**1. Introduction**

The rectified projection of tire yaw marks deposited on the road surface (often either from an orthophotomap, a point cloud or a total station survey) can be used in a macro-scale for:


For the latter—assuming the vehicle was moving on a horizontal and homogeneous surface along a curve of radius *r*—the critical speed formula (CSF) is most commonly used:

$$
v = \sqrt{\mu \cdot \mathbf{g} \cdot \mathbf{r}\_\prime} \tag{1}$$

where: *μ*—tire-to-roadway coefficient of friction, *r*—radius of the mark, *g* = 9.8 m/s2 gravitational acceleration.

Although this formula may seem simplistic, it has been shown repeatedly over the years that, under certain conditions, it can be considered as a quasi-empirical critical speed method (CSM). A broad overview of such conditions has been presented among others by Brach and Brach [1]. Sledge and Marshek [2] examined some refined forms of the CSF which account for the effects of, among others, vehicle weight distribution, slip angles, cornering stiffnesses and ABS. Cannon [3] has demonstrated that effective braking causes the CSF to overestimate the speed at the start of the yaw mark and that a 50 ft. chord appears to give acceptably low effective braking-related errors (<7%) for speeds of approximately 72 km/h (45 mph) and for speeds of approximately 97 km/h (60 mph) with light to moderate braking. Cliff et al. [4] have concluded that when using the peak coefficient of friction, both the CSF and simulation over-estimated the actual speed, whereas slide coefficient of friction under-estimated them. Braking tended to increase the results. Amirault and MacInnis [5] carried out a total of 29 tests at speeds of 80 to 95 km/h.

**Citation:** Wach, W.; Z ˛ebala, J. Striated Tire Yaw Marks—Modeling and Validation. *Energies* **2021**, *14*, 4309. https://doi.org/10.3390/ en14144309

Academic Editor: Aldo Sorniotti

Received: 16 June 2021 Accepted: 15 July 2021 Published: 17 July 2021

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**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

Bearing in mind non-braking and ABS braking tests, using 20 m chord measurements for the radius and the average braking coefficient of friction overestimated the measured speed by 4.1% ± 6.3% (±1σ), while using a center of gravity trajectory for the radius and the average braking coefficient of friction underestimated the measured speed by 2.0% ± 5.2% (±1σ).

Lambourn [6] conducted tests in which passenger cars were freely coasting, braking or under power when travelling in a curve at speed of 55 to 100 km/h, and proposed a procedure which makes it possible to limit the uncertainty of speed calculated from the CSF to ±10%. In [7] and [8] he concluded that his previously described CSM gives satisfactory results also in the event of light braking, heavy braking with ABS, acceleration and the operation of ESP. There was no sign of the cycling of the ABS. The yaw marks had the typical appearance, practically the same as in the case of low braking without ABS. Significantly less yaw or off-tracking were observed when compared with marks deposited with little or no braking. If the brakes were applied aggressively during the yaw, the amount of yaw would decrease. This feature—the reverse of usual yawing with the heavy non-ABS braking—is probably the result of the "select low" algorithm.

While most authors, notably Lambourn [6–8] and Amirault and MacInnis [5], limit their analysis to yawing with relatively low yaw rate, Cash and Crouch [9] derived a formula which accounts for a higher degree of vehicle yaw and any brake force (not only from driver input), resulting in a narrower error band than conventional CSMs. If the exact path and orientation of the collision vehicle are not readily apparent, their method allows the flexibility of considering ranges for the required values.

What distinguishes simulation methods is that they provide a deep view into the time histories of curvilinear movement parameters, including the critical speed, as well as identifying a set of data (e.g., steering angle, braking/accelerating level of particular wheels, yaw moment of inertia etc.) which enables a virtual vehicle to move along the actual tire marks.

The point of this article is the appearance of yaw marks in the micro-scale, that is from the perspective of the topology of striations forming the mark, which make it possible to infer the braking or acceleration status of the wheels (and sometimes even the steering angle of the front wheels). The yaw mark is essentially left by the entire tire footprint remaining in contact with the roadway (contact patch), but its blackness and distinctness depend on the local slip of the tread blocks relative to the road, and stress. The rule "the greater the stress, the more distinct the mark" applies both in the macro (when observing the entire yaw marks, the most pronounced are the marks of the external, loaded wheels) and in the micro scale (the most visible is the outer edge of a single yaw mark).

Yamazaki and Akasaka in their classic article [10] argue that deposition of striations is independent of the tread pattern and is caused by an in-plane bending moment that is transmitted from the roadway to the body of the tire via the tread contact patch during sharp cornering. They refer to this phenomenon as the bending buckling behavior of a steel-belted radial tire. Buckling occurs immediately in the contact patch in the presence of a large lateral reaction from the road and is specific to radial—not bias—tires. Yamazaki in [11] shows that sharp cornering turns on steel-belted radial tires often cause wavy wear along the periphery of the shoulder.

Beauchamp et al. in [12] summarize the literature concerning the yaw mark striations issue, analyze the differences in the mechanism in which striations are deposited, and discuss the relationship between tire mark striations and tire forces. They conclude that in the case of tires with pronounced shoulder blocks the striations are typically produced by these blocks whereas tires with very low pressure or without a tread pattern are more likely to deposit striations by buckling. In the absence of braking and acceleration the striation marks are parallel to the wheel rotation axis. When the brakes are applied aggressively, the striations will change to a direction more in line with the wheel trajectory but ABS prevents the tires from locking. Beauchamp et al. in [13] show examples where the striations reflect point loading of the tread shoulder blocks. In Figure 1 of their article they show a mark on which two stripes can be distinguished: lighter striations from the inside, being deposited by the tread, and darker striations from the outside, being deposited by the shoulder blocks. A scheme of formation of such a mark they demonstrate in Figure 3 of [13].

The authors of this work, in their research practice, have encountered all the types of yaw marks mentioned before—two examples are shown in Figure 1. In both cases the vehicle was yawing, but the tire mark (a) was deposited by a buckled, zero-pressure tire during a clockwise yaw, while the tire mark (b) was left by the leading shoulder blocks of the normal-pressure tire during a counter-clockwise yaw.

**Figure 1.** Yaw marks deposited during a full scale vehicle yaw testing performed by Z ˛ebala et al. [14]: (**a**) resulting from in-plane buckling of a tire with zero pressure; (**b**) left by the tread shoulder blocks of a tire with normal pressure.

The article by Beauchamp et al. [13] does an excellent job of analyzing yaw mark striations from the viewpoint of its geometry and deriving equations for the calculation of longitudinal slip *sx* using the striation marks angle *θ* and the slip angle *α* see analogous formulae (9) and (11) derived in this article). It was shown that the model offers insight into the braking actions of a driver at the time the tire marks were being deposited. The usefulness of such marks for accident analysis depends obviously on their quality and clarity, which affect the uncertainty in the measurement of the geometric parameters. Beauchamp et al. in [15] explore the sensitivity and uncertainty of the *sx* equation. They prove that at *α* = 5◦, the striations will change over 70 degrees between no braking and maximum braking, while at *α* = 85◦, less than 2 degrees separate no braking and maximum braking. In the first case braking will likely be easy to distinguish; in the second, changes in striation angle from braking are unlikely to be detected.

Undoubtedly, all researchers who focus on point loading of the tread shoulder blocks as well as in-plane buckling are right in their specific areas, as in general, the appearance of a striated mark left during curvilinear motion depends on many factors, the main ones being:


However, regardless of whether the striations occur from buckling or tread blocks, their direction always follows the direction of the resultant tire velocity vector in the contact patch (called the wheel slip velocity and hereinafter referred to as **v***IO*). The only difference lies in the pitch of the striations, which in the case of striations created by the tread shoulder blocks gives the opportunity to estimate the longitudinal slip *sx* from the topology of the striations (according to the formula of Beauchamp et al. [13], see also Equations (9) to (12) in this article), while in the case of buckling does not give the same possibility because of lack of a buckling wave pitch (a momentary and unique period of the buckling wave).

An in-depth look at the mechanics of the yaw mark creation is very interesting not only as a mathematical problem, but first of all crucial from the angle of the uncertainty of vehicle accident analysis (see e.g., [16]).

#### **2. Assumptions to the Model**

For the development of a model of creating a striated tire yaw mark the following assumptions have been made:

	- (a) partially, i.e., as to the tire mark course and the striations direction only; it is then sufficient to enter any pitch of the tread shoulder blocks;
	- (b) fully, i.e., as to the tire mark course, and striations direction and pitch, if the buckling pitch in the patch is known in some other way, which can be manually entered into the program.

### **3. Basic Terms Concerning Movement of a Wheel in a Bend**

A concise, computer-friendly vector-matrix notation has been used. The italic small letters (e.g., *v*) mean scalars, bold small letters (e.g., **v**)—vectors, and bold capital letters (e.g., **A**)—matrices. In the vector-matrix equations the Rill's subscript notation [18] has been used:


A characteristic point of the tire-road patch *Q*, which is the origin of the Cartesian coordinate system with unit vectors on its axes **e***x*, **e***y*, **e***<sup>n</sup>* (shown in Figure 2a), is referred to as the contact point. In the TMeasy tire model, the system of forces acting on the wheel is reduced to an equivalent system of forces (*Fx*, *Fy* and *Fz*) and their moments (*Mx*, *My* and *Mz*), whose directions of action coincide with the directions of the unit vectors. The position of the rim center plane in relation to the road is determined by the position vector **r***IO* and the unit vector **e***ky*, normal to this plane and defining the wheel rotation axis.

**Figure 2.** Velocities in the process of depositing striations by tire shoulder blocks during yaw: (**a**) perspective; (**b**) top view.

Camber *γ* as well as cornering cause the tire to deflect laterally and to offset the contact point *Q* against the rim center plane by the distance *ye* (see Section 5.3). The road surface geometry in {*I*} defines the function:

$$z\_s = z\_s(x, y). \tag{2}$$

The current position of the contact point *Q* in the global reference frame {*I*} is given by the formula:

$$\mathbf{r}\_{IQ,I} = \mathbf{r}\_{IO,I} + \mathbf{r}\_{OQ,I} \, \text{ \,\,} \tag{3}$$

where the vector **r***OQ*,*<sup>I</sup>* can be determined by the approach given in [18] or [21].

Figure 2a depicts a diagram of a wheel in lateral slip which rolls across the plane *π* with angular velocity *ω*. The following symbols have been adopted:

*vx* and *vy*—components of the wheel center velocity vector **v***IQ* parallel to *π*, the first of which lies on the direction of the longitudinal axis of the rim, and the other is perpendicular to it;

*γ*—camber angle;

*α*—wheel slip angle;

*ω*—wheel angular velocity about its spin axis given by the unit vector **e***ky*;

*vsx* and *vsy*—components of the absolute velocity vector of the contact point **v***IQ* (wheel slip velocity); *vsx* and *vsy* are parallel to *vx* and *vy*, and read:

$$
v\_{\\$x} = v\_x - r\_d \omega\_\prime \tag{4}$$

$$
v\_{\!\!\!\!y} = v\_{\!\!y}.\tag{5}$$

*rd*—dynamic tire radius;

*ϕ*—direction of the contact point velocity **v***IQ* against the longitudinal axis of the wheel (rim).

According to the ISO definition (see also Pacejka [19]), the longitudinal and lateral relative slips are:

$$s\_x = -\frac{v\_{sx}}{v\_x} = -\frac{v\_x - r\_d\omega}{v\_x},\tag{6}$$

and

$$s\_y = \tan \alpha = -\frac{v\_{sy}}{v\_x} \tag{7}$$

respectively. The vectors of the contact point velocity **v***IQ*, the relative slip **s** = *sx sy T* and the tangential force acting on the tire at the contact point **F** = *Fx Fy <sup>T</sup>* have the same direction, and satisfy the relationship:

$$
\tan \varphi = \frac{v\_{sy}}{v\_{sx}} = \frac{s\_y}{s\_x} = \frac{F\_y}{F\_x} \,\,\,\,\tag{8}
$$

wherein the components *Fx* and *Fy* are calculated according to the TMeasy (or any other) tire model.

It is easy to see that the relation (8) will also be fulfilled with other definitions of slips *sx* and *sy*, because they differ only in the denominator (e.g., at Rill *rd*|*ω*| [18] or at Mitschke max{*rdω*, *vx*} [22]), which will disappear when inserted into the formula (8).

#### **4. Wheel Velocities and Slips Versus Geometry of the Striated Tire Mark**

Figure 2 shows, schematically, the process of making a striated yaw mark on the road surface with the tread shoulder blocks of a tire. The direction of the velocity vector **v***IQ* is also the direction of the contact point *Q* displacement relative to the road and, consequently, the direction of striations deposited on the road by the tire during yaw.

Dividing the formula (7) by (8) gives:

$$s\_x = \frac{\tan a}{\tan q},\tag{9}$$

and because *sy* = tan *α*, hence

$$\mathbf{s}\_y = \mathbf{s}\_x \tan \varphi. \tag{10}$$

These relationships allow the wheel slips *sx* and *sy* to be calculated having only the striated yaw mark, as shown in Figure 3. Unlike the angle *θ* (representing the deviation of the striations direction from the tangent to the yaw mark, which is easy to measure on the mark), the slip and contact point velocity angles—*α* and *ϕ* respectively—are generally unknown because of the unknown orientation of the wheel relative to the mark. This

difficulty applies in particular to the front wheel marks, as the steering angle of these wheels against the vehicle body is variable, and may even change over time, and are therefore impossible to determine by the yaw marks topology alone.

**Figure 3.** Determination of velocities and angles configuration from the striated yaw mark disclosed on the road.

That is why Beauchamp et al. [13], using simple geometric relationships, derived the following formula for the slip angle:

$$\alpha = \arcsin \frac{S \sin \theta}{T} - \theta \ , \tag{11}$$

where the distances *S* and *T* shown in Figure 4 mean:

*S*—the striation pitch measured along the yaw mark;

*T*—the pitch of the tread shoulder blocks measured on the tire circumference.

**Figure 4.** Measurement of distances *S* on the yaw mark and *T* on the tire shoulder.

From Figure 3 it follows that:

$$
\varphi = \mathfrak{a} + \theta.\tag{12}
$$

To sum up, in order to calculate the wheel slippage at a selected point of the striated yaw mark one should:


The uncertainty of the results of such calculations related to non-uniform pitch of the striations S (as a consequence of the uneven tread pitch T aimed at reducing the noise generated by the tire) falls within the general uncertainty of this approach and, above all, the S and T measurement uncertainty.

#### **5. Model of Deposition of the Yaw Mark Striations on the Road Surface**

Figure 5a shows a diagram of tread shoulder blocks pitch, where:

*n*—number of blocks or grooves on the tread shoulder (consistently keeping to the chosen convention);

*Bk*, *k* = 0, . . . , *n* − 1—point indicating the *k*-th block (or groove);

*dk*, *k* = 0, ... , *n* − 1—distances between adjacent points *Bk* measured along an arc: with typical tires, without making a significant error, these can be measured in a straight line;

According to Figure 5a:

$$d\_k = \begin{cases} \text{distance between the points } B\_k \text{ and } B\_{k+1} & \text{for } \quad k = 0, \dots, n-2 \\ \text{distance between the points } B\_k \text{ and } B\_0 & \text{for } \quad k = n-1 \end{cases}$$

*r*0—unloaded tire radius.

**Figure 5.** The diagram of tread shoulder blocks: (**a**) the symbols used to define the pitch of the tread blocks; (**b**) configuration at the instant *i* = 0, *t* = 0.

The values of parameters *dk* and *r*<sup>0</sup> are constant throughout the simulation. In general, the pitch can be defined as non-uniform (*dk* = constant), but in the simplest case, the program may suggest by default a uniform pitch according to the formula:

$$d\_k = \frac{2\pi r\_0}{n}, \; k = 0, \ldots, n - 1. \tag{13}$$

Let's adopt additional symbols:

*i*—number of simulation step;

*t*—current simulation time;

*h*—simulation timestep.

The following calculation algorithm, repeated in each simulation step, can be proposed.
