Determination of the average torsional stiffness of tires of a double vehicle wheel during its interaction with the road surface

Annotation. Problem. In the scientific and methodological recommendations of forensic institutions and in the scientific and technical literature there are currently no universal methods for determining the braking parameters of cargo multi-axle vehicles that have wheels with double tires, which affects the results of drawing up the conclusions of the motor-technical expertise. The lack of universal methods is due to the difficulty of determining the actual braking, especially when the tires of dual wheels interact with the road surface. Goal. The goal is justification of the method of determining the average torsional stiffness of tires of a double vehicle wheel during its interaction with the road surface. Methodology. The approaches adopted in the work to achieve the set goal are based on the theoretical foundations of the deformation of elastic elements, which are located parallel to each other. Results. Equations are determined that allow you to calculate the value of the average torsional stiffness of the tire for wheels that have double, triple or quadruple tires. Originality. The results of the research provide a general idea of the effect of the pressure in the tires of a double wheel on the value of its average torsional stiffness. Practical value. The obtained results can be recommended to expert motor technicians when drawing up a conclusion of an expertise or an expert study. Besides, the results of the study can be used in the educational process during the training of specialists in the field of transport or mechanical engineering.


Introduction
Road safety to a large extent depends on the road conditions of operation of the vehicle and on the tires installed on its wheels, therefore, not only the result of determining the braking efficiency of the vehicle, but also the conclusions of the auto technical expertise in expert practice will depend on the modeling of the "tire-road surface" pair. It is known that thanks to the friction and adhesion of the tire of the vehicle wheel with the surface of the road surface, its stable movement in a given direction is ensured.
The amount of utilized adhesion between the tire and the surface of the road surface depends on many factors, including the speed of rotation of the vehicle wheel, the condition of the road surface, the parameters of the pneumatic tire, etc.
If you compare the tires of passenger cars, trucks and buses, they differ from each other in terms of dimensions, structure, material structure of individual layers of the tire. If we compare the appearance of such tires, passenger car tires have a lower tire profile height and a more complex tread pattern.
Most often, the tires of different vehicles are compared with each other according to the speed index and the load index, which depend on the stiffness properties of the tires, so the determination of the stiffness of the tire is a fundamental parameter for calculating the actual adhesion forces realized between the tire(s) and the road surface.

Analysis of publications
In the theory of the movement of wheeled vehicles, it is customary to characterize the process of interaction of the tire with the surface of the road surface by utilized adhesion, which occurs in the longitudinal and transverse directions relative to the plane of rotation of the vehicle wheel during the deformation of the tire. It is known that deformations of a vehicle wheel tire, which occur under the influence of external forces acting on the vehicle, can generally be divided into radial (normal), circumferential (tangential), angular and lateral deformation.
It should be noted that the indicated deformations of a pneumatic tire of a vehicle wheel in their pure form almost do not occur, practically all of them are in close connection with each other and appear simultaneously during the operation of a loaded tire, as a result of which the corresponding reactions acting in directions opposite to the forces acting on the vehicle. An increase in the number of wheels on the axles of the vehicle, accordingly, changes the ratio between the radial (normal), circumferential (tangential), angular and lateral deformation of the tire, but in the scientific and technical literature [1][2][3][4][5][6][7][8][9][10][11][12], as their analysis showed, not enough attention has been paid to the issue of this phenomenon for dual tires, so in this paper we will consider the interaction process of a dual vehicle wheel from the point of view of the deformation properties of its pneumatic tires.
Analysis of scientific and technical literature [12][13] showed that there are several models of interaction between a pneumatic tire and a road surface. The main models that characterize the longitudinal deformation of the tire during its braking can be presented in the form of the following diagrams (Fig. 1). The analysis of the features of the implementation of mathematical models of dual wheels of a vehicle [14][15] showed that most of them apply only to models of vertical load of tires of dual wheels during the analysis of their influence on the destruction of the road surface. In general, there are no models of the longitudinal interaction of dual-wheel tires with the road surface in the analyzed literature [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]. An exception is the paper [16], which considers not the model of the interaction of the tire with the road surface, but the method of determining the braking force between the double-wheel tires and the road surface.

Purpose and Tasks
The purpose of the work is a well-founded method of determining the average torsional stiffness of tires of a double vehicle wheel during its interaction with the road surface.
To achieve the goal, the following tasks must be completed: -to consider the scheme of circumferential deformation (tangential) deformation of the tire reduced to the plane of the surface of the road surface; -on the basis of the considered scheme, obtain an equation for determining the average torsional stiffness of tires; -to consider the possibility of applying the proposed method of determining the average torsional stiffness of dual tires for wheels with a greater number of tires than two; -carry out simulated modeling of the change in the average torsional stiffness of dual-wheel tires depending on the pressure in them.

Determination of the average torsional rigidity of tires of double automobile wheels
Let's present a mathematical model of dual-wheel tires in the form of two elastic elements (Fig. 2), which is based on the scheme (Fig. 1 a) during the twisting of the tire relative to the surface of the road surface. Figure 2 shows:  From the similarity of the triangles depicted on the X plane (Fig. 2), it can be seen that the average displacement of the tires of a wheel with double tires is defined as: and since the braking force realized between a vehicle wheel with double tires and the road surface is equal to the sum of the braking forces realized by each tire separately: then you can write from the sum of moments of forces: Автомобільний транспорт, Вип. 51, 2022 Rewriting equation (3) through the total braking force (2) realized between a vehicle wheel with double tires and the road surface, we get: and using Hooke's law, for elastic bodies, we write: By substituting equations (6) and (7) into equation (1), we write the equation (8).
and given that we will determine tire x xm R x С = ⋅ the total stiffness of tires installed on double wheels of wheeled vehicle according to the equation: In the case when 1 2 y y = , i.e. 1 2 (9) will take the known form for two parallel springs but such a case is possible only when the wheels with double tires are installed on the hub without distortions (Fig. 3), the tires have the same design and wear of the tread pattern, and also the same pressure is set in the tires, otherwise the equation (9) is valid, and not the equation (10). In the case of the location of one elastic element to the left and right of the braking force ( x R ) realized between the vehicle wheel and the surface of the road surface, we will use the Automobile transport, Vol. 51, 2022 diagram ( Fig. 4) of the vertical load of vehicle wheel tires with force to determine the distances. At the same time, the distribution of vertical loads between the tires of one wheel will take place relative to the so-called center of stiffness of the system (c.s.), the shift of which in the case of uneven tire stiffness can be taken into account by the coordinates of the position of the center of stiffness From the diagram shown in figure 4 and taking into account the methodology proposed in [17], we will write down the coordinates y 1 and y 2 for tires with double tires of the wheel and in the form: and the coordinates of the position of the center of stiffness of the system * 1 y , * 2 y and y ∆ are written as: ( ) where tire zi C -corresponding normal stiffness of the pneumatic tire, N/m. Determining the distances y 1 and y 2 for calculating the stiffness of parallel elastic elements, which cause their non-parallel compression or stretching, will be somewhat more complicated in comparison with the case of double arrangement of elastic elements (wheels with double tires), for example, for the cases of the implementation of vehicle tire tires, which are shown in the figure 5 and figure 6. Schemes for the implementation of tires on automobile wheels on such vehicles can have up to four reference points of contact (see Fig. 7 b), which will interact with the surface of the road surface, provided that it is not parallel to the horizon, at different levels from each other. To determine the unknown components y 1 and y 2 in equation (9) in the case of using triple or quadruple tires of the vehicle wheel, we will use the appropriate wheel tire inflation schemes (see Fig. 8, Fig. 9 and Fig. 10) taking into account the redistribution of weight between adjacent pneumatic tires by analogy with the method proposed in [17]. For the structured tire of the wheel, which has the load scheme shown in Figure 8, the equation for determining the coordinate y 1 and y 2 is written in the form: The coordinate of the position of the center of stiffness of such a system * 2 y is determined by equation (14) by calculating the coordinates * 1 y and y ∆ by the corresponding equation: tire r tire r tire r z z p z tire l tire r tire r z z z y C C y C y y C C C where _ 1 tire l z C and _ tire r zi C -normal stiffness's of the corresponding pneumatic tire of the wheel with triple tires (according to the diagram in figure 8), N/m.
The average torsional stiffness of the tires located to the right of the force P z is determined from a similar equation (9) in the form (20), which will determine the stiffness of the system shown in figure 8: If we accept the assumption that tires with stiffness For the structured tire of the wheel, which has the load diagram shown in figure 9, the equation for determining the coordinate y 1 and y 2 is written in the form: The coordinate of the position of the center of stiffness of such a system * 2 y is determined by equation (14) by calculating the coordinates * 1 y and y ∆ by the equation: ( ) Automobile transport, Vol. 51, 2022 Fig. 9. Scheme with triple tire (one tire is located to the right and two to the left of the braking force realized between the vehicle wheel and the road surface) where _ tire l zi C and _ 1 tire r z C -are the normal stiffness's of the corresponding pneumatic tire of the wheel with triple tires (according to the diagram in figure 9), N/m. The average torsional stiffness of the tires, which determines the stiffness of the system shown in figure 9, is determined from a similar equation (9) in the form: or if we assume that tires with stiffness Thus, it is obvious that for the quadruple wheel tire, which has the load scheme shown in figure 10, the equations for determining the coordinates y 1 and y 2 can be written in the form (28) and (29), respectively: The coordinate of the position of the center of stiffness of such a system * 2 y is determined by equation (14) where _ tire l zi C and _ tire r zi C -are the normal stiffness's of the corresponding pneumatic tire of the wheel with four tires (according to the diagram in Figure 10), N/m.
The average torsional stiffness of the tires, which determine the stiffness of the system shown in Figure 10, is determined from the similar equation (26) and (20), respectively, or by assuming that tires with torsional stiffness tire r x C move by the same value, the average tire stiffness can be determined from equation (27) and (21), respectively, to solve equation (9).

Автомобільний транспорт, Вип. 51, 2022
Torsional stiffness of the tire for loads from zero to 40,000 N can be determined by equation (10) [3,13,45,46], which are semi-empirical in nature. This dependence takes into account the effect on torsional stiffness only of the load on the tire and the pressure in the tire and does not take into account other factors: where tire p -the tire pressure, MPa; g -acceleration of free fall, m/s 2 ; max P x C -the experimental value of the torsional stiffness of the tire at the maximum allowable air pressure in the tire (determined at the maximum allowable load on the tire), Nm/rad; 0 B and 1 B -coefficients determined from equations: where min P x C -the experimental value of the torsional stiffness of the tire at the minimum permissible air pressure in the tire (determined at the maximum permissible load on the tire), N·m/rad. Thus, it can be argued that neglecting the features of the distribution of the vertical load between the tires of the wheels of a vehicle that has wheels with double, solid or quadruple tires can significantly affect the magnitude of the utilized adhesion force between the tires of such wheels and the road surface, therefore, during modeling it is necessary to take into account the peculiarities of the change in the torsional stiffness of the tire depending on the ratio y 1 and y 2 .
The normal stiffness of a pneumatic tire, in equations (15) -the coefficient of influence of pressure in the tire ( tire p ) on the normal stiffness of the tire, which is determined in [20]; 0.45 0.38 z tire n p ≈ + ⋅ -the coefficient of influence of pressure in the tire ( tire p ) on the normal stiffness of the tire, which is determined in [20]. As noted in work [20], the use of equation (35) gives an error in calculations of no more than 17% at pressures in the tire different from the nominal pressures by no more than 25%.
The results of modeling the nature of the change in the torsional stiffness of the tire according to equation (10) show that with an increase in the normal (vertical) reaction (R z ), which acts in the spot of contact of the tire with the road surface, a greater non-linearity of the torsional stiffness is manifested (Fig.  11) at R z > 20,000 N, than with vertical reactions R z < 20,000 N. Fig. 11. Modeling the nature of the change in the torsional stiffness of the 11.00 R20 wheel tire with a single tire as a function of the tire pressure and the normal reaction (R z ), which acts in the contact patch of the tire with the road surface Analyzing the effect of tire pressure on the torsional rigidity of the tire, it can be seen from figure 11 that with an increase in the normal reaction (R z ), which acts in the contact patch of the tire with the road surface, a significant nonlinearity when the pressure in the pneumatic tire decreases from 0.6 MPa to 0.1 MPa.
Automobile transport, Vol. 51, 2022 From figure 11, it can also be seen that at tire pressures from 0.4 MPa to 0.6 MPa, the torsional stiffness of a pneumatic tire with a single tire varies in a small range from 2500 N•m/deg to 3000 N•m/deg during the change of normal response (R z ) in the range from 25.000 N to 40.000 N.
Modeling the nature of the change in the normal stiffness of the tire (Fig. 12) according to equation (35) showed that the results of the simulation, in a wide range of pressure changes in the tire and the normal response in the spot of contact of the tire with the road surface, do not contradict the experimental studies given in work [20], therefore, the use of equation (35) is appropriate when determining the distances y, y 1 , and y 2 (Fig. 4, 8, and 10) for wheels with double, triple, or quadruple tires.  (Fig. 13) by 17% in case of a decrease in pressure in one of the tires of the wheel (left or right) and almost 100% loading of the tire of a double wheel in which the pressure is maintained at a level not lower 0.6 MPa.
In the case of simultaneous pressure reduction in both tires, the torsional stiffness tire x m C will decrease by 33% with an unchanged load on the double vehicle wheel. If the load on a double 11.00 R20 vehicle wheel is reduced, for example by two times (Fig. 14), the nature of the change in torsional rigidity remains the same as with a full normal load (Fig. 13), but the non-linearity of the torsional stiffness is expressed to a greater extent when the pressure in one of the tires of the double wheel (left or right) is reduced. The reduction in torsional stiffness occurs by no more than 14% at almost 100% loading of the dual-wheel tire, in which the pressure is maintained at a level of at least 0.6 MPa. the tire occurs in an insignificant range ( Fig. 13 and Fig. 14), as in the case of modeling the torsional stiffness of a vehicle wheel with a single strapping (Fig. 11). Such a change in the torsional stiffness of the 11.00 R20 double wheel tires indicates that the implementation of the traction properties of the tires of such a wheel will take place in a stable range than when the pressure in one of the tires is in the range from 0.15 MPa to 0.35 MPa. The non-linearity of the change in torsional stiffness ( tire x m C ) of dualwheel tires in the pressure range from 0.15 MPa to 0.35 MPa is caused primarily by the non-linearity of the change in distances y 1 and y 2 (Fig. 6) and the additional load of the tire (R z ) in which the pressure remains at the level of below 0.6 MPa when the pressure in the other tire drops below 0.35 MPa.
The non-linear change in distances y 1 and y 2 , as can be seen from figure 15, depends on the vertical load on the double wheel by no more than 10%, when the pressure in one of the tires drops to a level that is in the pressure range from 0.15 MPa to 0.35 MPa, and by 44% depending on the tire pressure. Fig. 15. Modeling of the nature of the change in distances y 1 and y 2 for a double wheel 11.00 R20 depending on the pressure in the right tire ( right tire p ) of the wheel during normal (vertical) reaction R z = 29430 N and R z = 58860 N When modeling the nature of the change in the distances y 1 and y 2 (Fig. 15), the redistribution of the vertical load between the dual wheel tires occurred according to a linear law, therefore the non-linearity of the distance change is related to the non-linearity of the influence of the pressure in the dual wheel tires on the normal stiffness (С zi ).

Conclusion
The performed theoretical study of the nature of the interaction of a double vehicle wheel with the road surface showed that the magnitude of the utilized adhesion force between the tires of such a wheel and the road surface depends on the nature of the change in the average angular stiffness of the tires. If there is a difference between the distances associated with the axis of symmetry of the wheel and the axis of symmetry of the corresponding tire, there is a decrease in the average angular stiffness of tires of wheels that have other than single tires, when the pressure in one of the tires of such a wheel is reduced.
A theoretical study of the nature of the effect of the average torsional stiffness of the tires of wheels that do not have a single tire showed that the value of the average stiffness of the tires of such wheels does not linearly affect the adhesion properties of the "tire-pavement surface" friction pair due to the full implementation of the angle of rotation of the tire relative to the road surface coating.
Analysis of the nature of the change in the torsional stiffness of the tire from the weight parameters of the vehicle showed that an increase in the mass of the vehicle leads to a decrease in the traction properties of its tires, and therefore to a decrease in the amount of deceleration. Reducing the mass of the vehicle, on the contrary, increases the traction properties between the tire and the road surface, which accordingly has a positive effect on the braking efficiency of the vehicle, since its deceleration will increase in proportion to the decrease in the weight of the vehicle.
If there is a difference between the distances associated with the axis of symmetry of the wheel and the axis of symmetry of the corresponding tire, there is a decrease in the average angular stiffness of the tires of wheels that have other than single tires by 14%, when the pressure in one of the tires of such a wheel is reduced by 80%.
A theoretical study of the nature of the influence of the average stiffness of tires of wheels that do not have a single tire showed that the value of the average stiffness of the tires of such wheels does not change when the tire pressure changes in the range from 0.15 MPa to 0.35 MPa. This is caused, first of all, by the non-linearity of the change in distances y1 and y 2 and the additional load of the tire (R z ) in which the pressure remains at a level not lower than 0.6 MPa when the pressure in the other tire drops to a level not lower than 0.35 MPa.

Motor vehicles
Automobile transport, Vol. 51, 2022 It was established that the non-linear change in the distances y 1 and y 2 depends on the vertical load on the double wheel by no more than 10%, when the pressure in one of the tires drops to a level that is in the pressure range from 0.15 MPa to 0.35 MPa, and by 44% depending on the tire pressure. In the case of a simultaneous decrease in pressure in both tires, the torsional stiffness will decrease by 33% with an unchanged load on the double vehicle wheel.