## 3 Results for the Simplest Two-Dimensional Cable Structure  
First, to illustrate the behavior of structures characterized by the self-stress state and infinitesimal mechanism, dynamic instability is analyzed. Three values of load are considered, i.e.,\(P=\{1\,\text{kN},3\,\text{kN},5\,\text{kN}\}\). Cables with diameter \(\phi=20\) mm and length \(l=1\) m are assumed. In turn, the material characteristics of steel, i.e., Young's modulus and density \(\rho\), are 210 GPa and 7860 kg/m\({}^{3}\), respectively.  
In this case, the minimum level of initial prestress does not depend on external loads, so \(S_{min}=0\) is assumed. In turn, the maximum level of initial prestress for such data equals \(S_{max}=70\) kN and corresponds to the effort of the structure \(\text{W}_{\text{max}}=0.85\). Assuming geometric and mechanical characteristics, the values of prestressing forces \(S\) are much lower than the longitudinal stiffness \(EA\): \(S_{max}=0.07\) MN \(<EA=66\) MN.  
The influence of the initial prestress level S and load on the main unstable region is shown. For example, Figure 4 shows the boundaries for three levels of initial prestress, i.e., \(S_{min}=0\) kN, \(S=30\) kN, and \(S_{max}=70\) kN. As can be seen, independent of the value of the prestress, the size of the regions mostly depends on the load. The greater the load, the larger the region. At the same time, the larger the load, the higher the frequency. The introduction of prestress (Figure 4b,c) decreases the size of the unstable regions and increases the resonant frequencies. The initial prestress level has a greater influence on the size of the unstable regions when smaller loads are applied. For example, in the case of \(P=1\) kN, introducing \(S_{max}=70\) kN results in an overlapping of the boundaries of unstable regions (Figure 4c).  
To better compare the influences of the initial prestress level \(S\) and the load, Figure 5a shows the areas of unstable regions \(A_{\eta}(S)\). Different behaviors are observed depending on the value of the load. In the case of \(P=1\) kN, the chart displays an almost exponential function. In turn, in the cases of \(P=3\) kN and \(P=5\) kN, the charts show similar behaviors. Comparing the load of \(P=5\) kN with \(P=1\) kN, the area of the unstable regions is larger by 1.7 (at \(S=0\) kN), 3.0 (at \(S=30\) kN), and 8.9 (at \(S=70\) kN) times. In turn, comparing the load of \(P=5\) kN with \(P=3\) kN, the area of the unstable regions is larger by 1.2 (at \(S=0\) kN), 1.3 (at \(S=30\) kN), and 1.7 (at \(S=70\) kN) times.  
To measure the changes in the areas of the unstable regions, the dimensionless parameter \(\lambda\) (15) was calculated (Figure 5b). In the case of \(P=1\) kN, the parameter \(\lambda\) is equal: \(\lambda=0.88\) (at \(S=10\) kN), \(\lambda=0.7\) (at \(S=30\) kN), and \(\lambda=0.12\) (at \(S=70\) kN), which mean that the unstable regions are, respectively, \(12\%\), \(30\%\), and \(88\%\) smaller than in the absence of initial prestress (\(S=0\) kN). In turn, in the case of \(P=5\) kN, the unstable regions for the considered prestress levels as previous are smaller by \(3\%\) (\(\lambda=0.97\)), \(8\%\) (\(\lambda=0.92\)), and \(38\%\) (\(\lambda=0.62\)). As can be seen, load causes differences in the changes in parameter \(\lambda\) with growth in the initial prestress level. For \(P=1\) kN, parameter \(\lambda\) decreases almost exponentially; in turn, for \(P=5\) kN, it is a polynomial of the second degree.