## 2 Analytical Investigation of the Experimental Parameters Range
The symmetry of the rigid-rigid case relative \(R/L\) = 0.5 was demonstrated previously [7]. The values of the rigid-pinned case are symmetric to those of the pinned-rigid, also with relative \(R/L\) = 0.5. The present study analyzed the dependence of \(K^{\prime}\) on \(0\leq u\leq u_{\sigma}\) using the normalized non-dimensional values \(K^{\prime}/K^{\prime}_{0}\) and \(u^{2}/u^{2}\sigma\). Here, \(K^{\prime}_{0}\) is non-dimensional stiffness at \(N\) = 0. This approach is convenient to investigate the deviation from linear dependence  
\[F=-V_{A}+M_{A}/R\, \tag{3}\]2  
For all connection cases, the graphs of normalized dependences pass through points (1, 0) and (0, 1). For the rigid-rigid connection (Figure 5), the graphs become close to the \begin{table}
\begin{tabular}{c c c c c} \hline  & & \multicolumn{4}{c}{\(u_{\sigma}\)} \\ \cline{3-5}  & & **Rigid–Pinned** & **Pinned–Rigid** & **Rigid–Rigid** \\ \hline  & 1.500 & 2.985 & 1.288 & 3.041 \\  & 1.667 & 2.464 & 1.512 & 2.622 \\ L/R & 2.000 & 1.915 & 1.915 & 2.400 \\  & 2.500 & 1.512 & 2.464 & 2.622 \\  & 3.000 & 1.288 & 2.985 & 3.041 \\ \hline \end{tabular}
\end{table} 
Table 1: Values of \(u_{\sigma}\) for the investigated end connection conditions. Figure 4: Dependence of roots \(u_{\sigma}\) on the \(R/L\) ratio. linear dependence, as the ratio \(L/R\) is closer to 2. For other values of \(L/R\) the dependence is nonlinear. The effect of \(L/R\) for the pinned-rigid and rigid-pinned cases is similar. In a similar way, we analyzed the dependence of \(\omega^{2}/\omega_{0}{}^{2}\) on \(u^{2}/u^{2}{}_{\mathbf{\sigma}}\), where \(\omega_{0}\) is the dominant vibration frequency at \(N=0\). According to Equation (1),  
\[\omega^{2}/\omega_{0}^{2}=K^{\prime}/K_{0}^{\prime}\frac{\alpha(0)+\delta}{ \alpha+\delta}. \tag{23}\]  
For \(1.5\leq L/R\leq 3\), in the experiments with constant \(R\) and variable \(L\), the coefficient \(\delta\) varies from 6.8 to 22.7. The coefficient \(\alpha\) for the same \(L/R\) and for \(u\leq u_{\mathbf{\sigma}}\) varies within a diapason between 0.14 and 0.28. As the values of \(\alpha\) are rather small, the nonlinearity of \(\omega^{2}/\omega_{0}{}^{2}\) vs. tensile force \(N\) is determined by that of \(K^{\prime}\). There is a range of \(L/R\) for which the dependence is close to linear with a certain accuracy. Considering a linear dependence of \(\omega^{2}\) on the rod tension,  
\[\omega_{Int}^{2}/\omega_{0}^{2}=1-\frac{u^{2}}{u_{\mathbf{\sigma}}^{2}}, \tag{24}\]  
the dependence of maximum deviation on \(\omega^{2}/\omega_{0}{}^{2}\) is analyzed according to Equations (23) and (24) in a range of \(0\leq u\leq u_{\mathbf{\sigma}}\):  
\[D\%=max(\omega^{2}/\omega_{0}^{2}-\omega_{lin}^{2}/\omega_{0}^{2})\%. \tag{25}\]  
The value of \(\omega^{2}/\omega_{0}{}^{2}\) according to Equation (23) is obtained considering the dependence \(\alpha(u)\) using Equation (15) for the given values of \(\delta\). The maximum deviation value occurs about at the middle of the interval \(0\leq u\leq u_{\mathbf{\sigma}}\). Table 2 and Figure 6 present the results of this analysis. The range of allowed \(L/R\) depends on the selected accuracy of the correspondence between the dependence and its linear approximation. As the accuracy of the experiment, dependent on measuring the frequency and using the squared frequency for extrapolation, was 2%, and the accuracy of force measuring was 0.5%, the total maximum error was assumed to be 2.5%. In this case, the range of the allowed \(L/R\) values was \(1.33\leq L/R\leq 4\) for the rigid-rigid connection, \(1\leq L/R\leq 3\) for the pinned-rigid case, and \(L/R\geq 1.5\) for the rigid-pinned one. The common range for all connections was \(1.5\leq L/R\leq 3\). Therefore, this interval was selected for further experiments. For all end  
Figure 5: Dependences of normalized values of non-dimensional stiffness \(K^{\prime}/K_{0}^{\prime}\) on normalized values \(u^{2}/u^{2}{}_{\mathbf{\sigma}}\) for the rigid–rigid connection.