## 4 Results  
The aim of the study is to indicate which of the plane (2D) trusses presented in the literature are wrongly classified as tensegrities. To do this, the qualitative analysis of selected trusses was carried out. Singular value decomposition of compatibility matrix **B** and spectral analysis of stiffness matrix with the effect of self-equilibrated forces \((\textbf{K}_{L}+\textbf{K}_{G})\) were used for the analysis. Since the characteristic features of tensegrity structures do not depend on geometrical and material characteristics, it was assumed that the constant longitudinal stiffness \(E_{e}A_{e}=1\) and constant lengths \(a=1\). Eighteen plane tensegrity structures were analyzed. The results were presented in two ways (mathematical and graphical)--for selected structures the following matrices were determined:  
* **L**--eigenvalues of matrix \(\textbf{B}^{\textsf{T}}\textbf{B}\),
* **x\({}_{i}\)**--eigenvectors (if any) of \(\textbf{B}^{\textsf{T}}\textbf{B}\) corresponding to the zero eigenvalues \(\lambda_{i}\) in **L** responsible for the existence of the mechanism (\(M\)),
* **M**--eigenvalues of matrix \(\textbf{B}\textbf{B}^{\textsf{T}}\),
* **y\({}_{i}\)**--eigenvectors of \(\textbf{B}\textbf{B}^{\textsf{T}}\) corresponding to the zero eigenvalues \(\mu_{i}\) (if any) in **M** responsible for the existence of the self-stress state (\(S\)),
* **O**--eigenvalues of matrix \((\textbf{K}_{L}+\textbf{K}_{G})\) with the self-stress taken from the singular value decomposition (SVD). In other cases, only the values of self-stress state \(S_{e}\) (values of self-stress were normalized such that \(\sum\limits_{i}^{n}S_{i}^{2}=1\); where \(n\)--number of elements) and forms of mechanisms (if any) were shown. The calculations were carried out in the Mathematica environment. For the purpose of this paper, the plane (2D) trusses defined in the literature as \"tensegrities\" were classified into three groups: \"Structures with tensegrity features\", \"pure tensegrities\" and \"non tensegrities\". Figure 2a shows the first structure under analysis. It is the most popular structure in the literature [8; 12; 15; 16; 17; 18; 37; 41; 42; 64; 67; 68; 69; 70; 71; 72; 73; 74; 75; 76; 77; 78], usually referred to as \"the simplest 2D tensegrity structure\" [77]. It is a truss (\(T\)) with six elements \((n=6)\) and five degrees of freedom (d.o.f.) \((m=5)\)--**q** = \(\left[ \begin{array}{cccc}q_{3}&q_{4}&q_{5}&q_{6}&q_{7}\end{array} \right]^{T}\). For this truss the compatibility matrix \(\textbf{B}\big{(}\in\mathbb{R}^{6\times 5}\big{)}\) is:  
\[\textbf{B}=\left[ \begin{array}{cccc}0&1&0&0&0\\ -1&0&1&0&0\\ 0&0&0&1&0\\ 0&0&0&0&1\\ -0.707&0.707&0&0&0.707\\ 0&0&0.707&0.707&0\end{array} \right].\]  
Figure 2: The “X” truss: (**a**) supported, (**b**) unsupported. The eigenvalues of both \(\mathbf{B}^{\mathsf{T}}\mathbf{B}\) and \(\mathbf{B}\mathbf{B}^{\mathsf{T}}\) matrices are respectively:  
\[\mathbf{L}=\text{diag}\Big{[} \begin{array}{cccccccc}2.89&2&1.35&1&0.25\end{array}  \Big{]};\ \mathbf{M}=\text{diag}\Big{[} \begin{array}{cccccccc}2.89&2&1.35&1&0.25&0 \end{array} \Big{]}.\]  
There is no zero eigenvalue in \(\mathbf{L}\), which means that no mechanism is identified. The zero eigenvalue in \(\mathbf{M}\) is responsible for the existence of the self-stress state (\(S\)):  
\[\mathbf{y}_{6}=\Big{\{} \begin{array}{cccccccc}0.35&0.35&0.35&0.35&-0.5&-0.5 \end{array} \Big{\}}.\]  
In this case, all eigenvalues of the matrix \((\mathbf{K}_{L}+\mathbf{K}_{G})\) are positive:  
\[\mathbf{O}=\text{diag}\Big{[} \begin{array}{cccccccc}2.71&2.38&1.49&0.77&0.29 \end{array} \Big{]},\]  
which means the structure is stable. For this truss the system of compression members (thick line) is discontinuous (\(D\)) and included inside the set of components in tension (\(I\)), which are cables (\(C\)). However, because of the absence of the mechanism, the truss shares only some features with a tensegrity. If unsupported \"X\" truss is analyzed (Figure 2b), there are eight d.o.f.