## 4 Parametric Excitation of the TSBD Due to Vertical Forcing  
The nonlinear equation of motion of the combined horizontal and vertical excited TSBD was derived in Section 2.2 (Equation (6)) and can be written in linearized form (_sin_\(\varphi\cong\varphi\) and _cos_\(\varphi\cong 1\)) as follows,  
\[ \begin{array}{l}\widetilde{\varphi}\ +\ 2\ \zeta_{TSBD}\ \omega_{TSBD}\ \dot{ \varphi}\ +\ \ \omega_{TSBD}^{2}\ \left[1+\frac{\ddot{v}}{g}\right]\ \varphi=-\omega_{TSBD}^{2}\frac{ \ddot{w}}{g}\ \ \.\end{array}  \tag{16}\]  
Neglecting the viscous damping term of the rolling body (\(\zeta_{TSBD}\) = 0) and assuming the absence of any horizontal excitation (\(\ddot{w}(t)=0\)), Equation (16) becomes the so-called Hill differential equation and can be written in the following form [32],  
\[ \begin{array}{l}\ddot{\varphi}\ +\ \ P(t)\ \ \varphi=0\ \cdots\text{with}\ \cdots\ P(t)\ =\omega_{TSBD}^{2}\left[1+\frac{\ddot{v}}{g}\right]\.\end{array}  \tag{17}\]  
[PERSON]'s differential equation is a linear differential equation of second order with a time-variant stiffness parameter \(P(t)\) and it is characterized by the fact that the parameter \(P(t)\) is a periodic function of time. In the case of an engineering structure, this could be, for example, the vertical periodic human-induced vibration of footbridges. In such vibrating systems, so-called parametric-excited vibrations occur, which are characterized by an unstable behavior of the vibration process in the case of parametric resonance, i.e., the vibration amplitude increases towards infinity despite the presence of structural damping of the dynamic system [32,36]. This fact is the fundamental difference to the conventional resonance phenomenon of linear dynamic systems, in which the vibration amplitude in the steady state reaches a stable constant value in the presence of structural damping. Another fundamental difference is the need for an initial disturbance of the dynamic system to raise parametric-excited vibrations and the infinite number of excitation frequencies at which parametric resonance can occur. A well-known and often-cited example of parametric-excited vibrations is the classical pendulum with a vertically moving suspension point [31,32]. For the special case of a time-harmonic vertical excitation function \(v(t)=v_{0}\cdot\cos(\Omega_{\text{p}}\ \dot{t})\), where \(v_{0}\) and \(\Omega_{\text{p}}\) are the excitation amplitude and the circular excitation frequency, Equation (17) becomes the so-called Mathieu differential equation which is not solvable in closed form. Its stability behavior, i.e., the statement about the limitedness or unlimitedness of the solution functions, has been studied, e.g., by [PERSON] [36], [PERSON] [37], and [PERSON] [38]. For a closer examination of the stability behavior, we transform Equation (17) into the mathematically more common normal form by introducing the dimensionless time \(\tau=\Omega_{v}\ t\)[38],  
\[\varphi^{\prime\prime}\ +\ \left[\lambda+\gamma\ \cos\tau\right]\ \varphi=0\quad\text{with}\ \ u^{\prime}=\frac{du}{d\tau}. \tag{18}\]  
In Equation (18) the dimensionless parameters \(\lambda\) and \(\gamma\) are defined as follows,  
\[\lambda=\frac{\omega_{\text{TSBD}}^{2}}{\Omega_{v}^{2}}\quad\text{and}\quad \gamma=-\ \omega_{\text{TSBD}}^{2}\frac{v_{0}}{g}. \tag{19}\]  
Depending on the choice of the two stability parameters \(\lambda\) and \(\gamma\), one obtains either stable or unstable solutions for [PERSON]'s equation, whereby an initial disturbance of the dynamic system is assumed. The calculation of the stability areas and their boundary lines can be approximated with the application of the perturbation calculation [36] or the method of slowly varying amplitude and phase according to Krylov and Bogoliubov [39]. The results of these calculations are presented in a \(\lambda\), \(\gamma\) diagram. Such a stability diagram, calculated for the first time by [PERSON] for an undamped dynamic system, is shown in Figure 4a.