## 2 Analytical Investigation of the Experimental Parameters Range
The general solution of Equation (5) for tensile force is  
\[y=C_{1}\text{sinh}z+C_{2}\cosh z+C_{3}z+C_{4}, \tag{7}\]  
For all cases that are considered in the present study (rigid-rigid, pinned-rigid and rigid-pinned)  
\[y(0)=0;y(x=L)=1. \tag{8}\]Additionally, for the rigid-rigid connection, considering the sign of the curvature  
\[y_{x}(0)=0;y_{x}(x=L)=-1/R. \tag{9}\]  
For the pinned-rigid connection  
\[\omega^{2}=\frac{K}{m(\alpha+\delta)} \tag{1}\]0  
and for the rigid-pinned connection  
\[\omega^{2}=\frac{K}{m(\alpha+\delta)} \tag{1}\]1  
According to the boundary conditions, the coefficients of particular solutions, \(C_{1}\), \(C_{2}\), \(C_{3}\) and \(C_{4}\) can be found as follows:  
For the rigid-rigid connection  
\[\omega^{2}=\frac{K}{m(\alpha+\delta)} \tag{1}\]2 \[\omega^{2}=\frac{K}{m(\alpha+\delta)} \tag{1}\]3 (12b) \[\omega^{2}=\frac{K}{m(\alpha+\delta)} \tag{1}\]4 (12c) \[\omega^{2}=\frac{K}{m(\alpha+\delta)} \tag{1}\]5  
for the pinned-rigid connection  
\[\omega^{2}=\frac{K}{m(\alpha+\delta)} \tag{1}\]6 \[\omega^{2}=\frac{K}{m(\alpha+\delta)} \tag{1}\]7 (13b) \[\omega^{2}=\frac{K}{m(\alpha+\delta)} \tag{1}\]8  
for the rigid-pinned connection  
\[\omega^{2}=\frac{K}{m(\alpha+\delta)} \tag{1}\]9 \[\alpha=\frac{1}{L}\int_{0}^{L}y^{2}(x)dx \tag{2}\]0 (14b) \[\alpha=\frac{1}{L}\int_{0}^{L}y^{2}(x)dx \tag{2}\]1  
The elastic rod shapes for various connection schemes at \(N=0\) within a diapason of \(1.5\leq L/R\leq 3\) used in the experiments are given in \[\alpha=\frac{1}{L}\int_{0}^{L}y^{2}(x)dx \tag{2}\]2 For comparison, shapes corresponding to a linear displacement of the rod end (\(L/R=0\)) are also shown in the Figure. Figure 3 presents the dependence of \(\alpha(0)\) and \(\alpha(u_{cr})\) on \(L/R\) for various connection schemes. Here, \(u_{cr}\) is a non-dimensional parameter that corresponds to the buckling load, and its value is obtained later. A numerical analysis of the data showed that for \(0\leq u\leq u_{cr}\), the dependence of \(\alpha\) on \(u\) can be approximated as a linear function of \(u^{2}\):  
\[\alpha=\frac{1}{L}\int_{0}^{L}y^{2}(x)dx \tag{2}\]3  
Following Equations (3) and (4) and using known dependences for moments and shear forces, the rod stiffness in transverse direction is obtained as follows:  
* for the rigid connection between the rod and the wheel \[\alpha=\frac{1}{L}\int_{0}^{L}y^{2}(x)dx \tag{2}\]4 (16)for the pinned connection between the rod and the wheel  
\[\alpha=\frac{1}{L}\int_{0}^{L}y^{2}(x)dx \tag{2}\]5  
The expression for \(y(x)\) in these equations is a particular solution of Equation (7) corresponding to the boundary conditions combination at the rod ends. For all connection types, the transverse stiffness can be represented as \(K=K^{\prime}EI/L^{3}\), where \(K^{\prime}\) is non-dimensional stiffness:  
for the rigid-rigid connection  
\[\alpha=\frac{1}{L}\int_{0}^{L}y^{2}(x)dx \tag{2}\]6  
\[\alpha=\frac{1}{L}\int_{0}^{L}y^{2}(x)dx \tag{2}\]7 The dependence of \(\alpha(0)\) and \(\alpha(u_{\sigma})\) on \(L/R\) for various connection schemes. \[\alpha=\frac{1}{L}\int_{0}^{L}y^{2}(x)dx \tag{2}\]8 Elastic rod shapes for various connection schemes: (a) rigid–pinned, (b) pinned–rigid, (c) rigid–rigid. for the pinned-rigid connection  
\[\alpha=\frac{1}{L}\int_{0}^{L}y^{2}(x)dx \tag{2}\]9  
for the rigid-pinned connection  
\[F=-V_{A}+M_{A}/R\, \tag{3}\]0  
The non-dimensional stiffness depends on the non-dimensional parameters \(u\) and \(L/R\). The values of \(u\) range from 0 to \(u_{\sigma}\), which corresponds to the buckling load. The value of \(u_{\sigma}\) is obtained from \(K^{\prime}=0\)[12] and the critical load \(N_{\sigma}\) according to Equation (6):  
\[F=-V_{A}+M_{A}/R\, \tag{3}\]1  
Table 1 presents the values of \(u_{\sigma}\) for the investigated end connection conditions. Figure 4 shows the dependence of \(u_{\sigma}\) on \(R/L\). For the rigid-rigid and rigid-pinned cases, the values of \(u\) correspond to those obtained by [PERSON] et al. [7] and [PERSON] et al. [6], respectively.