# Parametric Analysis of Tensegrity Plate-Like Structures: Part 1--Qualitative Analysis
## 2 Mathematical Description  
The space truss finite element of [PERSON]'s modulus \(E^{\prime}\), cross-sectional area \(A^{e}\) and length \(L^{e}\) (Figure 5a) are used to model the components of the tensegrity structure. In a global coordinate system \((x,\ y,\ z)\) a finite element is described by the compatibility matrix \(\mathbf{B}^{e}(\in\mathbb{R}^{1\times 6})\):  
\[\mathbf{B}^{e}=\left[ \begin{array}{cccc}-c_{x}&-c_{y}&-c_{z}&c_{x}&c_{y}&c_{ z}\end{array} \right], \tag{1}\]  
where \(c_{x}=\frac{x_{j}-x_{j}}{L^{e}}\), \(c_{y}=\frac{y_{j}-y_{j}}{L^{e}}\), \(c_{z}=\frac{z_{j}-z_{i}}{L^{e}}\). Additionally, due to the existing self-stress state \(S^{e}\), the standard truss finite element description needs to be complemented with the geometric stiffness matrix \(\mathbf{K}^{e}_{G}\big{(}\in\mathbb{R}^{6\times 6}\big{)}\):  
\[\mathbf{K}^{e}_{G}=\frac{S^{e}}{L^{e}}\Big{[} \begin{array}{cccc}\mathbf{I}& -\mathbf{I}&\mathbf{I}\end{array} \Big{]};\;\mathbf{I}=\left[ \begin{array}{ ccccc}1&0&0\\ 0&0&0\\ 0&0&0\end{array} \right]. \tag{2}\]  
The qualitative analysis is provided for an \(n\)-element space truss \((e=1,\ 2,\ldots,\ n)\) described by the elasticity matrix \(\mathbf{E}(\in\mathbb{R}^{n\times n})\):  
\[\mathbf{E}=diag\big{[} \begin{array}{cccc}\frac{E^{1}A^{1}}{L^{1}}&\frac{E^ {2}A^{2}}{L^{2}}&\ldots&\frac{E^{n}A^{n}}{L^{n}}\end{array} \big{]} \tag{3}\]  
Figure 5: (**a**) Space truss finite element, (**b**) global degrees of freedom of element \(e\). with \(m\) degrees of freedom \(\mathbf{q}(\in\mathbb{R}^{m\times 1})\):  
\[\mathbf{q}=\left[ \begin{array}{ccccc}q_{1}&q_{2}&\ldots&q_{m}\end{array}  \right]^{T}. \tag{4}\]  
The compatibility matrix \(\mathbf{B}(\in\mathbb{R}^{m\times m})\) and the geometric stiffness matrix \(\mathbf{K}_{\text{G}}(\in\mathbb{R}^{m\times m})\) for tensegrity structures are determined using the finite element formalism [50; 51; 52; 53; 54; 55; 56; 57; 58; 59]:  
\[\mathbf{B}=\left[ \begin{array}{c}\mathbf{B}^{1}\mathbf{C}^{1}\\ \mathbf{B}^{2}\mathbf{C}^{2}\\ \ldots\\ \mathbf{B}^{n}\mathbf{C}^{n}\end{array} \right], \tag{5}\]  
\[\mathbf{K}_{\text{G}}=\sum_{i=1}^{n}(\mathbf{C}^{\text{c}})^{T}\mathbf{K}_{ \text{G}}^{\text{c}}\ \mathbf{C}^{\text{c}}, \tag{6}\]  
where \(\mathbf{C}^{\text{c}}\big{(}\in\mathbb{R}^{6\times m}\big{)}\) is a Boolean matrix. It is assumed that the number of global degrees of freedom of each element \(q_{i}\)\((i=1,\ 2,\ \ldots,\ 6)\) corresponds to the number of global nodes of elements \(n_{1},\ n_{2}\), as shown in Figure 5b. Consequently, the non-zero elements of \(\mathbf{C}^{\text{c}}\) can be expressed as \(\mathbf{C}^{iq_{i}}=1\). The non-linearity equations of the equilibrium with the influence of the self-stress state in the structure can be written as:  
\[\mathbf{K}_{\text{T1}}\mathbf{q}=\mathbf{P};\ \mathbf{K}_{\text{T1}}=(\mathbf{K}_ {\text{L}}+\mathbf{K}_{\text{G}}), \tag{7}\]  
where \(\mathbf{K}_{\text{T1}}\) is the tangent stiffness matrix and \(\mathbf{K}_{\text{L}}=\mathbf{B}^{T}\mathbf{E}\mathbf{B}\) is the linear stiffness matrix. The qualitative analysis of tensegrity structures can be done through the singular value decomposition [60; 61; 62; 63; 64; 65; 66] of the compatibility matrix \(\mathbf{B}\)[58; 67; 68; 69; 70; 71]. The singular value decomposition (SVD) of matrix \(\mathbf{B}\) is a factorisation in the form:  
\[\mathbf{B}=\mathbf{Y}\mathbf{N}\mathbf{X}^{T}, \tag{8}\]  
where \(\mathbf{Y}(\in\mathbb{R}^{m\times n})=[ \begin{array}{ccccc}\mathbf{y}_{1}& \mathbf{y}_{2}&\ldots&\mathbf{y}_{n}\end{array} ]\) and \(\mathbf{X}(\in\mathbb{R}^{m\times m})=[ \begin{array}{ccccc}\mathbf{x}_{1}& \mathbf{x}_{2}&\ldots&\mathbf{x}_{m}\end{array} ]\) are orthogonal matrices and \(\mathbf{N}(\in\mathbb{R}^{m\times m})\) is a rectangular diagonal matrix.