*6.3. Stage 3. Verification of the Striation Creation Model*

In Figure 11 the striated yaw marks generated in the simulation have been superimposed on the actual ones (note that on the orthophotomap you can also see irrelevant marks from earlier runs). There is a strong convergence in both the paths of the marks and the direction of the striations, as shown in the close-ups of the time points *t* = 8.9 s, *t* = 9.8 s and *t* = 10.9 s. At *t* = 8.9 s, the virtual mark of the front right wheel is slightly shifted to the right of the actual mark, but this is due to fact that the authors refrained from fine-tuning the simulation indefinitely.

**Figure 11.** Comparison of striated yaw marks generated in simulation with actual ones (top view).

### *6.4. Example*

Figure 12 shows yaw marks generated in the simulation of a severe step steer in a left turn maneuver resulting in breaking the adhesion of the tires on the roadway and the vehicle yawing. Basically, this maneuver is similar to that of Section 6, but what is especially interesting here are typical yaw mark characteristics: variation of the width depending on the vehicle orientation with respect to the CG velocity direction, change of the striation angle with respect to the tangent to the yaw mark (straightening), interweaving and fading.

**Figure 12.** Yaw marks: (**a**) entire movement; (**b**) close-up of the end of yaw marks; (**c**) 3D view.

Remarks. The yaw mark creation model will still be valid on arbitrary 3D roads and typical road irregularities. Generally, bumps or potholes should result in gaps in the created yaw marks, but this problem has yet to be investigated.

In case of uncommon suspension systems and/or different tire types, e.g., forklift trucks [25,26] or other uncommon vehicles [27], the response of the unsprung masses is not expected to affect the striated yaw mark creation model effectiveness as long as the vehicle is moving along a non-deformable surface and the road wavelength is not larger than the tire circumference.

### **7. Conclusions**


**Author Contributions:** Conceptualization, W.W.; formal analysis, W.W.; methodology, W.W.; software W.W.; validation, W.W.; visualization, W.W; writing—original draft preparation, W.W.; resources, J.Z.; review and editing, W.W and J.Z. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable

**Acknowledgments:** This work was supported by the Institute of Forensic Research in Krakow— Project No. I/W/2017.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **Appendix A**

*Code 1* // Start of simulation int *n*; // Number of elements

```
int k; // Index
double t = 0; // Actual time of simulation
double α0 = 0; // Initial angular position of element 0
void CTire::Angular_position_of_block()
{
  from k = 1 to n
  {
     αk = αk−1 + dk
                   r0 [rad]; // Table α[k]; see equation (14)
     k = k + 1;
  }
}
Code 2
int q = n − 1; // index
from k = 0 to n − 1
{
  if (αk ≥ α
             t ) & 
                    αq < α
                           t

  {
     k first = k;
     exit;
  }
  else
  {
     q = k;
     k = k + 1;
  }
}
Code 3
int q = n − 1;
from k = 0 to n − 1
{
  if 
      αk > α
             t

                &

                   αq ≤ α
                           t

  {
     klast = q;
     exit;
  }
  else
  {
     q = k;
     k = k + 1;
  }
}
Code 4
from k = k first to klast
{
  // Determining the position of the tire-road contact rIQ,I from equation (3):
  r_IQ_I();
  // Determining the point Bk position, along the tire longitudinal direction
  // (see Figure 6):
  if (tan αk == π
                  2 )
     dx = 0;
  else
```

$$\begin{array}{l} d\_{x} = \frac{r\_{\flat}}{\tan a\_{k}}; //\text{sec (22)}\\ \rho = \frac{1}{2} \left( |y\_{\mathcal{E}}| + \frac{r\_{0}^{2}}{|y\_{\mathcal{E}}|} \right); //\text{sec (21)}\\ d\_{y} = \frac{l\_{y}}{2} - \sqrt{\rho^{2} - d\_{x}^{2}} + \rho; //\text{sec (25)}\\ \text{if } F\_{y} > 0\\ \mathbf{r}\_{I\mathcal{B}\_{b}I} = \mathbf{r}\_{IQ,l} + d\_{x}\mathbf{e}\_{x,l} - d\_{y}\mathbf{e}\_{y,l}; ~\text{// left turn}---\text{the yaw mark is drawn}\\ //\text{ by the right three shoulder; see (26)}\\ \mathbf{r}\_{I\mathcal{B}\_{b}I} = \mathbf{r}\_{IQ,l} + d\_{x}\mathbf{e}\_{x,l} + d\_{y}\mathbf{e}\_{y,l}; //\text{right turn}---\text{the yaw mark is drawn}\\ //\text{left 1111111} \end{array}$$

// by the left tire shoulder

}

*Code 5* { *α*0*<sup>i</sup>* = *α*0*<sup>i</sup>* + Ω*ih*; // [rad] if *α*0*<sup>i</sup>* ≥ 2*π α*0*<sup>i</sup>* = *α*0*<sup>i</sup>* − 2*π*; // [rad] Angular\_position\_of\_block(); };

#### **References**


**Krzysztof Ostrowski 1,\* and Marcin Budzynski <sup>2</sup>**


**Abstract:** Rural two-lane highways are the most common road type both in Poland and globally. In terms of kilometres, their length is by far greater than that of motorways and expressways. They are roads of one carriageway for each direction, which makes the overtaking of slower vehicles possible only when there is a gap in the stream of traffic moving from the opposite direction. Motorways and express roads are dual carriageways that are expected to support high speed travel mainly over long distances. Express roads have somewhat lower technical parameters and a lower speed limit than motorways. Two-lane highways are used for both short- and long-distance travel. The paper presents selected studies conducted in Poland in 2016–2018 on rural two-lane highways and focuses on the context of the need for their reliability. The research was carried out on selected short and longer road sections located in various surroundings, grouped in terms of curvature change rate CCR, longitudinal slopes and cross-sections (width of lanes and shoulders). The studies of traffic volumes, travel time and travel speed, as well as traffic density, will be used to analyze traffic performance and identify measures of travel time reliability. The analyzed roads were characterized by good technical parameters and significant variability of traffic volume throughout the day, week and year. Some roads experience congestion, i.e., situations in which traffic volume *Q* is close to or above respective road capacity *C*. In order to determine the form of the suitable reliability measures, it will be important to determine the extent to which a road's geometric and traffic characteristics impact travel speed and time. The paper presents well-known reliability measures for dual carriageways and proposes new measures, along with an evaluation of their usefulness in the assessment of the functioning of two-lane highways.

**Keywords:** reliability measures; two-lane highways; travel speed; travel time; empirical research
