## 3 Results
### Numerical Analysis on Small Vibrations  
As mentioned before, a linearization of vibrations around the equilibrium state was assumed. For a chosen pre-stress level, these vibrations are determined in numerical analysis using equations of motion as for a linear multi-degree-of-freedom system of the structure, i.e., by the system of ordinary  
Figure 4: The experimental distribution of member forces for the prestress levels: (a) the 4 th level; (b) the 13 th level. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \multicolumn{3}{c}{**Level**} & \multicolumn{3}{c}{**Cables [N]**} & \multicolumn{3}{c}{**Bars 2**} & \multicolumn{3}{c}{**Level**} & \multicolumn{3}{c}{**Cables [N]**} & \multicolumn{3}{c}{**Bars 2**} \\ \hline
**No** & \(p_{0}\)**[10\({}^{-6}\)]** & **Cross 1** & **Base 1** & **Base 2** & **[N]** & **No** & \(p_{0}\)**[10\({}^{-6}\)]** & **Cross 1** & **Base 1** & **Base 2** & **[N]** \\ \hline
1 & 618 & \(263\pm 27\) & \(75\pm 13\) & 55 & \(-282\) & 8 & 5082 & \(2164\pm 28\) & \(646\pm 20\) & 450 & \(-2322\) \\ \hline
2 & 1294 & \(551\pm 37\) & \(169\pm 17\) & 115 & \(-591\) & 9 & 6135 & \(2612\pm 35\) & \(766\pm 23\) & 542 & \(-2802\) \\ \hline
3 & 2104 & \(896\pm 40\) & \(263\pm 16\) & 187 & \(-961\) & 10 & 6877 & \(2928\pm 42\) & \(849\pm 24\) & 607 & \(-3141\) \\ \hline
4 & 2741 & \(1167\pm 38\) & \(345\pm 18\) & 243 & \(-1252\) & 11 & 7264 & \(3093\pm 43\) & \(892\pm 23\) & 641 & \(-3318\) \\ \hline
5 & 3368 & \(1434\pm 35\) & \(421\pm 18\) & 298 & \(-1538\) & 12 & 7788 & \(3316\pm 45\) & \(957\pm 25\) & 687 & \(-3558\) \\ \hline
6 & 3981 & \(1695\pm 31\) & \(500\pm 20\) & 352 & \(-1818\) & 13 & 8418 & \(3584\pm 40\) & \(1025\pm 25\) & 742 & \(-3845\) \\ \hline
7 & 4413 & \(1879\pm 29\) & \(559\pm 19\) & 391 & \(-2016\) & & & & & \\ \hline \hline \end{tabular} \({}^{1}\) Experimental values, 2 Calculated from (5).  
\end{table} 
Table 3: Member forces for the each prestress level. differential equations governing vectors of order \(n\) of the displacements \(\mathbf{u}(t)\) due to applied dynamic forces \(\mathbf{p}(t)\) in the matrix equation:  
\[\mathbf{M}\ddot{\mathbf{u}}(t)\mathbf{+}\mathbf{C}\ddot{\mathbf{u}}(t)\mathbf{+ }\mathbf{K}_{T}\mathbf{u}(t)\mathbf{=}\mathbf{p}(t) \tag{7}\]  
where \(n\) is the number of independent degrees-of-freedom (DOF), the square matrices of order \(n\mathbf{M}\), \(\mathbf{C}\), \(\mathbf{K}_{T}\) are the mass, damping and tangent stiffness matrices and \(\ddot{\mathbf{u}}\), \(\ddot{\mathbf{u}}\) are the resulting acceleration and velocity vectors. The complete solution \(\mathbf{u}(t)\) consists of two contributions: the initial part of the response induced by the initial conditions and the particular solution which satisfied the forcing function \(\mathbf{p}(t)\). When \(\mathbf{p}(t)=0\), a very frequently occurring problem in structural dynamics is the free vibration equation to be solved, usually by means of the finite element method as a linearized eigenproblem:  
\[\mathbf{K}_{T}\mathbf{\varphi}_{k}-\omega_{k}^{2}\mathbf{M}\mathbf{\varphi}_{ k}=\mathbf{0}, \tag{8}\]  
The solution lies in the spectral decomposition of the matrix \(\mathbf{M}^{-1}\mathbf{K}_{T}\). It gives the natural modes and frequencies of the analyzed structure. If the analyzed equlibrium state possess (n) DOFs, then the spectral decomposition of (8) provides (n) frequencies \(\omega_{k}(\omega_{1}<\omega_{2}\ldots<\omega_{n})\) and (n) corresponding eigenvectors \(\mathbf{\varphi}_{k}\). According to the finite element method (FEM), the system matrices \(\mathbf{K}_{T}\) and \(\mathbf{M}\) are typically build from the element matrices \(\mathbf{k}_{T}\) and \(\mathbf{m}\), which are determined in the local coordinate system. The geometric stiffness element matrix \(\mathbf{k}_{G}(S)\) dependent on the pre-stress level, and the elastic stiffness element matrix \(\mathbf{k}_{E}\), build the tangent element matrix \(\mathbf{k}_{T}\) by simple addition.