Comparative analysis of small size non-pneumatic tires and pneumatic tires - radial stiffness and hysteresis, selected parameters of the contact patch

Highlights Abstract ▪ The radial stiffness, hysteresis, contact patch of non-pneumatic (NPTs) and pneumatic tires (PTs) were compared. ▪ NPTs have a higher radial stiffness compared to PTs. ▪ The flexible cell structure increases the energy losses associated with the vertical displacements of the axle of the NPT. ▪ NPTs have a shorter length of the contact patch and the shape of the tread area allows to change the width of the contact patch. Nowadays, non-pneumatic tires are becoming an increasingly likely alternative to pneumatic tires. The function of compressed air has been taken over by the belt and the elastic structure (materials used and shape of the components). The research presented in the paper was carried out in quasi-static conditions. The research's aim was to compare the radial stiffness, hysteresis and selected parameters of the contact patch of two non-pneumatic tires and four pneumatic tires used interchangeably in ATVs/UTVs. The analyzed non-pneumatic tires are characterized by greater radial stiffness than pneumatic tires of the same size. Moreover the wheel with a cellular structure has the highest hysteresis of the radial characteristics of the tested wheels and the highest values of the unit pressure in the area of contact patch. The paper also verified two methods of calculating the contact patch length of non-pneumatic tires.


Introduction
Nowadays construction of non-pneumatic tires (NPTs) A detailed analysis of the most common construction solutions of NPTs is presented in [39,35]. In the construction of such the wheel, the following components are always detailed: rim, elastic structure, belt with a core and tread. The NPT's tread and rim perform the same tasks as in the case of pneumatic tires.
The belt consists of inextensible membranes with a core between them. The elastic structure connects the wheel rim with the belt. This component of non-pneumatic tires most often occurs in the form of radial spokes or as interconnected geometric figures (referred to as a cellular or layered structure).
It is also responsible for the mechanism of carrying multidirectional loads, including vertical forces. It follows that the function of the compressed air of a pneumatic tire in the NPT is mimic jointly by the elastic structure and the belt. The elasticity of those two components causes deformation (flattening) of the belt with the core in the contact patch when the wheel axle is loaded. The remaining non-deformed part of the belt stores mechanical energy (this action is compared in [14,15,21] to a bow). The method and range of allowable flexibility of the elastic structure are very important for the properties of the non-pneumatic tire in the radial direction. The vertical axial displacement of the non-pneumatic tire with single spokes is limited by the possibility of stretching the spokes located outside the contact patch, because the normal load is carried by the upper, undeformed part of the wheel ("top loader"). At the same time, as a result of loading with a normal force, the elastic structure located between the contact patch and the wheel axle is compressed and deformed (buckling), which means that it can carry only a small load or no load [4,7,17,39]. In addition, in this solution of elastic structure, the high stiffness of the spokes counteracted the circumferential deformations of the belt during the vertical load of the wheel, thus reducing the length of the contact patch [24]. In the case of the cellular structure, part of the vertical load is carried by the deformed structure under the wheel axle ("bottom loader") [2,39].Its load carrying capability is affected by incl. increasing the thickness of the components or a properly shaped structure (e.g. auxetic) [3,8,21]. In practice, the use of the specified NPT solution (e.g. type of elastic structure) is determined by detailed operational requirements or the target vehicle segment [33]. This is confirmed by the works presented in [34]. Simulation and experimental research were carried out there, and the process of manufacturing NPT equipped with radial spokes was described. The process of optimizing dimensions is presented, including the width and direction of the spoke. The aim was to minimize spoke weight (volume of material) and use NPT as the equivalent of a pneumatic tire on a commercial electric vehicle. The results of experimental and numerical research on radial stiffness can also be found in [27], where a wheel used in a skid steer was analyzed. In numerical research, the influence of spoke thickness changes on the trajectory of the radial characteristics and the value of the stress acting on the spoke was evaluated. The results of numerical calculations made it possible to find a spoke thickness that provides radial stiffness of the wheel comparable to the pneumatic tire (which was previously selected as the goal). The next stage of research [26] was the validation of the developed numerical model using the results obtained by rolling the NPT on the drum at a speed of approx. 3 m/s. In addition, deflections of individual spokes in specific places on the circumference of the wheel were analyzed. As was shown, the numerical model was highly compatible with the experiment results and will be applied to the assessment of rolling resistance and the dynamic NPT reaction during overcoming road unevenness. In [36], a comparison of the experimental and numerical research results of a pneumatic tire with the numerical research results of three NPTs was shown. The flexible structure was differentiated by the use of radial spokes, hexagonal cell structure, and grid type. Potential differences in NPT properties have been pointed out. Those equipped with the grid type cell structure were characterized by a significant resistance to buckling (this feature was strongly marked during the tests of the radial characteristics). The NPT equipped with radial spokes, on the other hand, had lower radial stiffness compared to the pneumatic and NPT with a hexagonal cell structure.
In [5], the analysis of influence the hexagonal structure changes (its single cell), i.e. its length, width, internal angle, wall thickness, density (total number of cells) on selected wheel properties (radial deflection, energy losses, pressures in the contact patch) was carried out. It was also shown that increasing the height of the cell increases the vertical displacement of the wheel axis (deformation of the elastic structure) during loading in the normal direction. The opposite phenomenon was observed during increasing the total number of cells (packing density) and the thickness of their walls. A similar nature of changes occurred in the area of energy losses related to the vertical deformation of the NPT. In addition, a significant concentration of pressures in the contact patch was observed at the connection point of the flexible structure with the belt, and the pressure distribution itself was significantly dependent on the length and thickness of the cell walls of the layered structure.
In [11], the influence of changing the geometric dimensions of a single hexagonal cell and its wall thickness on the radial characteristics, contact patch parameters, and rolling resistance was analyzed. It has been shown that the rolling resistance of an NPT with a cellular elastic structure depends on its volume (mass) and the range of its deformations. The lowest rolling resistance force values were obtained for the structure in which the smallest cell expanding angle was used. Increasing the wall thickness of the cell structure limited the range of vertical displacements of the NPTs. In [38], the wall thickness of a hexagonal cell was changed for three elastic NPT structures with defined dimensions and material data. The goal of this search was to obtain a radial stiffness equivalent to a pneumatic tire of size 205/55R16. The developed numerical models were also used to assess the directional stiffness of the NPTs (circumferential and lateral). Increasing the internal angle of the hexagonal cell resulted in an increase in the vertical displacements of the wheels. Higher circumferential deformations of the cellular structure (compared to pneumatic tires) resulted in a decrease in the value of the circumferential stiffness of the tested NPTs. The opposite trend was observed for the lateral direction.
Formally, the area of the NPT contact patch and the average value of the pressure in this area are shaped at the modeling stage by selecting the geometric dimensions of the NPT and the materials properties of the belt [22,31]. It is assumed that the product of the average pressure value of the contact path and the radius of the outer layer of the belt reinforcement (approximately the outer radius of the NPT) is equal to the product of the shear modulus and the height of the layer of the belt material (between the layers of reinforcement) for NPT equipped with a homogeneous core [4,22]. The above indicates that the average value of the pressure is an important design criterion of the NPT, which determines, including on the properties of the material used to create the core. Therefore, the introduction of a composite core in place of the homogeneous material core is currently being considered (aimed to obtain a structure characterized by a high dynamic shear modulus G, high elongation at break, and low energy losses during deformation). This will allow to reduce its thickness while maintaining the usable properties of the wheel [31].
Research [36], conducted with the numerical models, revealed the influence of the applied elastic structure also on the contact patch and contact pressure. The analysis did not take into account the influence of the tread pattern, and a significant concentration of pressure was observed at the contact point of the elastic structure with the belt/tread. In [6], in the numerical assessment of the pressures acting in the contact patch with the rigid ground, the shape of the tread was taken into account. The parameters of the contact patch, values and pressure distribution determined in numerical tests were consistent with those determined during the experiment on the test bench.
Other specific constructions of NPTs are also described in the literature. An example may be a structure named by the creators as ME-wheel (mechanical elastic wheel) [31]. In this wheel, steel hinges/joints with a different number of components were used to connect the belt with the wheel rim.
The use of three-piece hinges resulted in a reduction in circumferential stiffness compared to the results obtained for two-piece hinges. The increase in radial stiffness was also  [12,28] are often replaced by numerical research, which predominate in the available analyses.
Estimated features are usually compared with one pneumatic tire or proprietary NPT constructions. In addition, in analyzes of the contact patch parameters, the tread blocks are often omitted, and when they are included, their shape is simplified and the transverse outline of the tread is omitted. Therefore, it seems reasonable to carry out a comparative analysis using detailed results of experimental research on non-pneumatic tires and pneumatic tires. This article presents the results of experimental research related to the determination of the radial stiffness characteristics and the parameters of the wheels' footprints (NPTs and PTs), which can be used interchangeably in an ATV/UTV. The research included two non-pneumatic tires with different elastic structures (single spokes and hexagonal cell) and four pneumatic tires (radial and diagonal).

Methodology and research objects
Experimental research was carried out on the Universal Test Bench for Quasi-Static Tire Research ( fig. 1). A detailed description of the measurement capabilities of the stand was presented earlier in [13]. Experimental research included the determination of the radial stiffness characteristics, determination of energy consumption of the vertical deflection process of analyzed wheels, and measurement of the contact patch with a flat, non-deformable (rigid) ground. These research were designed to compare the important properties of commercial NPTs and their corresponding PTs. The initial comparative analysis was presented by the authors in [10], which included an analysis of the radial stiffness characteristics of NPTs and one pneumatic tire.  18,0 front/steering axle • * -mass measurement of the pneumatic tires was carried out after mounting on the wheel rim, the pressure inside the tire was equal to atmospheric pressure, the measurements were made using the Universal System for Weighing Vehicles with a measuring range of 4÷600 kg and an accuracy of 0,2 kg, • ** -load capacity and maximum speed were determined on the basis of information obtained from the distributor [29], The selection of research objects was preceded by an analysis of the availability of commercial non-pneumatic tires.
Their size and load capacity were used to indicate a group of comparative wheels made of 4 pneumatic tires. The wheels/tires accepted for testing, together with the assigned identification mark, are shown in fig. 2, and the basic features of their construction are listed in table 1.
Due to the lack of manufacturer's information regarding the NPT_2, its load capacity and maximum speed were determined on the basis of information obtained from the distributor [29] and a review of technical and operational parameters of ATVs and UTVs in which it can be used. Normal load values (table 2) were selected based on the ATV/UTV load variation range.
Factors such as curb weight, payload, and expected load changes resulting from vehicle operating conditions (acceleration, braking, driving on a slope, etc.) were taken into account during determining the force values.
The selected loads for some pneumatic tires exceed the value resulting directly from their load capacity index.
However, in accordance with the regulations of the Economic Commission for Europe of the United Nations (UN/ECE) [19,20], the load capacity index means the maximum permissible load on a pneumatic tire at a speed corresponding to the appropriate speed category symbol, while maintaining the conditions of use specified by the manufacturer. The experimental research was carried out in quasi-static conditions.
According to [19,20], the wheel load capacity index (whose linear velocity was 0 m/s) increases from 110% to 150%.     The measurements of the wheel contact patch with the nondeformable ground made it possible to calculate additional auxiliary coefficients [32]: • tread pattern density coefficient • coefficient of the contact patch shape The second method used to calculate the contact patch length (method no. 2) is presented in [23]. In this model, it was assumed that the radial force increases in a linear manner and is described by the radial stiffness coefficient and can be expressed as the product of the average pressure value and the area of the contact patch [23]: ▪ -the rectangular area of the contact patch described by the length lS and the width bS, ▪ -the average value of the contact pressure on the rectangular contact patch area.
In this method of estimating the contact patch length, it is assumed that the contact patch area is rectangular, and the contact patch width is similar to the width of the tire (tread blocks are not taken into account). Equation (10) has been transformed in order to connect to the data that will be measured from the experiment. After substituting equation (6) and assuming that the rectangular contact patch can be replaced with the AC area, the following expression was obtained: The average value of the contact pressure resulting from the total contact patch area was also assumed as: The results of the contact patch length calculations using methods 1 and 2 in the "Discussion" section were compared with the results of experimental research.       Radial deflection [mm]

Discussion
The analysis of direct and indirect wheel research results was divided into three parts. This approach made it possible to systematize the reasoning while revealing the interconnections between the individual areas of consideration.

Hysteresis -energy losses
The to 30%. On the other hand, the NPT_2 was characterized by the highest value of the hysteresis coefficient among the analyzed wheels. Depending on the normal load, energy losses ranged from 30% to even 40%. The reason for high energy losses is that, part of the normal load is transferred through the elastic structure located between the NPT_2 axis and the road.
Changing the maximum value of the normal load of NPTs resulted in a reduction of energy losses by approx. 10% (almost linear course, reducing losses by 5% for each 1 kN increase in normal load). This direction of changes may be caused by the increasing influence of the increasingly deformed belt with the core and confirms the conclusion presented in [14,15,21].
The increase in the inflation pressure of pneumatic tires has obviously resulted in a beneficial reduction in energy losses.  This results in an increase in the volume of material deformed during tire exploitation.

Radial stiffness
Compared to pneumatic tires, non-pneumatic tires are characterized by a clearly higher radial stiffness and a similar to linear course of the center lines of the radial stiffness characteristics ( fig. 10). The NPT_1, equipped with radial spokes, showed significantly lower radial stiffness and lower energy losses during deflection in comparison to the NPT with a hexagonal structure. This is due to a different way of loading the elastic structure, the lower spokes of which are subject to buckling and carry the normal load to a small extent, while the upper spokes (above the wheel axis) are stretched [39].
Observing the changes in the value of the radial stiffness coefficient with the increase in the normal load ( fig. 11 The authors' experience also shows that the non-uniformity of production is difficult to assess for tires with a very low density of the tread pattern built of considerable height blocks.

Contact patch
The analysis of the registered contact patches ( fig. 18) made it possible to determine the parameters that were used for their mutual comparison (table 4). The first noticeable feature is that the PTs (compared to NPTs) have always had a longer footprint.
The NPT_1 contact patch width, due to the flat shape of the tread front, was practically insensitive to normal load changes.
On the other hand, the rounded shape of the NPT_2 tread front  All these factors taken together show the forecast for the cooperation of the analyzed wheels with a non-deformable (traction and intensity of tread wear) and deformable ground. In this case, the number of the tread blocks hooking edges related to this type of tread pattern with the value of the tread density coefficient, the shape of the tread blocks and the total contact patch area determining the pressure exerted on the ground (parameters partially determined in the described experiment but not yet analyzed) will be additionally important. driving on rough ground, NPTs can be a source of significant loads, which will also negatively affect driving comfort (vehicle users' loads).
4. The source of driving comfort decreases (at higher speeds) will also be radial non-uniformity, which in the analyzed test objects is mainly due to low density and significant tread blocks.