## 3 Experimental Investigation
### Experimental Verification of the Proposed Dependencies  
The possibilities of experimental verification are limited by the minimal natural vibration frequency of the system. As the tensile force becomes higher, the resistance of the bearing increases, the rod stiffness in transverse direction decreases, which yields high damping, and vibrations significantly decrease during a very short time that is lower than one natural vibration period; this prevents finding the natural frequency. It is experimentally shown that to obtain the natural vibration frequency with an accuracy of 1%, this frequency for the investigated system should be at minimum 0.3 Hz. A corresponding value of the maximal tensile load achieved in the experiment, \(N_{max}\), is presented in Table 3.  
It is shown in the experiments that, in order to obtain the critical load by extrapolating the measured values up to zero, the minimal number of experimental values of frequency should be 5. If the number is lower, extrapolation accuracy decreases. Increasing the number of the measured values does not increase the accuracy due to the errors in single measurements. To evaluate the accuracy of dependence of \(N\) on \(f\)[2] and to find the value of \(N_{\sigma}\), linear extrapolation of the measured values was used:  
\[N=N_{ce}-\mu f^{2}, \tag{31}\]  
Figure 11: (**a**) Vibration time history of the tested structure; (**b**) corresponding spectra, obtained using FFT for 100 vibration periods. where \(N_{cr}\) is the extrapolated value of the critical force. The parameters \(N_{cr}\) and \(\mu\) were obtained by approximating Equation (31) by the least squares method [26]. Figure 12 presents the dependence of \(N\) on \(f\)2 for rigid-rigid end connections and \(L/R\) ratio of 1.5 and a corresponding linear approximation. Footnote 2: The \(N_{cr}\) and \(\mu\) were calculated using the same method as in the previous section. Estimation of the linear dependence error was carried out according to the absolute value of the deviation:  
\[D_{lin}\%=abs\bigg{[}\frac{N-N_{lin}}{N}\bigg{]}\%, \tag{32}\]  
where \(N_{lin}\) is calculated following Equation (31), corresponding to the obtained values of \(N_{cr}\) and \(r\) and the experimental frequency values. The maximum deviation values are shown in Table 3.  
Figure 12: Dependence of the squared natural vibration frequency on load \(N\) for \(L/R\) = 1.5 and rigid–rigid end connections. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \(L/R\) & \(N_{max}\) [N] & \(N_{cr}\) [N] & \(N_{max}\)/\(N_{cr}\) & \(N_{cr}\) [N] & \(D_{N}\) \% & \(D_{lin}\) \% \\ \hline \multicolumn{8}{c}{rigid–rigid} \\ \hline
1.50 & 35.2 & 41.4 & 0.85 & 40.5 & 2.1 & 2.3 \\
1.67 & 20.2 & 24.9 & 0.81 & 24.4 & 2.2 & 1.7 \\
2.00 & 11.2 & 14.5 & 0.77 & 14.1 & 2.7 & 1.3 \\
2.50 & 7.6 & 11.1 & 0.69 & 10.7 & 3.1 & 2.3 \\
3.00 & 6.9 & 10.4 & 0.67 & 10.1 & 3.5 & 3.2 \\ \hline \multicolumn{8}{c}{pinned–rigid} \\ \hline
1.50 & & 8.0 & \textless{}0.6 & & & \\
1.67 & 5.5 & 8.9 & 0.62 & 8.7 & 3.2 & 2.3 \\
2.00 & 6.5 & 10.0 & 0.65 & 9.7 & 2.8 & 2.8 \\
2.50 & 7.1 & 10.6 & 0.67 & 10.3 & 2.7 & 3.1 \\
3.00 & 7.3 & 10.8 & 0.68 & 10.5 & 2.4 & 3.3 \\ \hline \multicolumn{8}{c}{rigid–pinned} \\ \hline
1.50 & 38.7 & 45.0 & 0.86 & 44.1 & 2.3 & 2.2 \\
1.67 & 20.1 & 24.9 & 0.81 & 24.2 & 2.8 & 2.5 \\
2.00 & 7.2 & 10.4 & 0.69 & 10.7 & 3.1 & 2.8 \\
2.50 & & 4.2 & \textless{}0.6 & & & \\
3.00 & 2.1 & \textless{}0.6 & & & & \\Evaluation of the error in obtaining \(N_{\sigma}\) was performed using the following dependence:  
\[D_{N}\%=abs[(N_{\sigma}-N_{\sigma\sigma})/N_{\sigma}]\%. \tag{33}\]  
The calculated and experimental values of the critical load as well as estimation of measurements' accuracy are given in Table 3. The experimental values of \(N_{\sigma\sigma}\) are given only for cases in which the maximal tensile force is at least 60% of the calculated critical value.