## 2 Materials and Methods  
Structure of the Linear Model Used in the Simulation Tests and Equations of Motion in the Time and Frequency Domains  
As in publications [4; 11; 12; 15; 16; 17; 23; 26; 27; 34; 37; 38; 39; 40; 41], the linear 'quarter-car' model presented in Figure 2a was used (cf. with that in [20; 28]). In Figure 2b, the inertia of the tester's vibration plate was taken into account after [17; 27; 34; 35]. The model shown in Figure 2b consists of three mass elements, i.e.,'sprung mass' m\({}_{1}\) [kg], 'unsprung mass' m\({}_{2}\) [kg], and vibration plate m\({}_{3}\) [kg]. The suspension and tyre stiffness have been denoted by k\({}_{1}\) [N/m] and k\({}_{2}\) [N/m], respectively. Parameters c\({}_{1}\) [N\(\cdot\)s/m] and c\({}_{2}\) [N\(\cdot\)s/m] represent the suspension and tyre damping coefficients, respectively. The quantity denoted by \(\zeta(t)\) [m] is the kinematic excitation resulting from the motion of the vibration plate (exciter), which supports the vehicle wheel. The measuring system measures the dynamic component F\({}_{\text{md}}\) [N] of the force applied from below to the vibration plate, the static load N\({}_{\text{stm}}\) resulting from the weight of the mass elements of the model (including the weight of the vibration plate m\({}_{3}\cdot\)g) and the force of inertia of the vibration plate F\({}_{\text{bp}}\) (negative). The static forces are as follows:  
\[\text{N}_{\text{st}}=(\text{m}_{1}+\text{m}_{2})\cdot\text{g} \tag{1}\]  
\[\text{N}_{\text{stm}}=(\text{m}_{1}+\text{m}_{2}+\text{m}_{3})\cdot\text{g} \tag{2}\]  
The tyre-plate interaction force F\({}_{\text{op}}\) [N] is equal to the sum of the static load N\({}_{\text{st}}\) [N] and the dynamic component of the vertical force F\({}_{\text{dz}}\) [N] in the tyre; the latter is equal to the sum of the dynamic spring force F\({}_{\text{dso}}\) [N] (measured in relation to the state of static equilibrium, i.e., for the radial deformations of the tyre relative to the static deformation) and the force F\({}_{\text{two}}\) [N] of viscous damping in the tyre:  
\[\text{F}_{\text{op}}=\text{F}_{\text{dz}}+\text{N}_{\text{st}} \tag{3}\]  
\[\text{F}_{\text{dz}}=\text{F}_{\text{dso}}+\text{F}_{\text{two}} \tag{4}\]  
The diagnostic tester measures the force F\({}_{\text{opm}}\) [N], which includes the dynamic component F\({}_{\text{md}}\) [N] and the static reaction force N\({}_{\text{stm}}\) [N] recorded at the beginning of the test. Figure 2: Schematic diagram of the ‘quarter-car’ model (for the notation, see the text): (**a**) general form of the linear model; (**b**) linear model for vehicle vibration testing on a diagnostic tester; (**c**) non-linear model for vehicle vibration testing on a diagnostic tester (with representation of the forces acting in the suspension and the tyre). The vibration plate's weight \(\text{m}_{3}\cdot\text{g}\) [N] and the force \(\text{F}_{\text{bp}}\) [N] of its inertia (negative) are also taken into account. \[\text{F}_{\text{opm}}=\text{F}_{\text{op}}-\text{F}_{\text{bp}}+ \text{m}_{3}\text{g}= \text{F}_{\text{op}}+\text{m}_{3}\frac{\text{d}^{2}\zeta(\text{t})}{ \text{dt}^{2}}+\text{m}_{3}\text{g}= \text{F}_{\text{dz}}+\text{N}_{\text{stm}}+\text{m}_{3}\frac{\text{d }^{2}\zeta(\text{t})}{\text{dt}^{2}}=\text{F}_{\text{md}}+\text{N}_{\text{stm}} \tag{5}\] \[\text{F}_{\text{bp}}=-\text{m}_{3}\frac{\text{d}^{2}\zeta(\text{t} )}{\text{dt}^{2}}\] (6) \[\text{F}_{\text{md}}=\text{F}_{\text{dz}}+\text{m}_{3}\frac{\text{d }^{2}\zeta(\text{t})}{\text{dt}^{2}} \tag{7}\]  
As mentioned above, \(\text{F}_{\text{md}}\) [N] is a result of the measurement of the dynamic component (where the static load \(\text{N}_{\text{stm}}\) [N] is not taken into account) of the force \(\text{F}_{\text{dz}}\) [N] in the tyre. The result of measurement of the force \(\text{F}_{\text{dz}}\) (i.e., the value of this force) and, in consequence, the value of force \(\text{F}_{\text{op}}\) as well, are distorted by the force of inertia \(\text{F}_{\text{bp}}\) [N] and weight \(\text{m}_{3}\cdot\text{g}\) [N] of the vibration plate.