Refined assessment of tire rolling resistance in the starting mode

Annotation. Problem. Rolling resistance is one of the most important operational characteristics of a car tire. First of all, the fuel consumption, durability and service life of the tire depend on the rolling resistance. Of course, the higher the rolling resistance, the faster and more intensively the tire will wear out. Also, car braking parameters, such as deceleration and, accordingly, the braking distance, may indirectly depend on the rolling resistance characteristics. In turn, rolling resistance is influenced by the properties of the tire itself, as well as the condition of the road surface and weather conditions. It is especially important to evaluate the operational characteristics of a car tire in the starting period of movement, because at this time the behaviour of the tire is unpredictable. Goal. The purpose of this work is to improve the methodology for evaluating the rolling resistance of a car tire in the starting period of movement. Knowing how the rolling resistance changes during the starting movement, it is possible to evaluate the operational properties of the car. Methodology . Refined assessment is carried out with the help of dependencies obtained in previous scientific works. But in order for the result of the rolling resistance assessment to be correct, it is necessary to determine the necessary characteristics of the tire. Namely, we are talking about the characteristics of radial stiffness. In the work, these characteristics are determined both by conducting experimental studies and by using FEM. Results. Thanks to the conducted research, significant results were obtained on the change in rolling resistance during the starting motion. It was established that with an increase in internal pressure from 0.17 MPa to 0.225 MPa, as a result of the starting motion, the rolling resistance decreases by approximately 20-25%. Originality. From the analysis of the literature, it can be concluded that in most of the studied works, the peculiarities of the interaction of automobile tires with the supporting surface in the starting period are not taken into account during various studies. It can be said that the originality of this work, on the one hand, is that the rolling resistance is investigated during the starting motion, and on the other hand, that its assessment is carried out both on the basis of the results of experimental studies and with the help of FEM. Practical value. In this work, it is possible to highlight the obtained results, thanks to which we can estimate the rolling resistance of a car tire with sufficient accuracy. The methods of determining the operational characteristics of a car tire, presented in the work, can be used during its operation and at the design stage.


Introduction
The condition and behavior of a car tire in the period from the start of driving on "cold" tires to the stabilization of its operational characteristics are unpredictable. The above-mentioned movement period is usually called the "starting period" [1]. At this time, the main characteristics of the tire change, such as the internal pressure of the gas filler (mainly air), the temperature of the tire, etc. [1][2].
After analyzing the scientific publications in which the operational characteristics of car tires are studied, it can be concluded that the starting period is not taken into account when evaluating various properties of tires and the car. That is, in most works, the tire is considered already warmed up (it has constant values of internal pressure, temperature, etc.). Also, quite often the research does not take into account the weather conditions under which the experiments are conducted, namely the ambient temperature. It should be noted that the condition and behavior of the tire significantly depends on the temperature of the surrounding environment.
In the starting period of movement, it is important to know how the operational Automobile transport, Vol. 51, 2022 characteristics of a car tire will change, among which the most important characteristic is rolling resistance. Of course, this characteristic depends on a combination of many factors, such as: the condition of the car tire and bearing surface, environmental conditions, etc. The rolling resistance of a tire directly affects the durability and service life of the tire, as well as fuel economy. In one of the previous works, the methods of indirect assessment of the rolling resistance of a car tire in the starting period have already been given. In the current work, a refined assessment of rolling resistance is carried out. For this assessment, parameters are used, which are determined as a result of conducting experimental studies and due to the application of the finite element method (FEM).
Currently, it can be confidently stated that FEM is the most common in solving many engineering problems of varying complexity [3][4]. This is primarily due to the high accuracy of calculations and relative ease of application of this method. There are a large number of software systems based on FEM, sometimes it is not even necessary to fully understand this method, because all calculations are performed automatically. In this work, FEM will be used to determine the radial deflection, otherwise we can say to evaluate the stiffness of the car tire. But it is also worth noting that when solving the problem, some assumptions will be made that, on the one hand, greatly simplify the problem, and on the other hand, do not entail significant errors. In more detail, the application of FEM to solving the problems of car tire mechanics will be indicated in the corresponding section.
Therefore, it can be said that the question of evaluating the rolling resistance of a car tire when driving in the starting mode is relevant and deserves careful research, especially with the use of modern technologies.

Analysis of publications
As it has already been repeatedly stated, the issues related to the safety of road transport are very important and require detailed research to further ensure traffic safety. Traffic safety directly depends on the condition of car tires [5]. Accordingly, there is a large number of scientific works devoted to the study of these issues.
It should be recalled that this work is a continuation of the scientific work [2], which presents the methods of indirect assessment of the rolling resistance of a car tire during the start-up period. The work [2] also presents the results of experimental studies of changes in the internal pressure of the gas filler and the temperature of the tire.
The value of the increase in the internal air pressure in the tire was determined and the starting time for the cars under study was approximately determined. It was found that the most intense temperature increase occurs in the shoulder zone of the tires. Also, the starting time of the studied cars was determined. Namely, the internal pressure in the tire, in the mode of starting movement, stabilizes after about 20-30 minutes of uniform movement, the temperature of the surface layers of the tire stabilizes after about 30-40 minutes. The temperature of the surface layer of winter tires increases less than that of summer tires, and accordingly, driving on winter tires is safer at temperatures close to 0 o C and below. This confirms the well-known statement that it is necessary to replace summer tires with winter tires in a timely manner (when the ambient temperature can reach +5 o C and below), thanks to this, road safety can be significantly increased. As a result of an indirect assessment, it was established that the rolling resistance during the starting movement changes by approximately 15% [2]. But still, it is worth noting that during the indirect assessment, not all the physical and mechanical properties of the investigated tires were indicated. In this work, it is planned to carry out a refined assessment of rolling resistance in the starting period, taking into account the properties of the tire material.
Works [6-7] present a method for estimating the rolling resistance coefficient, which is based on the elliptic-degree model of the absorbing property of the tire. This model is built by mathematically describing the experimental characteristics of normal stiffness. As you know, the stiffness characteristic, in the form of the area of a closed loop, reflects the energy spent during deformation due to the action of inelastic resistance forces (hysteresis losses). The stiffness characteristics of the tires obtained in the dynamic loading mode are presented in Fig. 1.
Based on the obtained characteristics and taking into account the functional dependence of the values of the inelastic resistance force on the amplitude values of the radial deformation of the tire, the mathematical description of the ellipticdegree model of the absorbing property of the tire can be determined by equation (1). The proposed method of determining the rolling resistance coefficient is based on the characteristics of the normal stiffness of the tire. This approach is more reliable than experimental methods of direct measurement of rolling resistance forces. Note that for the correct use of equation (1), it is necessary to determine the necessary tire parameters. To find them, both the results of experimental studies and the results of modeling the interaction of a car wheel with the supporting surface of the FEM will be used. The methodology of experimental research on determining the area of the contact zone is well covered in the work [8].
Problems related to the study of the radial stiffness of tires are quite common. In [9], the radial and tangential stiffness of the tire is determined. Two different load schemes were used to determine the vertical (radial) stiffness. The difference between the schemes lies in the contact area of the outer cylindrical surface of the tread with the load, i.e. in the first case the contact area constantly increases (with increasing load), in the second case the contact area remains constant. As a result of the conducted research, functional dependences for the radial and tangential deformation of the tire from the external load were obtained.
The method of determining the radial rigidity is also presented in works [5,10]. Thus, the work [5] describes the stand that was developed at the KNAHU. This stand allows you to determine the dynamic radial stiffness and damping coefficient of car tires. This stand uses the method of free oscillations of the system, the elastic element of which is a pneumatic.

Purpose and Tasks
The main goal of this work is to evaluate the rolling resistance of a car tire when driving in the starting mode. As has been repeatedly stated, rolling resistance is an extremely important operational characteristic. In order to determine this characteristic with sufficient accuracy, it is necessary to clarify the already existing methods of predicting the amount of rolling resistance.
To apply already known dependencies, it is necessary to know the relevant parameters of the tire, including stiffness characteristics and geometric parameters. That is, to determine the specified parameters, experimental studies should be conducted, and the stiffness of the tires can also be estimated by using the FEM. For its correct application, it is necessary to create a model of the investigated car tire with appropriate (allowable) simplifications.

Automobile transport, Vol. 51, 2022
Experimental studies of car tire stiffness The purpose of conducting experimental research is to determine the stiffness characteristics of a car tire, namely, the radial deflection. Having obtained the dependence of the radial deflection on the internal pressure of the tire gas filler, it is possible to indirectly predict how the rolling resistance coefficient will change. To determine the rolling resistance, using equation (1), it is necessary to determine the area and length of the contact zone of the tested tire with the support surface at different values of internal air pressure.
In fig. 2 presents the process of obtaining an impression of the contact zone. The experiment was conducted on a technically sound Lexus RX300 car equipped with 235/55R18 winter tires. The air temperature in the room where the measurements were taken was +20°C, humidity -80%. It is also worth noting that the study was carried out on a "flat" support surface to ensure the correct result of measuring the geometric characteristics of the contact zone. In addition to the car, a jack, a sheet of A3 format (preferably Whatman paper) and copy paper are needed to conduct the research. It can be immediately noted that this experiment is quite simple from the point of view of the necessary tools for its implementation.
As a result of the research, a number of impressions of the tire contact zone with the support surface at different values of internal pressure (from 0.16 to 0.225 MPa) were obtained. One of the prints is shown in fig. 3.   The application of the finite element method to solving the problems of the mechanics of a car tire The finite element method is a progressive tool for solving a wide range of engineering problems. The fundamental principle of FEM consists in dividing the studied area into elementary areas of finite dimensions (finite elements) [11]. In each such element, the unknown function is approximated by a polynomial. Currently, there are many software systems based on FEM for solving various engineering problems. The most developed and well-known is the "Ansys" system. However, it can be said that other, more affordable systems can be used to solve relatively simple problems. In this work, we will solve the problem of evaluating the radial stiffness of the tire using the "Autodesk Inventor" program.
To solve the given problem, it is first necessary to create a model of the tire under study. At first glance, it may seem that a car tire is a fairly simple element of a car wheel. But in fact, it is an extremely complex structure in the form of a shell made of composite material (cord) with a connecting rubber (Fig. 5). 1 -innerliner; 2 -carcass; 3 -bead filler; 4 -chafer; 5 -bead wire wrapping; 6 -sidewall; 7 -under-belt corner; 8 -belt; 9 -tread.
It is clear that each of the elements of the tire has different physical and mechanical properties, about which, by the way, it is not always easy to find information. So, to solve the problem, you need to make certain assumptions about the car tire model. At the first stage of FEM modeling, we can assume that the tire consists of one continuity, isotropic and homogeneous material. Moreover, the specified material will have the physical and mechanical properties of the sidewall. After all, in our case, when it is necessary to determine the radial deflection, the main role is played by the sidewall of the tire.
To apply FEM, it is necessary to know the following properties of the tire material: composite modulus of elasticity, shear modulus, Poisson's ratio, density, yield strength, and strength limit. The definition of the composite modulus of elasticity is considered in detail in [12][13].
But again, in order to determine the composite Young's modulus, it is necessary to know many parameters, for example, the curl angle of the frame cord rope, the density of threads per unit length, etc. Therefore, we finally choose the material properties according to [5].
Before creating a model of a 235/55R18 car tire, it is necessary to determine the geometry of the meridional section. From the marking, the mounting diameter, width and height of the tire profile are known. It is clear that the manufacturers do not provide complete information regarding the geometric parameters of the tire. Therefore, it is necessary to independently create a simplified drawing of the meridional section based on the known dimensions (Fig. 6). Based on the drawing (Fig. 6), we create a model of a car tire in the Autodesk Inventor environment (Fig. 7), and set the above properties for its material. Next, we create an approximate model of a car disk and connect it to the tire, and also create an element that will imitate the supporting surface and then position the car wheel relative to the supporting surface. We run the simulation with the same inputs as in the actual experiment detailed in the previous section. . In this case, the radial deflection is -mm 85 , 3 = h , which almost coincides with the result of a real experimental study. The dependence of the area of the contact zone on the value of the internal pressure, according to the results of FEM simulation, will be close to the graph shown in Fig. 4.

Evaluation of the rolling resistance of a car tire
Based on the results of experimental research and simulation of FEM, it is possible to estimate the rolling resistance of a car tire using equation (1).
To do this, we calculate the necessary parameters that will change as a result of the change in internal pressure: l k -the proportionality factor between the length of the contact zone and the normal deflection of the tire; ш H -the proportionality factor between the value of the inelastic resistance force and the normal deflection; max z h -full deflection of the tire in the radial direction, mm; d r -dynamic wheel radius, mm.
Recall that the load in each case remains constant N 5000 = P . Based on the results of the calculations, we plot the dependence of the rolling resistance coefficient on the internal pressure. In fig. 9 shows the dependence built on the basis of the results of an experimental study. Comparing the results, it can immediately be noted that higher values of the rolling resistance coefficient are observed in the FEM simulation. It is worth noting that the dependence (Fig. 9) is closer to the results of determining the rolling resistance coefficient obtained in [2]. It can also be said that the general trend of change in rolling resistance is quite obvious (as the internal pressure increases, the rolling resistance coefficient decreases). But the shape of the curves (Figs. 9, 10) differs from those given in [2].

Conclusions
In the work, an analysis of the literature is performed, which examines the issue of the interaction of a car tire with a supporting surface. Based on the results of the analysis of the research works, it can be said that most of the works do not cover the peculiarities of the mechanics of the car tire when driving in the starting mode. That is, when studying operational characteristics, it is considered that the tire has already reached optimal parameters (temperature, internal pressure, etc.), that is, in other words, it can be said that the presence of the starting period of movement is not taken into account. In previous works [1][2], it was already proven that in the period from the start of movement (on "cold" tires) to the stabilization of the operational characteristics of the tire, the internal pressure and temperature change quite significantly (by approximately 20-35%), and accordingly other properties of the car tire also change.
In this paper, the dependences of rolling resistance on internal pressure, based on the elliptic-degree model of the absorbing property of the tire, were obtained [6]. To apply this method of determining rolling resistance, it is necessary to evaluate the stiffness characteristics of the car tire and the geometric parameters of the contact zone of the car tire with the supporting surface. Stiffness assessment (determination of radial deflection), as well as determination of geometric parameters of the zone of contact with the support surface, was carried out by conducting a real experiment and using the FEM. It is worth noting that a simplified tire model was used to simulate the interaction of a car wheel with the road surface.
The following conclusions can be drawn on the basis of the research carried out in the work on the assessment of rolling resistance: -the results of simulation of the interaction of a car tire with the road surface of the MSE almost coincide with the results of experimental studies, which indicates the sufficient accuracy and efficiency of the application of the FEM to solve the problems of the mechanics of the car tire; -the rolling resistance of a car tire decreases with an increase in the internal pressure of the gas filler, with an increase in the internal pressure in the range from 0.17 to 0.225 MPa, the coefficient of rolling resistance decreased by approximately 20-25%; -when studying the operational characteristics of automobile tires, it is necessary to take into account the presence of a starting period of movement to ensure safety during this period.