applied sciences Article Assessment of Tire Features for Modeling Vehicle Stability in Case of Vertical Road Excitation Vaidas Lukoševiˇcius*, Rolandas Makaras and Andrius Dargužis Faculty of Mechanical Engineering and Design, Kaunas University of Technology, Studentu˛Str. 56, 44249 Kaunas, Lithuania; [email protected] (R.M.); [email protected] (A.D.) * Correspondence: [email protected] Abstract: Two trends could be observed in the evolution of road transport. First, with the traffic becoming increasingly intensive, the motor road infrastructure is developed; more advanced, greater quality, and more durable materials are used; and pavement laying and repair techniques are im- proved continuously. The continued growth in the number of vehicles on the road is accompanied by the ongoing improvement of the vehicle design with the view towards greater vehicle controllability as the key traffic safety factor. The change has covered a series of vehicle systems. The tire structure and materials used are subject to continuous improvements in order to provide the maximum possi- ble grip with the road pavement. New solutions in the improvement of the suspension and driving systems are explored. Nonetheless, inevitable controversies have been encountered, primarily, in the efforts to combine riding comfort and vehicle controllability. Practice shows that these systems perform to a satisfactory degree only on good quality roads, as they have been designed specifically for the latter. This could be the cause of the more complicated car control and accidents on the lower-quality roads. Road ruts and local unevenness that impair car stability and traffic safety are not Citation: Lukoševiˇcius,V.; Makaras, avoided even on the trunk roads. In this work, we investigated the conditions for directional stability, R.; Dargužis, A. Assessment of Tire the influence of road and vehicle parameters on the directional stability of the vehicle, and developed Features for Modeling Vehicle recommendations for the road and vehicle control systems to combine to ensure traffic safety. We Stability in Case of Vertical Road have developed a refined dynamic model of vehicle stability that evaluates the influence of tire tread Excitation. Appl. Sci. 2021, 11, 6608. and suspensions. The obtained results allow a more accurate assessment of the impact of the road https://doi.org/10.3390/app11146608 roughness and vehicle suspension and body movements on vehicle stability and the development of recommendations for the safe movement down the road of known characteristics. Academic Editors: Flavio Farroni, Andrea Genovese and Keywords: directional stability; road profile; road unevenness; vehicle-road interaction; vertical Aleksandr Sakhnevych vehicle excitation; tire models; tire tread Received: 29 June 2021 Accepted: 16 July 2021 Published: 18 July 2021 1. Introduction Publisher’s Note: MDPI stays neutral The vehicle-road subsystem includes various aspects of the vehicle-road interac- with regard to jurisdictional claims in tion. This subsystem is emphasized where the technical part of transport operations is published maps and institutional affil- investigated. Considerable emphasis has already been placed on the description of the iations. vehicle-road interaction, vehicle movement under various conditions, and substantiation of the requirements for vehicles both in the theoretical studies and in the experimental investigations where the theory has not provided any viable solutions [1–3]. Basic solutions could mistakenly be considered to have been found, with further inquiry allegedly being limited to observations that are necessary to correct the solutions. Nonetheless, certain Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. limitations have been introduced increasingly by a lot of countries in response to the This article is an open access article aggressive development of this type of transport. These limitations have an effect both distributed under the terms and on individual components of the subsystem and on the subsystem itself and promote the conditions of the Creative Commons improvement of both technical and organizational aspects thereof. The road and vehicle Attribution (CC BY) license (https:// as the elements of the vehicle-road subsystem may be described individually as they are creativecommons.org/licenses/by/ viewed and may be explored as individual elements of the system in an investigation of 4.0/). transport systems [4]. Appl. Sci. 2021, 11, 6608. https://doi.org/10.3390/app11146608 https://www.mdpi.com/journal/applsci Appl. Sci. 2021, 11, 6608 2 of 34 Pavement unevenness values are not completely random. An attempt was made to systematize these values. A method enabling assessment of the unevenness height dispersion and correlation function that has been used for a fairly long time is already more appropriate for the modeling [5–8]. The specifics of unevenness lie in that certain patterns between the unevenness height and unevenness wavelength become established during the pavement laying process and use. Hence, the random process model cannot be applied to the road description, as waviness and ruts are registered in the road pavement, in particular, after the use [9–12]. In the stability tasks, two types of unevenness have the greatest effect: road micro- profile that causes excitation of the suspension operation [13,14] and road roughness due to its relation to the grip ratio [15–17]. The vehicle-road interaction is determined by two intermediate links: the tire and the suspension [18,19]. Vehicle stability modeling tasks involve multiple parameters and need to be split into several individual tasks during the analysis. Nonetheless, a more detailed inquiry has shown that the tasks are interrelated, and integrated analysis is needed when dealing with more complex tasks. Analysis of the evolution of trends in traffic accidents and vehicle design has shown that controllability is one of the key operational properties of a vehicle that influences safety. Vehicle controllability is the ability to move following the trajectory designed by the driver [20,21]. The driver sets the desired direction using the steering wheel mechanism, and the controllability tasks are focused on the steering wheel mechanism [22,23]. Deeper inquiry into the potential excitation caused by road unevenness has shown that it is not enough to know the slip characteristics, as the wheel direction vector is influenced by the wheel spatial position, i.e., camber, toe alignment, etc. The dynamic equations cannot be used directly due to the specifics of the tasks and to the fact that the characteristics of individual elements are clearly nonlinear. Semi-empirical formulae are used to describe the effect of the change of the forces acting on the wheel and the spatial position thereof on the direction vector. Dependencies of the link between the slip angle and lateral force, lateral force and relative wheel slippage, slip angle, and stabilizing moment are required in order to be able to perform the modeling. During the modeling, the dependencies are expressed by either linear or higher-degree dependencies [24–30]. The majority of the currently available methodologies employ simplified models. These models individually assess the kinematics of the steering wheel mechanism, sus- pension kinematics, stiffness of the suspension elements, and forces acting on the vehicle while making a turn [31–35]. This kind of analysis does not enable the identification of the key factors. Hence, the models consisting of multiple elements, the characteristics of which can be known only after specific vehicle design details are available, need to be used for the integrated analysis. This complicates the integrated analysis and generalization in the investigation of directional stability viewed as the criterion for assessment of the technical condition of the vehicle and the condition of the pavement. Hence, the focus was first turned to another driver-vehicle-road system element, i.e., road, prior to designing the integrated directional stability model for the vehicle and performing the analysis. In this context, the road is associated with the effects which may cause vehicle instability, while the purpose of improvement of stability is related to the solution of the vehicle-road interaction. A moving vehicle is subject to external effects and reacts to the driver’s actions (Figure1 ). In the investigation of vehicle directional stability, all these factors need to be considered. However, the vehicle model becomes very complex, making the analysis of the effect of individual factors more difficult. The present paper does not account for the driver’s actions, environmental impact, aerodynamic effects, and vehicles design features. Appl.Appl. Sci. Sci. 20212021, ,1111, ,6608 6608 3 3of of 35 34 FigureFigure 1. 1. InfluenceInfluence of of external external factors factors on on vehicle vehicle directional directional stability. stability. This paper provided a comprehensive review of the key parameters of road-vehicle This paper provided a comprehensive review of the key parameters of road-vehicle interaction that have an effect on vehicle stability and design the movement model, con- interaction that have an effect on vehicle stability and design the movement model, con- sisting of a finite number of parameters, for investigation of stability. The required tire sisting of a finite number of parameters, for investigation of stability. The required tire parameters were determined