## 2 Materials and Methods
For the positions of the extremes of the curves under analysis relative to each other to be described (in terms of both the time and phase of the process), the dynamic components of the said resultant forces are important. Therefore, the authors focused mainly on forces F\({}_{\text{dz}}\) and F\({}_{\text{md}}\) (see Figure 2b,c and Formulas (4) and (7)). Relation (20) presents the equations of motion of the non-linear 'quarter-car' model. The equations have also been derived from the principle of dynamic force analysis. \[ \begin{cases}\ddot{\zeta}_{1}=\frac{\text{F}_{\text{S}_{1}}+\text{F}_{\text{S }_{1}}+\text{F}_{\text{H}_{1}}}{\text{m}_{1}}-\text{g}\\ \ddot{\zeta}_{2}=\frac{-\text{F}_{\text{S}_{1}}-\text{F}_{\text{S}_{1}}-\text {F}_{\text{H}_{1}}+\text{F}_{\text{S}_{2}}+\text{F}_{\text{H}_{2}}}{\text{m}_ {2}}-\text{g}\end{cases}  \tag{20}\]  
where \(\ddot{\zeta}_{1}\) and \(\ddot{\zeta}_{2}\) are accelerations of the sprung and unsprung mass, respectively; F\({}_{\text{S}_{1}}\), F\({}_{\text{S}_{1}}\), and F\({}_{\text{H}_{1}}\) are forces of elasticity, sliding friction, and viscous damping in the suspension system, respectively; F\({}_{\text{S}_{2}}\) and F\({}_{\text{t}_{2}}\) are tyre spring and damping (substitute viscotic) forces, respectively; and g is gravitational acceleration. The spring force in the suspension (21) was approximated by a linear function and a third-degree polynomial. \[\text{F}_{\text{S}_{1}}= \begin{cases}\text{A}\text{1}\text{s}_{1}\cdot\text{u }_{1}&\text{for $\text{u}_{1}\leq\text{u}_{1\text{gr}}$}\\ \text{A}\text{2}\text{s}_{1}\cdot\text{u}_{1}^{3}+\text{B}\text{2}\text{s}_{1 }\cdot\text{u}_{1}^{2}+\text{C}\text{2}\text{s}_{1}\cdot\text{u}_{1}+\text{D }\text{2s}_{1}&\text{for $\text{u}_{1}>\text{u}_{1\text{gr}}$}\end{cases}  \tag{21}\]  
where A\(\text{1}\text{s}_{1}\) and A\(\text{2}\text{s}_{1}\) through D\(\text{2}\text{s}_{1}\) are coefficients of the functions that describe (as appropriate) the force of suspension elasticity at small and large deflections, respectively; u\({}_{1\text{gr}}\) is the threshold of applicability of different formulas describing the spring force; u\({}_{1}\) is the suspension deflection defined by Formula (22). \[\text{u}_{1}=\zeta_{2}-\zeta_{1}+\text{om}_{0} \tag{22}\]  
where \(\text{om}_{0}\) is the difference between coordinates \(\zeta_{1}\) and \(\zeta_{2}\), corresponding to \(Fs_{1}=0\). The sliding friction force in the suspension has been expressed by Equation (23). \[\text{F}\text{ts}_{1}= \begin{cases}\text{A}\text{ts}_{1}\cdot\text{sgn}\dot{ \text{u}}_{1}&\text{for}\big{|}\dot{\text{u}}_{1}\big{|}>\dot{\text{u}}_{1 \text{gr}}\\ \text{A}\text{ts}_{1}\cdot\frac{\dot{\text{u}}_{1}}{\text{u}_{1\text{gr}}}& \text{for}\big{|}\dot{\text{u}}_{1}\big{|}\leq\dot{\text{u}}_{1\text{gr}}\end{cases}  \tag{23}\]  
where \(\text{A}\text{ts}_{1}\) is the amplitude of the sliding friction force in the suspension system; \(\dot{\text{u}}_{1\text{gr}}\) is the threshold value of velocity \(\dot{\text{u}}_{1}\) of the suspension deflection, according to (24). \[\dot{\text{u}}_{1}=\dot{\zeta}_{2}-\dot{\zeta}_{1} \tag{24}\]  
The viscous damping force in the suspension has been expressed by Equation (25). \[\text{F}\text{t}_{1}= \begin{cases}-\text{i}_{x}\cdot\text{c}_{0}\cdot\big{(}- \dot{\text{u}}_{1}\cdot\text{i}_{2}\big{)}^{\text{W}\text{o}}&\text{for $\dot{\text{u}}_{1}<0$}\\ \text{i}_{x}\cdot\text{c}_{u}\cdot\big{(}\dot{\text{u}}_{1}\cdot\text{i}_{2} \big{)}^{\text{W}\text{u}}&\text{for $\dot{\text{u}}_{1}\geq 0$}\end{cases}  \tag{25}\]  
where i\({}_{x}\) is a coefficient that describes the kinematic relations in the suspension; c\({}_{0}\) and c\({}_{u}\) are coefficients of rebound and compression damping in the shock absorber, respectively; Wo and Wu are exponents of the functions describing the rebound and compression damping in the shock absorber, respectively. The spring force in the pneumatic tyre was approximated by a third-degree polynomial (26).