## 2 Methods of Analysis
### Discrete Model of Geiger Dome  
The components of Geiger domes are modeled as a space finite tensegrity element \(e\) of [PERSON]'s modulus \(E_{e}\), density \(\rho_{e}\), cross-sectional area \(A_{e}\), and length \(L_{e}\) (Figure 1). This element is described by the elongation matrix  
\[\mathbf{B}^{e}=\begin{bmatrix}-\mathbf{c}_{x}&-\mathbf{c}_{y}&-\mathbf{c}_{z }&\mathbf{c}_{x}&\mathbf{c}_{y}&\mathbf{c}_{z}\end{bmatrix}\in\mathbb{R}^{1 \times 6}, \tag{2}\]where  
\[\mathrm{c}_{x}=\frac{L_{ex}}{L_{e}},\mathrm{c}_{y}=\frac{L_{xy}}{L_{e}},\mathrm{c} _{z}=\frac{L_{ez}}{L_{e}}\]  
are the directional cosines, where \(L_{ex}=x_{n_{2}}-x_{n_{1}}\), \(L_{ex}=y_{n_{2}}-y_{n_{1}}\), and \(L_{ex}=z_{n_{2}}-z_{n_{1}}\). Additionally, the inclusion of the initial stress \(\sigma_{0}\) is considered, and the influences of the external load on the static and dynamic behavior of the structure are considered. We take into account the following matrices describing the element:  
\[\begin{split}\mathbf{K}_{L}^{\epsilon}=\frac{E_{e}A_{e}}{L_{e}} \begin{bmatrix}\mathbf{I}_{0}&-\mathbf{I}_{0}\\ -\mathbf{I}_{0}&\mathbf{I}_{0}\end{bmatrix},\quad\mathbf{K}_{G}^{\epsilon}= \frac{A_{e}\epsilon_{0}}{L_{e}}\begin{bmatrix}\mathbf{I}&-\mathbf{I}\\ -\mathbf{I}&\mathbf{I}\end{bmatrix},\;\mathbf{K}_{GN}^{\epsilon}=\frac{N_{e}}{ L_{e}}\begin{bmatrix}\mathbf{I}&-\mathbf{I}\\ -\mathbf{I}&\mathbf{I}\end{bmatrix}\\ \mathbf{K}_{NL}^{\epsilon}=\frac{E_{e}A_{e}}{L_{e}^{2}}\begin{bmatrix} \mathbf{I}_{1}&-\mathbf{I}_{1}\\ -\mathbf{I}_{1}&\mathbf{I}_{1}\end{bmatrix}+\frac{E_{e}A_{e}}{L_{e}^{2}} \begin{bmatrix}\mathbf{I}_{2}&-\mathbf{I}_{2}\\ -\mathbf{I}_{2}&\mathbf{I}_{2}\end{bmatrix},\;\mathbf{M}^{\epsilon}=\frac{ \rho_{e}A_{e}}{\delta}\begin{bmatrix}\mathbf{2I}&\mathbf{I}\\ \mathbf{I}&\mathbf{2I}\end{bmatrix},\end{split} \tag{3}\]  
where \(\mathbf{K}_{L}^{\epsilon}\in\mathbb{R}^{\epsilon\times 6}\) is a linear stiffness matrix, \(\mathbf{K}_{G}^{\epsilon}\in\mathbb{R}^{6\times 6}\) is a geometric stiffness matrix depending on the initial stress \(\sigma_{0}\), \(\mathbf{K}_{GN}^{\epsilon}\in\mathbb{R}^{6\times 6}\) is a geometric stiffness matrix depending on the axial forces that result from the external load, \(\mathbf{K}_{NL}^{\epsilon}\in\mathbb{R}^{6\times 6}\) is a non-linear stiffness matrix, \(\mathbf{M}^{\epsilon}\in\mathbb{R}^{6\times 6}\) is a consistent mass matrix, and  
\[\mathbf{I}=\begin{bmatrix}1&0&0\\ 0&1&0\\ 0&0&1\end{bmatrix},\;\mathbf{I}_{0}=\begin{bmatrix}1&0&0\\ 0&0&0\\ 0&0&0\end{bmatrix},\;\mathbf{I}_{1}=\begin{bmatrix}2\Delta_{u_{1}}&\Delta_{u_{ 2}}&\Delta_{u_{2}}\\ \Delta_{u_{2}}&0&0\\ \Delta_{u_{1}}\Delta_{u_{2}}&\Delta_{u_{2}}\Delta_{u_{2}}&(\Delta_{u_{2}})^{2} \end{bmatrix}\]  
where \(\Delta_{u_{i}}=q_{i}^{(n_{2})}-q_{i}^{(n_{1})}(i=1,2,3)\) are the displacement increases between nodes of the element, i.e., \(q_{i}^{(n_{1})}\) is a displacement of second node, and \(q_{i}^{(n_{1})}\) is a displacement of the first node (Figure 1).  
The \(n\)-element structures \((e=1,2,\ldots,n)\) with \(m\) degrees of freedom described by a displacement vector \(\mathbf{q}(t)\) and loaded by forces applied at the nodes \(\mathbf{P}\) are considered:  
\[\mathbf{q}(t)=\begin{bmatrix}q_{1}&q_{2}&\ldots&q_{m}\end{bmatrix}^{T}\in \mathbb{R}^{m\times 1},\qquad\mathbf{P}=\begin{bmatrix}P_{1}&P_{2}&\ldots&P_{m} \end{bmatrix}^{T}\in\mathbb{R}^{m\times 1}. \tag{4}\]  
The elasticity of the structures is described by an elasticity matrix  
\[\mathbf{E}=diag\Big{[}\frac{E_{1}A_{1}}{L_{1}}\quad\frac{E_{2}A_{2}}{L_{2}} \quad\cdots\quad\frac{E_{e}A_{e}}{L_{e}}\Big{]}\in\mathbb{R}^{n\times n}, \tag{5}\]  
whereas the other matrices describing the structure, i.e., \(\mathbf{B}\in\mathbb{R}^{n\times m}\), \(\mathbf{K}_{L}\in\mathbb{R}^{m\times m}\), \(\mathbf{K}_{G}\in\mathbb{R}^{m\times m}\), \(\mathbf{K}_{NL}\in\mathbb{R}^{m\times m}\), and \(\mathbf{M}\in\mathbb{R}^{m\times m}\), are determined using finite element formalism [27; 28].