## 4 Discussion  
The first frequency depends on the self-stress state. This is due the fact that in the simplex one infinitesimal mechanism can be identified. The phenomenon is known in the literature. In the absence of prestressing, the first natural frequency should be zero, and after introducing the self-stress state, it increases. The tendency of rising the natural frequency is clearly visible in the experimental test in the first rotational mode. Successive frequencies are practically insensitive to changes of the prestress level. The rotational modes do depend mainly on the geometrical stiffness matrix, which is a function of the prestress level. Other modes are dependent only on the elastic stiffness matrix, which is dependent on the material, cross section, geometry and boundary conditions. Figure 8: The stabilization diagram for the 4 th level of the prestress, 4x direction of acceleration signal and the impulse at node \(P4\) in direction: (**a**) 120 degrees and (**b**) 330 degrees. If the equilibrium equations of the representative node \(P_{6}\) of the simplex are considered under the applied torque \(M_{t}=F_{t}r/3\) associated to the rotation \(\varphi\) and under the uniform axial loading \(F_{Z}\), the summation of all the forces acting in current configuration can be written as:  
\[ \begin{array}{l}g_{1}=x_{1}-x_{3}/2+(x_{1}+3x_{2}+x_{3})\cos(\varphi+\pi/3)- M_{t}\sin(\varphi-2\pi/3)/L^{2}=0,\\ g_{2}=-x_{3}\sqrt{3}/2+(x_{1}+3x_{2}+x_{3})\sin(\varphi+\pi/3)+M_{t}\cos( \varphi-2\pi/3)/L^{2}=0,\\ g_{3}=-h(x_{1}+x_{3})+F_{Z}/3=0,\end{array}  \tag{13}\]  
where the vectors of the total vertical force \(F_{Z}\) and of the moment \(M_{t}\) are acting along the \(Z\)-axis. Solving the system of Equation (13) with respect to the torque \(M_{t}\) and for \(F_{Z}=0\) leads to the following constitutive law for the angle \(\varphi=\theta+5\pi/6\):  
\[M_{t}=-x_{1}\sqrt{3}L^{2}\cos(\varphi-\pi/3)=x_{1}\sqrt{3}L^{2}\sin\theta=k_ {1}(1-s_{N}/s)\sqrt{3}L^{2}\sin\theta, \tag{14}\]  
where the linear elastic stiffness and the geometrically non-linear constitutive model of the cross-cable \(x_{1}=N_{s}/s=k_{1}\left(1-s_{N}/s\right)\) are assumed. If the cross-cables are elastic, while the horizontal triangles are rigid, the length \(L=L_{0}=const\), and the response of the rigid-elastic model of the simplex is obtained in (14) in the form of the system of a single degree of freedom, which is a function of the self-stress level \(p_{0}\). The same constitutive law as in (14) was derived in [35] starting from the model of a single degree of freedom by means of the energy expression. The derivative of (14) \(M_{t}{}^{\prime}(\theta)\) with respect the angle \(\theta\) gives the tangent torsional stiffness of the model and the initial modulus of the elastic response \(K_{t}\) as:  
\[ \begin{array}{l}M_{t}{}^{\prime}(\theta)=k_{1}L^{2}\sqrt{3}(1-s_{N}/s)\cos \theta+k_{1}L^{4}\frac{s_{N}}{s^{3}}\sin^{2}\theta\\ K_{t}=k_{1}L^{2}\sqrt{3}(1-s_{N}/s)=k_{1}L^{2}\sqrt{3}\frac{p_{0}}{1+p_{0}} \end{array}  \tag{15}\]  
where the initial tangent torsional modulus \(K_{t}\) is zero if there is no prestress. Using the modulus \(K_{t}\), the first rotational frequency of the simplex at small amplitude can be written as:  
\[f_{0}(N_{s})=\frac{1}{2\pi}\sqrt{\frac{K_{t}}{I_{0}}}=\frac{1}{2\pi}\sqrt{ \frac{\sqrt{3}N_{s}}{m_{r}s_{N}[1+N_{s}/\left(E_{s}A_{s}\right)]}}, \tag{16}\]  
where the effective torsional stiffness \(K_{t}(15_{1})\) is written depending on the prestress tension force in the cross-cable \(N_{s}=E_{s}A_{s}p_{0}\), the mass moment of inertia about the simplex vertical axis for the three nodal masses is \(I_{0}=m_{r}L^{2}\) and \(m_{r}\) denotes _the reduced mass_ of one node. Figure 9 presents the comparison between the experimental results and the formula (16) and the results from the FEM. Additionally, the relationship between the square frequency and the force in the cross-cable is shown. The curve (16) is fitted into the experimental results by means of the curve fitting procedure performed in Matlab [28].