## 2 Mechanical Model
### Free-Body Diagram of the TSBD  
The substructure method is used to derive the coupled nonlinear equations of motion of a biaxial-excited two-degree-of-freedom (2-DOF) main system with a single TSBD attached. For this purpose, in the first step, the free-body diagram of the TSBD is considered in a general plane deflected position (Figure 2). Figure 3 illustrates the free-body diagram of the biaxial-excited 2-DOF main system with applied reaction Forces \(F_{x}\) and \(F_{y}\) resulting from the movement of the rolling body in the runway of the TSBD. In Figure 2, the chosen reference point \(A\) is shifted by the combined horizontal and vertical dynamic displacement of the 2-DOF main system \(w(t)\) and \(v(t)\) with respect to the origin O. The coordinates of the origin are denoted by \((x,y)\) and the moving coordinates are denoted by \((x^{\prime},y^{\prime})\). \(m_{R}\) is the mass of the rolling body and \(r\) is the corresponding radius. The runway is assumed to be semicircular with a radius of \(R\). The distance of the chosen reference point \(A\) to the center of gravity of the rolling body is \((R-r)\). The degrees of  
Figure 3: Free-body diagram of the biaxial-excited 2-DOF main system with applied reaction Forces \(F_{x}\) and \(Fy\) resulting from the movement of the rolling body in the runway of the TSBD. Figure 2: Free-body diagram of the TSBD in a general plane deflected position. freedom of the rolling body are defined by \(\varphi(t)\) and \(\alpha(t)\) and are linked via the rolling condition (pure rolling assumed) as follows,  
\[\dot{\alpha}=\dot{\varphi}\ \ \frac{(R-r)}{r}\ . \tag{1}\]  
The nonlinear equation of motion of the rolling body in the runway is determined by applying [PERSON]'s equation [31],  
\[\frac{d}{dt}\left(\frac{\partial T}{\partial\varphi}\right)-\frac{\partial T}{ \partial\varphi}+\frac{\partial V}{\partial\varphi}=Q_{\varphi}\ . \tag{2}\]  
In Equation (2), \(T\) defines the kinetic energy, \(V\) the potential energy, and \(Q_{\varphi}\) the generalized forces. In the generalized forces \(Q_{\varphi}\), the energy dissipation occurring during the rolling process of the rolling body due to friction effects (Coulomb friction) is considered and introduced as equivalent viscous damping in the equation of motion [1]. The kinetic energy \(T\) of the horizontally and vertically excited TSBD is composed of the rotational and translational components as follows,  
\[T=\frac{1}{2}I_{S}\ \dot{\varphi}^{2}+\frac{1}{2}m_{R}\ v_{S}^{2}+\frac{1}{2}I_{S }\ \dot{\alpha}^{2}\ . \tag{3}\]  
In Equation (2), \(I_{S}\) defines the mass moment of inertia in relation to the center of gravity of the rolling body and \(v_{S}\) its corresponding velocity which results from the time derivative of the displacement variables in \(x\)- and \(y\)-directions,  
\[v_{S}^{2}=\dot{w}^{2}+2\dot{w}\ \dot{\varphi}(R-r)\cos\varphi+\dot{v}^{2}+2 \dot{v}\ \dot{\varphi}(R-r)\sin\varphi+(R-r)^{2}\dot{\varphi}^{2}\ . \tag{4}\]  
The potential energy \(V\) of the TSBD is given by,  
\[V=m_{R}\ g\left[v-(R-r)\cos\varphi\right]\ . \tag{5}\]  
Substituting the center of gravity velocity (Equation (4)) into the kinetic-energy expression (Equation (3)) and performing the derivatives of \(T\) and \(V\) according to [PERSON]'s equation (Equation (2)) yields the nonlinear parametric-excited equation of motion of the TSBD given below,  
\[\ddot{\varphi}\ +\ \ 2\ \zeta_{TSBD}\ \omega_{TSBD}\ \dot{\varphi}\ \ +\ \ \omega_{TSBD}^{2}\left[1+\frac{\ddot{v}}{g}\right]\ \ \sin\varphi=-\omega_{TSBD}^{2}\frac{\ddot{w}}{g}\ \ \cos\varphi\ . \tag{6}\]  
For small vibration amplitudes, i.e., small rotation angles \(\varphi(t)\) and, hence, \(sin\ \varphi\cong\varphi\) and \(cos\ \varphi\cong 1\); Equation (6) becomes a linear equation of motion with constant natural angular frequency \(\omega_{TSBD}\), see Equation (12). The friction damping of the rolling body in the runway is introduced in Equation (6) by the equivalent viscous damping term \(\zeta_{TSBD}\).