## 2 Mechanical Model
### Substructure Synthesis of the Biaxial-Excited 2-DOF Main System with TSBD Attached  
For the determination of the coupled nonlinear equations of motion of the considered 2-DOF main system--degrees of freedom denoted as horizontal and vertical dynamic displacement amplitudes \(w(t)\) and \(v(t)\)--with a single TSBD attached; the derived reaction forces \(F_{x}\) and \(F_{y}\) (Equation (10)) resulting from the motion of the rolling body in the runway are applied in the selected reference point \(A\) of the deflected free-body diagram of the main system (Figure 3).  
In the considered general plane case the main system is excited horizontally and vertically to forced vibrations. The horizontal and vertical dynamic excitation forces \(F_{h}\) and \(F_{\
u}\) act at the center of gravity of the 2-DOF main system. It is assumed that the center of gravity coincides with the center of stiffness of the main system. Therefore, no coupled bending and torsional vibrations occur due to the external dynamic forces and inertia forces. The two displacement coordinates of the main system are denoted by \(w(t)\) and \(v(t)\). Application of d'Alembert's principle [31] to the free-body diagram of the main system (Figure 2) in the \(x\)- and \(y\)-directions yields the following two coupled equations of motion,  
\[\begin{split}&\ddot{w}+2\zeta_{H,h}\ \omega_{H,h}\ \dot{w}+\omega_{H,h}^{2}\ w=\frac{1}{m_{H}}[F_{h}-F_{x}]\ \,\\ &\ddot{v}+2\zeta_{H,\
u}\ \omega_{H,\
u}\ \dot{v}+\omega_{H,\
u}^{2}\ v= \frac{1}{m_{H}}[F_{\
u}-F_{y}]\ \.\end{split} \tag{11}\]The energy dissipation of the main system was introduced into the equations of motion in the horizontal and vertical direction of vibration as linear viscous damping \(\zeta_{H,h}\) and \(\zeta_{H,v}\), \(\omega_{H,h}\) and \(\omega_{H,v}\) define the natural angular frequencies of the main system in the horizontal and vertical vibration directions. The kinetically equivalent moving mass of the main system \(m_{H}\) is assumed to be identical for both vibration directions. Inserting the derived reaction forces of the rolling body \(F_{x}\) and \(F_{y}\) into Equation (11) and including the equation of motion of the TSBD (Equation (6)) yields the following three coupled nonlinear differential equations of motion for describing the motions of the biaxially excited main system with a single attached TSBD,  
\[\begin{array}{l}(1+\mu)\ddot{w}+\mu(R-r)\ \ddot{\varphi}-\mu(R-r)\ \dot{\varphi}^{2} \varphi+2\zeta_{H,h}\ \omega_{H,h}\ \dot{w}+\omega_{H,h}^{2}\ w=\frac{F_{0}}{m_{H}}\ \,\\ (1+\mu)\ddot{v}+\mu(R-r)\ \ddot{\varphi}\ \varphi+\mu(R-r)\ \dot{\varphi}^{2}+2 \zeta_{H,p}\ \omega_{H,v}\ \dot{v}+\omega_{H,v}^{2}\ v=\frac{F_{0}}{m_{H}}\ \,\\ \ddot{\varphi}\ +\ \omega_{TSBD}^{2}\frac{\ddot{w}}{\delta}\ \ +\ 2\ \zeta_{TSBD}\ \omega_{TSBD}\ \dot{\varphi}\ +\ \ \omega_{TSBD}^{2}\left[1+\frac{\ddot{v}}{\delta}\right]\ \ \varphi=0\ \.\end{array} \tag{12}\]  
In Equation (12) the nonlinear functions \(sin\ \varphi\cong\varphi\) and \(cos\ \varphi\cong 1\) were linearized for simplification of the equations of motion. However, the time-variant stiffness parameter and the quadratic terms of the angular velocity of the rolling body are still left in the equations for further analysis of the damping effectiveness using the parametric resonance effect of the TSBD (see Section 6) in the case of vertical forcing of the main system and TSBD, respectively. \(\mu\) defines the ratio of the damper mass \(m_{R}\) (i.e., rolling body mass) to the kinetically equivalent mass of the main system \(m_{H}\),  
\[\mu=\frac{m_{R}}{m_{H}} \tag{13}\]