## 3 Mathematical Model
\tag{8}\]  
Taking the above into account, the adopted model can be presented in matrix form:  
\[\dot{x}= \begin{bmatrix}\dot{x}_{1}\\ \dot{x}_{2}\\ \dot{x}_{3}\\ \dot{x}_{4}\end{bmatrix} = \begin{bmatrix}0&1&0&0\\ -\frac{k_{u}+k_{s}}{m_{u}}&-\frac{c_{u}+c_{s}}{m_{u}}&\frac{k_{u}}{m_{u}}& \frac{c_{s}}{m_{u}}\\ 0&0&0&1\\ \frac{k_{u}}{m_{s}}&\frac{c_{s}}{m_{s}}&-\frac{k_{s}}{m_{s}}&-\frac{c_{s}}{m_{ s}}\\ \end{bmatrix}\begin{bmatrix}x_{1}\\ x_{2}\\ x_{3}\\ x_{4}\end{bmatrix} + \begin{bmatrix}0&0\\ \frac{k_{u}}{m_{u}}&\frac{c_{s}}{m_{u}}\\ 0&0\\ 0&0\end{bmatrix}\begin{bmatrix}u_{1}\\ u_{2}\end{bmatrix} . \tag{9}\]  
After minor transformations, the model can be written as follows:  
\[\dot{x}= \begin{bmatrix}\dot{x}_{1}\\ \dot{x}_{2}\\ \dot{x}_{3}\\ \dot{x}_{4}\end{bmatrix} = \begin{bmatrix}x_{2}\\ -\frac{c_{u}}{m_{u}}(x_{2}-u_{2})-\frac{k_{u}}{m_{u}}(x_{1}-u_{1})+\frac{c_{ s}}{m_{u}}(x_{4}-x_{2})+\frac{k_{s}}{m_{u}}(x_{3}-x_{1})\\ x_{4}\\ -\frac{c_{s}}{m_{s}}(x_{4}-x_{2})+\frac{k_{s}}{m_{s}}(x_{3}-x_{1})\end{bmatrix} . \tag{10}\]The first of the mentioned methods is inconvenient in practical implementation. In this case, the spring element and damper should be dismantled to obtain their characteristics on a special test stand. Moreover, an accurate suspension scheme is needed for the determination of the suspension ratio. In the second approach, the measurements are realized on the entire vehicle. This test does not require penetration into the vehicle or knowledge of the suspension structure. Additionally, performing the measurements on a complete car allows for the consideration of forces and torques resulting from friction in the suspension joints. Therefore, such a measurement better reflects the real working conditions of suspension. The suspension stiffness coefficient \(k_{s}\) (as well as the tire radial stiffness coefficient \(k_{u}\)) may be determined relatively easily in this way (static experiment). The damping coefficients are much more difficult to determine because dynamic conditions for testing are required (in the adopted model, the damping forces depended on the speed of the moving mass). Therefore, for the purposes of this study, the values of these coefficients were taken from the literature. The suspension stiffness characteristic and tire radial stiffness characteristic were determined in accordance with Polish standard BN-57/3615-06 [42]. The wheels of the analyzed axle were placed on the scales (Figure 3). Suspension deflection is usually accompanied by changes in the wheelbase or the toe-in. Rotary-sliding plates between the wheels and the plates of the scales ensured the elimination of tangential forces (the suspension was only then loaded with vertical force). During measurement, the considered suspension should be overloaded by an amount equal to 50% of the static load (passenger car). The wheels of the considered axle were loaded approximately equally. The changes in the vertical forces acting on each wheel were realized by a screw jack, which was used to lift the vehicle body. In every measurement step, the following parameters were noted: normal forces (masses denoted from the scales), heights of previously defined points on the bodywork (right and left sides; Figure 4), and heights of previously defined points on the rims (right and left sides; Figure 4). The obtained data allowed for the performance of the stiffness characteristics for suspension and tires in the radial direction. The characteristics presented the dependence of the vertical force on the deflection of the suspension or tires. On this basis, the stiffness coefficients (\(k_{u}\) and \(k_{s}\)) could be determined. Figure 5 depicts the tested vehicles during measurement of the suspension stiffness coefficient. Figure 4: Test stand for determining suspension stiffness characteristics: 1, rotary-sliding plate; 2, screw jack; 3, electronic scales; HP and HL, heights of points on the bodywork (right and left side, respectively); hP and hL, heights of points on the rims (right and left side, respectively).