## 2 Materials and Methods
\[\mathrm{F_{dz}(s)=c_{2}\cdot[\dot{\zeta}(s)-\dot{z}_{2}(s)]+k_{2}\cdot[\zeta(s )-\dot{z}_{2}(s)]} \tag{14}\]  
On the grounds of Equations (10)-(14), with necessary transformations, the final concise form (15) of the transmittance of the dynamic vertical tyre-plate interaction force was obtained (remembering that \(\mathrm{q_{1}=z_{1},\,q_{2}=z_{2}}\)):  
\[\mathrm{H_{Fdz}(s)=\frac{F_{dz}(s)}{\zeta(s)}=(c_{2}\cdot s+k_{2})\cdot[1-H_{ q_{2}}(s)]} \tag{15}\]  
The difference between the dynamic component \(\mathrm{F_{md}}\) of the force measured in the test facility and the vertical dynamic component \(\mathrm{F_{dz}}\) of the tyre-plate interaction force is caused by the force of inertia of the plate (the last summand in (16)). \[ \begin{array}{l}\mathrm{H_{Fmd}(s)=\frac{F_{md}(s)}{\zeta(s)}=\frac{F_{dz}(s) +m_{3}\cdot\ddot{\zeta}(s)}{\zeta(s)}=\frac{F_{dz}(s)+m_{3}\cdot s^{2}\cdot \zeta(s)}{\zeta(s)}=\frac{F_{dz}(s)+m_{3}\cdot s^{2}}{\zeta(s)}=\\ =(c_{2}\cdot s+k_{2})\cdot[1-H_{q_{2}}(s)]+m_{3}\cdot s^{2}=H_{Fdz}(s)+m_{3} \cdot s^{2}\end{array}  \tag{16}\]  
The values of spectral transmitances \(\mathrm{H_{Fdz}(i\cdot\omega)}\) and \(\mathrm{H_{Fmd}(i\cdot\omega)}\), where \(\omega\) is the radian frequency of kinematic excitation \(\zeta\), are defined for \(s=i\cdot\omega\). Their moduli are equal to the ratios of the amplitudes of forces (\(\mathrm{F_{dz}}\) and \(\mathrm{F_{md}}\), as appropriate) to the amplitude of excitation \(\zeta\), and the arguments are defined by the phase shifts. By comparing the moduli of the transmitances \(\mathrm{H_{Fdz}}\) and \(\mathrm{H_{Fmd}}\) with each other (for identical excitation), it will be possible to assess indirectly the impact of mass \(\mathrm{m_{3}}\) of the tester's vibration plate on the differences between forces \(\mathrm{F_{dz}}\) i \(\mathrm{F_{md}}\). In their publications [27; 35], the authors have presented the formulas that define the natural radian and Hertz frequencies of the undamped system (resonance radian frequencies \(\omega_{01}\), \(\omega_{02}\) [rad/s] and Hertz frequencies \(\mathrm{\SIUnitSymbolMicro 1}\), \(\mathrm{f_{02}}\) [Hz]) as well as suspension's critical damping \(\mathrm{c_{kr1}}\) [N\(\cdot\)s/m] and relative damping \(\mathrm{\SIUnitSymbolMicro 1}\) [-]. \[\omega_{01/02}^{2}=\frac{k_{1}\cdot m_{2}+(k_{1}+k_{2})\cdot m_{1}}{2\cdot m_{1 }\cdot m_{2}}\mp\sqrt{\left[\frac{k_{1}\cdot m_{2}+(k_{1}+k_{2})\cdot m_{1}}{2 \cdot m_{1}\cdot m_{2}}\right]^{2}-\frac{k_{1}\cdot k_{2}}{m_{1}\cdot m_{2}}} =(2\pi\cdot f_{01/02})^{2} \tag{17}\]  
\[\mathrm{c_{kr1}=2\cdot\sqrt{\frac{k_{1}\cdot k_{2}\cdot m_{1}}{k_{2}+k_{1} \cdot\left(1+\frac{m_{2}}{m_{1}}\right)}}\approx 2\cdot\sqrt{\frac{k_{1}\cdot k_{2} \cdot m_{1}}{k_{2}+k_{1}}}\text{ for }m_{1}\gg m_{2} \tag{18}\]  
\[\mathrm{\SIUnitSymbolMicro 1}=\frac{c_{1}}{c_{kr1}} \tag{19}\]  
where \(c_{1}\) [N\(\cdot\)s/m] is the viscous damping coefficient of the suspension system in the linear model defined previously. Structure of the Non-Linear Model Used in the Simulation Tests and Its Equations of Motion in the Time Domain  
The structure of the non-linear 'quarter-car' model (Figure 2c) is similar to that of the linear one (see also [3; 5; 35]). It differs from the latter in the description of the spring and damping forces in the suspension system and pneumatic tyre. It reflects the actual characteristic curves measured in laboratory conditions and represents shock absorber and tyre damping forces \(\mathrm{F_{t1}}\) [N] and \(\mathrm{F_{t2}}\) [N], respectively, suspension and tyre spring forces \(\mathrm{F_{s1}}\) [N] and \(\mathrm{F_{s2}}\) [N], respectively, sliding friction in the suspension \(\mathrm{F_{ts1}}\) [N], and tyre bouncing ('wheel hop') phenomenon. The non-linear model describes changes in the resultant forces \(\mathrm{F_{op}}\) and \(\mathrm{F_{opm}}\), taking into account the constant values of the gravity forces of the model masses (N\({}_{\text{st}}\) and N\({}_{\text{stm}}\), respectively, according to Equations (1) and (2)).