## 3 Mathematical Model  
The developed model adopted a two-degrees-of-freedom (DOF) quarter-car suspension system. The quarter-car suspension model has been widely used in the literature since the 1970s [38; 39; 40]. The implemented model described vertical vibrations of the sprung mass (vehicle body) and unsprung mass (including wheels, elements of suspension, and brake system). The general equations of motion for the nonlinear quarter-car model can be presented as follows [41]:  
\[m_{s}\ddot{x}_{s}=-F_{se}-F_{sd}, \tag{1}\]  
\[m_{u}\ddot{x}_{u}=-F_{ue}-F_{ud}+F_{se}+F_{sd}, \tag{2}\]  
where:  
\(m_{s}\) is the sprung mass;  
\(m_{u}\) is the unsprung mass;  
\(x_{s}\) is the vertical displacement of the sprung mass;  
\(x_{u}\) is the vertical displacement of the unsprung mass;  
\(F_{se}\) is the elasticity force between the sprung and unsprung masses (suspension elasticity);  
\(F_{sd}\) is the damping force between sprung and unsprung masses (suspension damping);  
\(F_{ue}\) is the elasticity force between the road and unsprung mass (tire elasticity);  
\(F_{ud}\) is the damping force between the road and unsprung mass (tire damping). The elasticity and damping forces are usually approximated by polynomials, the degrees of which depend on the suspension and tire characteristics. A suspension system with asymmetric (different for compression and rebound) and progressive elastic-damping elements may even require a 5 th-degree polynomial:  
\[F_{se}=A_{se}u_{s}^{5}+B_{se}u_{s}^{4}+C_{se}u_{s}^{3}+D_{se}u_{s}^{2}+E_{se}u _{s}+F_{se}, \tag{3}\]  
\[F_{sd}=A_{sd}u_{s}^{5}+B_{sd}u_{s}^{4}+C_{sd}u_{s}^{3}+D_{sd}u_{s}^{2}+E_{sd}u _{s}+F_{sd}, \tag{4}\]  
where \(u_{s}\) is the suspension deflection, and \(A_{se}\div F_{se}\) and \(A_{sd}\div F_{sd}\) are coefficients of the polynomials. Similar formulas can be given for tires. However, it should be noted that pneumatic tires mostly show linear characteristics with negligibly low damping forces. The presented model was adapted to the case considered during the research experiment. The one-sided amplitude of the test plate (showed in previous section) was equal to 3 mm. Thus, the deflections of the suspension and tires were very limited. For this reason, the application of a simple linear model was fully justified in further analysis (confirmed by simulations). The mass elements of the system were connected by a spring-damper linear module representing properties of the wheel suspension (Figure 3). The spring-damper linear module between the unsprung mass and the ground represented the wheel properties in the radial direction. Figure 3: Two-degree-of-freedom quarter-car suspension system. The linear system with two-DOF can be described using Newton or Euler-Lagrange methodology by a system of two differential equations in the following form [39]:  
\[m_{s}\ddot{x}_{s}+c_{s}\big{(}\dot{x}_{s}-\dot{x}_{u}\big{)}+k_{s}(x_{s}-x_{u})=0, \tag{5}\]  
\[m_{u}\ddot{x}_{u}+c_{u}\big{(}\dot{x}_{u}-\dot{x}_{p}\big{)}+k_{u}\big{(}x_{u}-x_ {p}\big{)}-c_{s}\big{(}\dot{x}_{s}-\dot{x}_{u}\big{)}-k_{s}(x_{s}-x_{u})=0, \tag{6}\]  
where:  
\(c_{s}\) is the suspension damping coefficient;  
\(c_{u}\) is the tire damping coefficient;  
\(k_{s}\) is the suspension stiffness coefficient;  
\(k_{u}\) is the tire radial stiffness coefficient;  
\(x_{p}\) is the vertical displacement of the suspension test plate. In the above model, two assumptions must be noted:  
1. The tire always remains in contact with the ground (suspension test plate);
2. There are no stroke limits for the suspension components. The adopted equations of motion can be presented in a more expedient state-space form:  
\[\dot{x}=Ax+Bu, \tag{7}\]  
where \(x\) as a state vector and u as a control input can be written as follows:  
\[x= \begin{bmatrix}x_{1}\\ x_{2}\\ x_{3}\\ x_{4}\end{bmatrix} = \begin{bmatrix}x_{u}\\ \dot{x}_{u}\\ x_{s}\\ \dot{x}_{s}\end{bmatrix} ,\hskip 14.226378 ptu= \begin{bmatrix}u_{1}\\ u_{2}\end{bmatrix} = \begin{bmatrix}x_{p}\\ \dot{x}_{p}\end{bmatrix} .