3.3.2. Forces in the Vehicle-Trailer Coupling Device

The generalized external forces *Fsi* also include the coupling device force *Rs*. The coupling device equation, being at the same time an equation of the constraints imposed upon the trailer, has been written as follows:

$$\begin{aligned} \mathcal{R}\_{AX} + \mathcal{R}\_{BX} &= \Delta \mathcal{R}\_{1zad} \\ \mathcal{R}\_{AY} + \mathcal{R}\_{BY} &= \Delta \mathcal{R}\_{2zad} \\ \mathcal{R}\_{AZ} + \mathcal{R}\_{BZ} &= \Delta \mathcal{R}\_{3zad} \end{aligned} \tag{12}$$

where:

*RAX*, *RAY*, *RAZ*, *RBX*, *RBY*, *RBZ*—components of the coupling device force for motorcar *A* and trailer *B*, as appropriate, calculated in every step of the integration of model equations and expressed in the global coordinate system {*O*};

Δ*R*1*zad*, Δ*R*2*zad*, Δ*R*3*zad*—acceptable values of the differences between components of the coupling device force.

#### 3.3.3. Model of the Tire–Road Interaction

The tire–road interaction has been described with the use of the non-linear TMeasy model [33–36]. This model makes it possible to determine the external forces *FxT* and *FyT*, generated in the tire-road contact area and acting from the road via the suspension system onto the vehicle. These forces are functions of the normal tyre-road contact force *FzT*(*t*) and of the longitudinal tyre slip ratio *sxT* and lateral tyre slip ratio *syT*; for individual wheels, they are calculated from equations:

$$F\_{\underline{x}T}(\mathbf{s}\_{\underline{x}T},t) = \mu \,\mu\_{\mathbf{x}}(\mathbf{s}\_{\underline{x}T}) \, F\_{\underline{z}T}(t) \tag{13}$$

$$F\_{yT}(s\_{yT}, t) = \mu \,\,\mu\_y(s\_{yT}) \,\, F\_{zT}(t) \tag{14}$$

where:

*μ*—local tire-road adhesion coefficient;

*μx*(*sxT*), *μ<sup>y</sup> syT* —unit longitudinal and lateral forces as characteristics describing the properties of a specific tire model as functions of tire slip ratio;

*FzT*(*t*)—current value of the normal tire-road contact force for each wheel.

Figure 7 shows, inter alia, the tire velocity vectors, which are necessary for determining the tire slip ratio. The position of the center of the tire-road contact area (point *OT*) has been defined using the local coordinate systems attached to vehicle bodies {*Cs*} and to vehicle wheels {*OTxTyTzT*}, shown in Figure 6.

**Figure 7.** Kinematics and dynamics of the steerable wheel.

Based on Figure 7, the longitudinal tire slip ratio *sxT* and lateral tyre slip ratio *syT* have been determined for the *u*th wheel:

$$s\_{iT} = \frac{v\_{\text{uxT}}}{(v\_T + v\_{\text{uxT}})} \tag{15}$$

$$s\_{yT} = \frac{v\_{uyT}}{\left(v\_T + v\_{uxT}\right)}\tag{16}$$

where:

*vu*—*u*th wheel slip velocity, resulting from longitudinal and lateral slip velocities, i.e., *vuxT* and *vuyT*, respectively;

*vT* + *vuxT*—longitudinal component of the wheel center velocity vector;

*vT*—circumferential velocity of the tire, resulting from the rotational wheel motion.

The tire sideslip angle has been determined from the equation:

$$\kappa = -\arctan{\frac{\upsilon\_{\text{u}\text{y}T}}{(\upsilon\_{T} + \upsilon\_{\text{u}\text{x}T})}}\tag{17}$$

The values of the tire slip ratios according to (15) and (16) are necessary to calculate the forces *FxT* and *FyT* tangential to the road surface. The elastodynamic tire characteristics have a significant impact on the CT unit's behavior in the road situation under consideration (high traveling speed with high values of longitudinal slip ratio and sideslip angle of vehicle tires).

The tire model parameter values taken for this study have been based on the results of the experimental testing of tires 185R14C and 235/60R16 [37–39].

Figure 8 shows an example comparison of tangential force curves *Fx*(*sx*) and *Fy sy* for tests carried out on road surfaces with adhesion coefficients of *μ*<sup>1</sup> = 0.8 (dry asphalt concrete) and *μ*<sup>2</sup> = 0.8 (wet road). The *Fy*(*α*) curve has been plotted for only one road surface type. The curves presented show that the maximum values of the tangential reactions at the tire-road contact area occur at a slip ratio of about 0.15 and a tire sideslip angle of about 8 deg. Further growth in the slip ratio causes a reduction in the tangential reactions and, in consequence, increasing deviation of the vehicle's motion from the trajectory planned.

**Figure 8.** Non-linear characteristics of the dependence of tangential reactions on tire slip ratio and sideslip angle.

#### **4. Validation of the Model of a CT Unit**

The model validation was preceded by a parametrization process based on the results of measurements of mass distribution, as well as dimensions and characteristics of the suspension system and tires, carried out on the vehicle combination under test. The vehicle combination under test can be seen in Figure 9, when it was performing a dynamic lanechange maneuver according to the ISO 3888-1 standard [40]. Results of the measurements carried out were compared with results of simulation of a double lane-change maneuver.

**Figure 9.** Experimental tests.

During the validation tests, a road infrastructure model according to the ISO 3888-1 standard and a CT unit control system model according to Figures 2, 3 and 5 were also used. The trajectory of the center of mass of the motorcar model *yM*(*x*) was specially selected for the steering wheel angle *δH*(*x*) obtained to be in conformity with the curve recorded during the experimental tests. The result of such a model validation procedure has been presented in Figure 10.

**Figure 10.** Comparison of the steering wheel angle curves *δH*(*x*) obtained in the model tests and experimental tests carried out to validate the CT model.

Example comparisons of results of the experimental and simulation tests have been presented in Figure 11.

**Figure 11.** *Cont*.

**Figure 11.** Comparison of results of the experimental and simulation tests of the CT unit performing a double lane-change maneuver with a speed of *v* = 60 km/h (left: motorcar; right: trailer).

Figure 11 shows the validation results. For model validation purposes, the following results of the model and experimental tests have been compared with each other, separately for the motorcar (*s* = *A*) and for the trailer (*s* = *B*): trajectory *yCs*(*x*) and lateral acceleration *ayCs*(*x*) of the centre of vehicle mass, vehicle roll angle *<sup>ϕ</sup>s*(*x*) and velocity . *ϕs*(*x*), and vehicle yaw angle *<sup>ψ</sup>s*(*x*) and velocity . *ψs*(*x*). The conformity between the extreme and mean characteristic values at various stages of the maneuver performed was analyzed. The validation process, based on the results of parametrization of the CT unit and control system models, resulted in the obtaining of good consistency between the kinematics of the model and the actual motion of the real object (i.e., the motorcar and the trailer). The results of this highly multi-aspect assessment may be considered as confirming good agreement between the profiles of individual physical quantities, which means good agreement between the characteristics of the object and its model.
