## 2 Analytical Investigation of the Experimental Parameters Range  
The stand for testing (Figure 1) consists of a wheel 1, placed on bearing 2, an elastic rod 3, a sliding guide 4 that moves straight along a line, passing through the wheel center (line OC in the figure). One end of the rod is connected to the wheel, and the other moves along the direction slide. The rod connection with the wheel and the direction slide enables changing the deformable rod length. It also allows pinned-rigid, rigid-rigid, or rigid-pinned connections. The axial load in the rod was applied by a weight connected to the clamp through a block by a cable. Figure 1: (**a**) Scheme of a stand for testing systems, including a tensile rod with an end sliding along a circular guide: 1—curved guide, placed on bearing 2; 3—elastic rod; 4—sliding guide; (**b**) forces in the rod section and forces acting on the curved guide. Note that \(V_{A}\) and \(M_{A}\) are internal forces acting in the rod at its connection to the curved guide, \(N_{A}\) is the tensile force in the rod, \(F\) is the external force applied to obtain the transverse stiffness of the system. According to the Rayleigh method [17], it was assumed that the system's mode shape corresponds to that under a static force [18] applied to the mass. The dominant modal frequency of the system was obtained as follows:  
\[\omega^{2}=\frac{K}{m(\alpha+\delta)} \tag{1}\]  
where \(K\) is the rod static stiffness in transverse direction, \(\delta=M/m\), \(M\) is the equivalent mass at the end of the rod, \(m\) is the elastic rod mass, \(\alpha\) is equivalence coefficient of the elastic rod mass and that of the lumped muss at the rod end:  
\[\alpha=\frac{1}{L}\int_{0}^{L}y^{2}(x)dx \tag{2}\]  
Here, \(y(x)\) is the shape of the elastic rod due to unit static displacement at its end, corresponding to the trajectory; \(L\) is the length of the deformed part of the rod. According to possible rod end displacement, the rod stiffness in transverse direction is defined as a force \(F\) that yields a unit displacement at the rod end along the circle with a radius \(R\) (see Figure 1). According to the possible displacement at the end A of the rod, the above-mentioned force direction is tangent to the trajectory (perpendicular to the radius, connecting point A with the rotation center). From the equilibrium of the curved guide (Figure 1), the load \(F\), required for deviation \(\varphi=I/R\), corresponding to rigid connection, is  
\[F=-V_{A}+M_{A}/R\, \tag{3}\]  
where \(V_{A}\) and \(M_{A}\) are shear force and bending moment in the rod, respectively. At \(F=0\), Equation (3) corresponds to the equilibrium of a curved clamp [7; 14]. For a pinned connection between the wheel and the rod end  
\[F=-V_{A}+N_{A}(\varphi-\gamma), \tag{4}\]  
where \(N_{A}\) is an axial force in the rod, \(\gamma\) is the angle between the rod axis and the line OC, and \(\varphi\) is the wheel rotation angle, corresponding to unit displacement of the rod end along the circular trajectory (see Figure 1). To obtain the dominant vibration frequency of the system, it was assumed that the equivalent mass of the rod was lumped at its end [19]. The general equivalent mass of the system includes that of the rod and that of the curved guiding elements. The shape of the elastic rod and the stiffness in transverse direction are obtained by integrating the differential equation of the rod deformation under tension  
\[d^{4}y/dz^{4}-d^{2}y/dz^{2}=0, \tag{5}\]  
where \(z=u\ x/L\), \(u=L\ (N/EI)^{0.5}\), \(E\) is the modulus of elasticity, \(I\) is the moment of inertia of the rod section, \(N\) is the axial force. As  
\[N=u^{2}\ EI/L^{2}, \tag{6}\]  
then \(u^{2}\) is a non-dimensional parameter of the axial force in the rod.