energies Article Striated Tire Yaw Marks—Modeling and Validation Wojciech Wach * and Jakub Z˛ebala Institute of Forensic Research in Kraków, 31-033 Kraków, Poland; [email protected] * Correspondence: [email protected]; Tel.: +48-12-61-85-763 Abstract: Tire yaw marks deposited on the road surface carry a lot of information of paramount importance for the analysis of vehicle accidents. They can be used: (a) in a macro-scale for establishing the vehicle’s positions and orientation as well as an estimation of the vehicle’s speed at the start of yawing; (b) in a micro-scale for inferring among others things the braking or acceleration status of the wheels from the topology of the striations forming the mark. A mathematical model of how the striations will appear has been developed. The model is universal, i.e., it applies to a tire moving along any trajectory with variable curvature, and it takes into account the forces and torques which are calculated by solving a system of non-linear equations of vehicle dynamics. It was validated in the program developed by the author, in which the vehicle is represented by a 36 degree of freedom multi-body system with the TMeasy tire model. The mark-creating model shows good compliance with experimental data. It gives a deep view of the nature of striated yaw marks’ formation and can be applied in any program for the simulation of vehicle dynamics with any level of simplification. Keywords: yaw marks striations; tire model; multibody dynamics; vehicle accident simulation 1. Introduction Citation: Wach, W.; Z˛ebala,J. The rectified projection of tire yaw marks deposited on the road surface (often either Striated Tire Yaw Marks—Modeling from an orthophotomap, a point cloud or a total station survey) can be used in a macro-scale and Validation. Energies 2021, 14, for: 4309. https://doi.org/10.3390/ • determination of subsequent vehicle positions and orientation during critical move- en14144309 ment by matching the position of individual wheels to the corresponding marks; • estimation of the vehicle’s speed at the start of yawing. Academic Editor: Aldo Sorniotti For the latter—assuming the vehicle was moving on a horizontal and homogeneous Received: 16 June 2021 surface along a curve of radius r—the critical speed formula (CSF) is most commonly used: Accepted: 15 July 2021 p Published: 17 July 2021 v = m·g·r, (1) r g 2 Publisher’s Note: MDPI stays neutral where: m—tire-to-roadway coefficient of friction, —radius of the mark, = 9.8 m/s — with regard to jurisdictional claims in gravitational acceleration. published maps and institutional affil- Although this formula may seem simplistic, it has been shown repeatedly over the iations. 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 Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. braking causes the CSF to overestimate the speed at the start of the yaw mark and that This article is an open access article a 50 ft. chord appears to give acceptably low effective braking-related errors (<7%) for distributed under the terms and speeds of approximately 72 km/h (45 mph) and for speeds of approximately 97 km/h conditions of the Creative Commons (60 mph) with light to moderate braking. Cliff et al. [4] have concluded that when using the Attribution (CC BY) license (https:// peak coefficient of friction, both the CSF and simulation over-estimated the actual speed, creativecommons.org/licenses/by/ whereas slide coefficient of friction under-estimated them. Braking tended to increase the 4.0/). results. Amirault and MacInnis [5] carried out a total of 29 tests at speeds of 80 to 95 km/h. Energies 2021, 14, 4309. https://doi.org/10.3390/en14144309 https://www.mdpi.com/journal/energies Energies 2021, 14, 4309 2 of 21 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 Figure1 of their article they show a mark on Energies 2021, 14, x FOR PEER REVIEW 3 of 21 Energies 2021, 14, 4309 3 of 21 striations reflect point loading of the tread shoulder blocks. In Fig. 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 thewhich shoulder two stripes blocks. can A bescheme distinguished: of formation lighter of such striations a mark from they the demonstrate inside, being in deposited Fig. 3 of [13].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]. TheThe authors authors of of this this work, work, in in their their research research practice, practice, have have encountered encountered all allthe the types types of yawof yaw marks marks mentioned mentioned before—two before—two examples examples are are shown shown in in Figure Figure 1.1 .In In both both cases the vehiclevehicle was was yawing, yawing, but but the the tire tire mark mark (a) (a) was was deposited deposited by by a a buckled, buckled, zero-pressure zero-pressure tire tire duringduring a a clockwise clockwise yaw, yaw, while while the the tire tire mark mark (b) (b) was was left left by by the the leading leading shoulder shoulder blocks blocks of of thethe normal-pressure normal-pressure tire tire duri duringng a a counter-clockwise counter-clockwise yaw.