*5.2. Calculation of the Characteristic Dimensions of the Contact Patch*

Assuming that the tire remains in full contact with the road over the entire tread width *Lb*, and the contact patch has a rectangular shape with length *Lx* and width *Ly*, these dimensions can be calculated using the approximation proposed by Rill in [21]:

$$L\_X = L\_X(t) \approx 2\sqrt{r\_0 \Delta z} = 2\sqrt{r\_0 \frac{F\_z}{c\_R}} \, , \tag{15}$$

$$L\_y = L\_y(t) \approx \frac{L\_b}{\cos \gamma} \,\, \, \, \, \tag{16}$$

where:

Δ*z* is the total tire deflection given by the formula:

$$
\Delta z = \Delta z(t) = r\_0 - r\_s = \frac{F\_z}{c\_R} \tag{17}
$$

*Fz* = *Fz*(*t*)—the normal reaction of the roadway to the tire at the contact point;

*cR* = const—the radial stiffness of the tire;

*rs* = *rs*(*t*)—the loaded (static) tire radius;

*γ* = *γ*(*t*)—the tire camber angle, i.e., the inclination of the rim center plane against the roadway normal.

For example, for a tire 205/55 R16 at *Fz* = 4700 N and the pressure *p* = 2.5 bar, after adopting the data *cr* = 265000 N / m and *r*<sup>0</sup> = 0.317 m, one gets *Lx* ≈ 0.15 m.

#### *5.3. Calculating the Geometry of Striations*

In a single simulation step the coordinates of the points at which the shoulder blocks contact the road should be determined. They are shown in the global Cartesian coordinate system {*I*}.

In reality, the deposition of a yaw mark on the road surface depends on many local or temporary factors that are difficult to discover after the accident. As the main one is a sufficiently high lateral tire force, the following formula can be used to define the condition when the yaw mark should be created in the program:

$$F\_y \ge \frac{p\_{\%}}{100} \,\mu F\_{z\_{\%}} \,\tag{18}$$

where:

*Fy* = *Fy*(*t*)—current lateral force acting on the tire, calculated according to the TMeasy (or any other) tire model;

*Fz* = *Fz*(*t*)—current normal force to the roadway acting on the tire;

*μ*—tire-road friction coefficient; in general *μ* = *μ*(*x*, *y*);

*p*%—percentage of the maximum horizontal force *μFz* at which the yaw mark is to be made; by default *p*% = 95% can be adopted.

The length of the contact patch is limited by two boundary angles *α <sup>t</sup>* and *α <sup>t</sup>* indicating the first and last point, respectively, of the tire circumference being in contact with the road (not to be confused with the point *Bk* indicating the tread element). They are illustrated in Figure 6 and given by the formulae:

$$\alpha\_t' = \arccos \frac{L\_x}{2r\_0} = \arctan \frac{2r\_s}{L\_x} \tag{19}$$

and

$$
\alpha\_t'' = \frac{\pi}{2} + \arcsin\frac{L\_x}{2r\_0}.\tag{20}
$$

**Figure 6.** Configuration at the instant *i* = 0, *t* = 0 (note: the deflection of the tire is here exaggerated for the case of normal pressure).

In order to find the indexes *k* of all points *Bk* in contact with the road, to begin with the indexes of the first and last contact points have to be determined using the following short algorithms.

Determining the index of the first tread block in the contact patch *k first*—see *Code 2* in Appendix A.

Determining the index of the last tread block in the contact patch *klast*—see *Code 3* in Appendix A.

Now the coordinates of the points forming the striations of the yaw mark on the road in {*I*} can be calculated. Figure 7 shows a simplified diagram of the lateral deflection of a tire while driving on a curvilinear track.

In fact, the deflection of the tire in the top view has a slightly more complex shape, but taking into account the assumptions made earlier, only the outer arc-shaped contour of the contact patch will be relevant (see Figure 4a,b in [10]). Thus, without making a significant error, it can be assumed that it is a fragment of a circle with a radius *ρ* determined by chord *ye* and middle coordinate 2*r*<sup>0</sup> from the formula:

$$\rho = \frac{1}{2} \left( |y\_c| + \frac{r\_0^2}{|y\_c|} \right). \tag{21}$$

In Figure 7 the distances *dx* and *dy* lying on the road plane are indicated, which are distances from the tire-road contact point *Q* to the point *Bk* measured along and across the wheel, respectively. The first one is:

$$d\_{\mathcal{X}} = \frac{r\_{\mathcal{s}}}{\tan \omega\_{\mathcal{k}}} \tag{22}$$

and the other

$$d\_y = \frac{L\_y}{2} + y\_c - \delta.\tag{23}$$

Since

$$
\delta = \sqrt{\rho^2 - d\_{\rm x}^2} - \rho + y\_{\varepsilon\,\,\prime} \tag{24}
$$

hence the formula (23) takes the form:

$$d\_y = \frac{L\_y}{2} - \sqrt{\rho^2 - d\_x^2} + \rho \,. \tag{25}$$

**Figure 7.** A simplified diagram of the tire lateral deflection while driving on a curvilinear track.

Finally, the position of the point *Bk* is:

$$\mathbf{r}\_{IB\_k,I} = \mathbf{r}\_{IQ,I} + d\_x \mathbf{e}\_{x,I} - d\_y \mathbf{e}\_{y,I} \, , \tag{26}$$

where: **e***x*,*I*, **e***y*,*<sup>I</sup>* are the unit vectors of the wheel in the point *Q*, shown in Figures 2a and 7. The algorithm for calculation of the vector **r***IBk*,*<sup>I</sup>* components is outlined in the section

*Code 4* in Appendix A.

The next step of the simulation will be:


$$a\_{0\_i} = a\_{0\_{i-1}} + \omega\_i h \, | \, \text{rad} \rangle\_{\prime \prime}$$

where *ω<sup>i</sup>* is the current angular velocity of the wheel about its spin in [rad];


A single striation is drawn by connecting the positions of the point *Bki* with lines in successive, adjacent steps *i*, as long as the condition (18) is satisfied. The striated yaw mark can be being drawn as the simulation proceeds or exported at the end as a drawing file (e.g., dxf).

#### **6. Validation and Discussion**

*6.1. Stage 1. Measurement of Time Histories of Vehicle Motion Parameters (Actual Data)*

The model was validated using the results of one of the full scale vehicle yaw tests performed as part of the Research Project No. VII/W-2014 of the Institute of Forensic Research [14]. The test vehicle was a 2003 Volkswagen Passat 2.0 TD station wagon, with Firestone FireHawk 195/65R15 91T tires with their normal inflation pressure.

Experiments were performed in the summer, on a level, horizontal and dry asphalt road surface. In the test selected for validation, having established the speed of 53 km/h in a straight line the test driver steered the vehicle hard to the left causing the tires to break the grip on the roadway and to deposit striated yaw marks.

As a result of kinematic transformations of the raw measurement data the following parameters were obtained:


For example, assuming the notations as in Figure 8, the absolute velocity of the wheel center *O*, expressed in the local, rim fixed reference frame {*O*}, can be calculated from the vector equation:

$$\mathbf{v}\_{IO,O} = \mathbf{A}\_{OA} \mathbf{A}\_{IA}^{\mathrm{T}} \mathbf{v}\_{IO,I} = \mathbf{A}\_{OA} \left[ \mathbf{A}\_{IA}^{\mathrm{T}} \mathbf{v}\_{IA,I} - (\boldsymbol{\omega}\_{IA,A} \times \mathbf{r}\_{OA,A}) \right],\tag{27}$$

where:


**Figure 8.** Coordinate systems: {*I*}—global, {*A*}—local vehicle-fixed (GPS receiver and/or IMU location), {*O*}—local rim fixed.

Thus, all parameters describing the movement of the vehicle (including all points, especially wheels) in the domains of time and space are known.

#### *6.2. Stage 2. Vehicle Movement Simulation and Its Verification*

The results of the measurements described in the Stage 1 were used to verify the yaw marks creation model, which is the very core of this article. It was implemented by the author-developed multibody dynamics simulation program Model.exe, briefly outlined in the Appendix of [24]. It was assumed that the vehicle is a multibody system with 36 degrees of freedom (DoF), composed of rigid bodies connected by geometric constraints, divided into the following partial-systems: basic—body with wheel suspensions, steering

and drive. The equations of motion were generated using the Jourdain's principle. The tire forces were determined with the TMeasy model described in [17,18].

The vehicle motion was simulated, the model of which was parametrized according to the basic measurements data listed in Table 1. In simulation the measured time histories of the parameters shown in Figure 9 were adopted:



**Table 1.** Technical data of the tested Volkswagen Passat

**Figure 9.** Inputs realized in simulation: (**a**) steering wheel angle; (**b**) torque acting on the front wheels.

The simulation results represented in Figure 10a,b of the time histories of various dynamic parameters, show good agreement with the actual data. In Figure 10c the simulated and actual positions and orientations of the vehicle have been compared. For as long as approximately 12 s they are almost identical, and some divergence of the CG paths occurs only at the very end of the movement. The differences may arise from the approximate torque waveform shown in Figure 9b, and partly from the simplified parametrization of the steering and drive systems and the neglected pavement unevenness.

In Figure 10a, a large discrepancy can be observed for the slip *sx* of the front right wheel at the end of the simulation. The relative longitudinal slip *sx* expressed by the formula (6) is the absolute slip (defined by the numerator) related to the speed *vx* of the wheel center (denominator). Although the absolute slip within the entire time range 6– 13 s is small, the low speed just before the vehicle stops (12–13 s) caused a sharp and disproportionate increase in the value of the relative slip *sx*. In other words, the reason for the sudden increase in the measured *sx* are low values of *vx* and *rdω* together with inaccuracies in the independent measurement of this parameters.

Simulation, as a theoretical process, is free of measurement flaws, hence even small numbers are precise and correlated enough that dividing them gives reasonable results. That is why at the end of the simulation, at low speeds, a disturbing difference between the measured and simulated slips *sx* occurred. This issue should not be overestimated.

**Figure 10.** Comparison of the simulation and measurement results: (**a**) time histories of the longitudinal and lateral accelerations, yaw rate and slips; (**b**) vehicle positions and orientations in top view and in perspective.
