\mathbf{y}=0}{ \rightarrow}\ \mathbf{\mu}= \begin{bmatrix}2&0\end{bmatrix} \rightarrow\mathbf{y}_{S}= \mathbf{y}_{2}(\mu_{2}=0)= \begin{bmatrix}1\\ 1\end{bmatrix} . \tag{17}\]  
Figure 3: Two-element structure: (**a**) geometry, (**b**) scheme of loading regime, (**c**) actual configuration. Spectral analysis of matrix \(\mathbf{B}^{T}\mathbf{B}\) leads to obtaining two eigenvalues, one of which is also zero \((\lambda_{2}=0)\). The zero eigenvalue is related to the existence of one mechanism considered as an eigenvector related to the zero eigenvalue:  
\[\mathbf{B}^{T}\mathbf{B}= \begin{bmatrix}2&0\\ 0&0\end{bmatrix}\begin{pmatrix}\mathbf{B}^{T}\mathbf{B}-\Delta I\mathbf{)} \mathbf{x}=0\\ \rightarrow\end{pmatrix} \mathbf{\lambda}= \begin{bmatrix}2&0\end{bmatrix}  \rightarrow\ \mathbf{x}_{S}=\mathbf{x}_{2}(\gamma_{2}=0)= \begin{bmatrix}0\\ 1\end{bmatrix} . \tag{18}\]  
_Step 3--Compute the stiffness matrix \(\mathbf{K}_{L}+\mathbf{K}_{G}(\mathbf{S})\)_  
The elasticity matrix (5) and the self-equilibrated normal forces (6) are as follows:  
\[\mathbf{E}=\frac{EA}{l} \begin{bmatrix}1&0\\ 0&1\end{bmatrix} \!,\quad\mathbf{S}=S \begin{bmatrix}1\\ 1\end{bmatrix} \!. \tag{19}\]  
Due to this, the stiffness matrix takes the following form:  
\[\mathbf{K}_{L}=\frac{2 EA}{l} \begin{bmatrix}1&0\\ 0&0\end{bmatrix} \!,\ \ \mathbf{K}_{G}(\mathbf{S})=\frac{2S}{l} \begin{bmatrix}1&0\\ 0&1\end{bmatrix} \!. \tag{20}\]  
_Step 4--Compute the geometric stiffness matrix \(\mathbf{K}_{GN}(\mathbf{N})\)_  
The non-linear equation of a static equilibrium (8) for a considered truss takes the following form:  
\[\left\{\frac{2 EA}{l} \begin{bmatrix}1&0\\ 0&0\end{bmatrix} +\frac{2S}{l} \begin{bmatrix}1&0\\ 0&1\end{bmatrix} +\frac{EA}{l^{3}} \begin{bmatrix}q_{1}^{2}&q_{1}q_{2}\\ q_{1}q_{2}&q_{2}^{2}\end{bmatrix} \right\} \begin{bmatrix}q_{1}\\ q_{2}\end{bmatrix} = \begin{bmatrix}0\\ -P\end{bmatrix} \!. \tag{21}\]  
Due to the symmetry of the structure and load, the displacement \(q_{1}\) is zero. To simplify the notation \(q{=-q_{2}}\), Equation (21) takes the following form:  
\[[S+N(P)]q=\frac{Pl}{2}, \tag{22}\]  
where  
\[N(P)=\frac{EA}{2}\Big{(}\frac{q}{l}\Big{)}^{2} \tag{23}\]  
is a longitudinal force caused by load \(P\).