Comparative study of static and dynamic characteristics of non-pneumatic tires with gradient honeycomb structure

 Abstract: The static and dynamic properties of the honeycomb non-pneumatic tires (NPTs) are strongly influenced by the spoke structure. Due to the complexity of the honeycomb structure, an in-depth understanding of the influences of the design parameters related to the honeycomb structure on its mechanical properties is essential, particularly for designing NPT of desired properties. Inspired by the concept of functionally graded structure, this paper aims to design a novel non-pneumatic tire with honeycomb-spoke graded thickness. Firstly, the in-plane mechanical characteristics of the thickness-graded honeycomb structures were investigated theoretically. On this basis, the finite element technique was developed for the NPTs using the corresponding thickness-graded honeycomb structures were established, and their static and dynamic mechanical properties were investigated using simulations and experimental tests. The results show that a reasonable thickness design can effectively enhance the load-bearing capacity of the NPT. The deformation features of the spoke were analyzed under the static state, and the contribution of different honeycomb structure edges deformation on the spokes is also discussed. The stress of the spoke and the tread under the static and dynamic loading conditions were studied, and comparison with the NPT-4 with a uniform thickness honeycomb structure, the results show that the thickness-graded honeycomb structure in NPT-3 significantly amplifies its load-bearing capability while also providing effective cushioning and shock absorption properties. This work would provide a basis for innovative design and performance optimization of NPTs.


Introduction
Pneumatic tires have been receiving attention since their invention due to their excellent cushioning and damping performance, as well as their high load-bearing capacity.
With the continuous development of design, manufacturing, and materials, the performance of pneumatic tires has also been greatly improved, leading the tire market for over a decade [1].However, pneumatic tires have several fatal drawbacks, such as catastrophic tire explosions, complex manufacturing processes, and the need to maintain internal inflation pressure [2].In addition, pneumatic tires are often unable to withstand severe impacts and uneven road surfaces in harsh road conditions such as mines and military operations, thereby affecting the performance of vehicles [3][4][5][6].In order to improve and overcome the shortcomings of traditional pneumatic tires, major tire companies and research institutes have increased their investment in research and development of safety tires.Therefore, nonpneumatic tires (NPTs) have emerged.NPTs differ significantly from conventional pneumatic tires in that they do not require air pressure to support the tires, so NPTs will not experience gas leakage and tire explosion accidents.
They have the advantages of no maintenance of air pressure and explosion-proof tires.
Over the past 10 years, well-known companies and universities have released a wide variety of NPTs, including the Tweel, Air Free concept, and the Honeycomb tire.A typical NPT typically consists of a wheel rim, flexible spokes, shear bands, reinforcement layer, and tread.Among them, spokes are vital components that absorb impact and provide tire compliance characteristics under static and dynamic conditions [7].The spokes of NPTs need to meet certain stiffness and elasticity requirements under alternating tensile and compressive loads, this presents a significant barrier to the structural design and spoke material choice.These conditions may be satisfied by honeycomb materials and the structures made from them, and honeycomb spokes in NPTs offer a lot of promise.In addition to having a high in-plane elasticity and out-of-plane stiffness and strength, honeycomb has a comparatively low mechanical resistance [8] [9].As a result, biomimetic honeycomb NPTs have received widespread attention, and many researchers have analyzed and designed the structure of honeycomb NPTs with superior performance.Umesh and Amith Kumar et al. have found that honeycomb spoke tires offer uniform traction and low wear performance, making them a better alternative to conventional pneumatic tires [9].
Different spoke configurations' effects on tire contact pressure were compared by Aboul Yazid et al.Findings from the study show that the honeycomb spoke structure reduces the contact pressure and the stress on the spokes of the tire [11].A NPT with honeycomb spoke has a complex design, According to the results, the vertical stiffness of honeycomb structures is significantly affected by the density, thickness, and internal angle of the honeycomb cells, and internal stress can be significantly decreased by both a decrease in the length of honeycomb cells and increasing cell density [15].
As the honeycomb spoke structure is a key component affecting the performance of the NPT, an in-depth understanding of the honeycomb structure is vital for design and optimization.In a regular honeycomb structure, the hexagonal cells have the same dimensions and are combined into rows and columns, which absorb energy over a restricted range.However, for irregular structures, absorption can reach a broader range [16] [17].Gradient honeycomb structure is the name given to this lack of regularity in a particular direction.[18].The ability of functionally graded structures to absorb energy under    1.
The static and dynamic response characteristics of a NPT are significantly affected by their spoke structures, which serve as a critical of load bearing and energy-absorbing component.spoke could be evaluated based on the theory reported in [22][23][24][25][26][27], as: ( ) where r E  , E   and r G   are the effective moduli of the 2D spoke in the radial, circumferential and shear directions, respectively.Moreover, h equals to 3 cos 2 l  , while l is taken as the length of spoke 2 l and 4 l , respectively, and  is taken as 3 Based on the geometric relationships in Fig. 2, The angles ∠1 and ∠2 formed between the walls of the spoke and the vertical direction can be determined using Eqs.( 4) and ( 5), respectively, as: The lengths of the diagonal edges 4 l , 3 l and 2 l projected to the vertical direction (T1, T2, T3) could thus be determined as: The total length of the honeycomb structure ( D ) in the radial direction is: The length of each edge of the honeycomb spokes is a crucial parameter that changes the structure of the honeycomb and affects the performance of the NPT.
According to Eq. ( 9), the length of each edge can be determined based on its radial projection length, thereby quantitatively determining the specific structure of honeycomb spokes.The specific length parameters of each structure are shown in Table 2.  l thickness gradients are developed, as shown in Table 3.

Material parameters
The aluminum alloy (AI 7075-T6) is used to construct the wheel rim, while the two reinforcement layers are made of the high-strength steel (ANSI 4340), the mechanical properties of which are given in Table 4 [12].Polyurethane is used to construct honeycomb spokes, shear bands, and inner and outer covers, while the tread is made of synthetic rubber.The hyper-elastic and viscous properties of these materials were defined using the Ogden model and the Prony series, respectively, the material constants of which were identified via experimental tests [12], as summarized in Table 5.

Finite element (FE) analysis method
A number of simplifications and assumptions were made when developing the FE models for the NPTs.These are considered less influential in view of simulation accuracy and are listed as follows: 1) Ignore the friction between the contacting components.
2) Neglecting the effect of the tread patterns on the performance of the honeycomb NPTs.
3) Assuming that all material are isotropic, and the tire is completely symmetrical about the wheel center plane (longitudinal-vertical) that bisects the NPT structure.
4) Ignore the influence of temperature during tire rolling.3), when subjected to gradually increasing displacement loads (up to 10 mm).These could be used to characterize the radial stiffness properties of the NPTs.In this figure, higher reaction forces of the NPTs are invariably observed with an increase in the vertical displacement, as expected.Moreover, the NPT-1 reveals comparable reaction force responses as the NPT-3, when the road is displaced less than 7mm.
Greater force response and thus radial stiffness, however, were noted in NPT-1 with a further increase in the displacement load.On the other hand, NPT-2 and NPT-4 reveal nearly same radial stiffness characteristics, within the entire displacement load range.Relative close similarity was also noted between those of NPT-5 and NPT-7.
When the displacement value of the road surface reaches the maximum value of 10mm, NPT-1 exerts the maximum force on the road surface (i.e., NPT-1 has the maximum stiffness), while NPT-7 exerts the minimum force on the road surface (i.e., NPT-7 has the minimum stiffness).While the radial stiffness value of NPT-4 is 582.8mm/N, that of NPT-1 and NPT-3 is 637.441mm/N and 612.236mm/N, respectively.NPT-1 and NPT-3 have radial stiffnesses that are 9.38% and 5.05% higher than NPT-4, respectively.
Overall, the stiffness of NPT-1 and NPT-3 with a thickness of 5mm in spoke 2 l is greater than that of NPT-4, NPT-2, and NPT-6 with a thickness of 4mm in spoke 2 l ; At the same time, the stiffness of NPT-4, NPT-2, and NPT-6 with spoke 2 l thickness of 4mm is also greater than that of NPT- 5 and NPT-7 with spoke 2 l thickness of 3mm.It can be inferred that spoke  2 ′ thickness significantly impacts the radial stiffness of non-pneumatic tires.
The stress distribution in the spokes of all NPTs under a static load of 5000N is shown in Fig. 6.In the structure of all NPTs, the part with the highest stress is the inner reinforcement layer, followed by the outer reinforcement layer.As this article focuses on the honeycomb spoke structure, Fig. 6 does not show the stress distribution of the inner and outer reinforcement layers and the wheel rim part.
Moreover, only the bottom part of each NPT was shown owing to its relatively higher stresses and deformations.6.
From the deformation data in Table 6, it can be concluded that NPT-5 and NPT-7 have the most significant total deformation of the honeycomb structure, corresponding to the radial stiffness graph, and NPT-5 and NPT-7 have less radial stiffness.Among the honeycomb structures of all NPTs, the least amount of deformation is observed in spoke 4 l , making a small contribution to the overall deformation of the honeycomb structure.Except for NPT-1, among the honeycomb spoke structures of other NPTs, the deformation of spoke 2 l is relatively largest, making a significant contribution to the overall deformation of the honeycomb structure.At the same time, it can be found that the proportion of the deformation of spoke 2 l to the total deformation of the honeycomb structure is approximately linearly related to the radial stiffness of each structure, this is consistent with the analysis results of radial stiffness in Fig. 5.The proportion of the deformation of spoke 2 l to NPT-1 and NPT-3 is relatively small, while the radial stiffness is relatively large; The proportion of the deformation of spoke 2 l in NPT-5 and NPT-7 is relatively large, while the radial stiffness is relatively small.The significant difference between NPT-1 with the highest radial stiffness and other NPTs in the deformation of each spoke is that the deformation of the spoke 3 l and 2 l of NPT-1 accounts for a larger proportion of the overall deformation of the honeycomb structure.

Experimental verification of NPTs with gradient honeycomb spokes
The rim of the NPT is connected to the shear band through the spokes, which are key components that carry the weight of a car [28].The load-bearing capacity of the NPTs varies when associated with different spoke structures.
Measurements were conducted to verify the effect of spoke     This may be due to the dynamic effects in the direction normal to the road.At the same time, it is not only the fluctuation of reaction force in NPT-3 is smaller than that in NPT-4, but also the average reaction force of NPT-3 is also smaller than NPT-4.Meanwhile, the variances of the reaction force for NPT-3 and NPT-4 are 32938.73and 36227.02respectively, and it means that NPT-3 can improve the comfort of a vehicle compared with NPT-4.Corresponding to the spokes, there are also four tread stress peaks that appear periodically.Unlike the stress of the spokes, the overall stress amplitude of the tread is smaller than that of the spoke.The peak stress of NPT-4 tread is greater than that of NPT-3 tread, however, the stress changes of the tread far from the contact area are more smoothly.

Dynamic performance of the NPTs
and its mechanical properties are closely related to the honeycomb geometry.The influence of honeycomb structure factors on the mechanical characteristics of honeycomb tires was investigated by Jin et al.The findings show that the tire's load-bearing capability is inversely correlated with the honeycomb unit's expansion angle [12].Z. Zheng et al. investigated the bending stiffness characteristics of honeycomb tires under various unit expansion angles.The results indicate that the shear layer, tread geometry, and material characteristics significantly impact the turning stiffness of honeycomb tires [13].Ju et al. found that spokes with larger unit expansion angles have lower local stress and mass under the same load.This characteristic proves advantageous for the fatigue resistance design of spokes [14].Ganniari et al. conducted in-depth research on the influence of honeycomb structure design parameters on the performance of NPTs.
impact loads was investigated by Xu et al.According to research, grading systems perform better at absorbing energy than conventional uniform structures [19].A study conducted by Tao et al., graded design boosted the specific energy (SEA) of honeycomb structures by 70% when compared to conventional honeycomb designs [20].Sahu et al. used 3D printing technology to manufacture graded honeycomb structure samples and conducted cyclic compression and free vibration experiments.According to the laboratory findings， the hybrid gradient honeycomb structure has a high damping capacity and is an effective energy absorption element [18].Therefore, introducing graded and gradient structures into honeycomb structures can significantly improve the physical properties of structures made of honeycomb.The concepts of grading and gradient are also applied in NPTs' design.The finite element approach was used by Wu et al. to develop an NPT with a gradient anti-tetrarchical structure and analyze its mechanical characteristics.The findings show that this type of NPT has a good load-bearing capacity [21].The above studies show that graded and gradient honeycomb structures have great potential for application in NPTs.However, NPTs currently mainly use spokes with a homogeneous structure， which confirms the novelty of the research in this paper.At the same time, the majority of present study on NPTs focuses on the analysis of static performance, with less analysis of dynamics, and most of it remains theoretical, with no actual performance experiments.This article conducts finite element simulation on the static and dynamic performance of NPTs with different thickness gradient honeycomb structures.In particular, a thickness gradient honeycomb structure was introduced, and dynamic analysis was conducted.The structure of this article is as described below: Section 2 introduces the models used for simulation, mainly including geometric parameters, material properties, and finite element analysis methods.In Section 3, the load-bearing capacity, stress distribution, and deformation performance of honeycomb structure NPTs with various thickness gradients were investigated.Section 4 summarizes the research results.

Fig. 1
Fig. 1 Structural diagram of the NPT with honeycomb spokes

Fig. 2
depicts a representative sector of the periodic cellular structure.It shows a reference cell of a honeycomb spoke structure, which achieves the honeycomb support spoke for NPTs by uniformly distributing the reference cell at 25 equal cycles in the circumferential direction.The in-plane effective moduli of a 2D honeycomb

 and 4 
, respectively.s E stands for the Young's modulus of the base material of the spokes and t is taken as the corresponding spoke thickness.It needs to be noted that the thickness dimensions of the walls of each spoke may be different.The effective moduli corresponding to spoke 2 l and 4 l respectively are then averaged as the in-plane effective modulus of the honeycomb structure.
In order to investigate the effects of the honeycomb thickness gradient on the performances of the NPT, the length of each edge is fixed constant, and the thickness dimensions of 5 l and 1 l are specified as 4mm and 8mm, respectively.Moreover, the average value of the thicknesses corresponding to 4 l , 3 l and 2 l is taken as 4mm, although each dimension is allowed to alter.Seven FE models of the NPTs with different combinations of 4 l , 3 l and 2

Fig. 2
Fig. 2 Structure diagram of a 2D honeycomb spoke

Fig. 5
Fig. 5 Load-deflection curves of the NPTs Fig. 5 shows the reaction force responses of the abovestated NPTs with 7 different spoke designs (Table3), when

2 l
Observing the spoke structure, it can be observed that different degrees of stress concentration have occurred on the spoke 1 l in contact with the wheel rim, and the locations of the stress concentration are all located on the side of the spoke 1 l away from the vertical centerline of the wheel.The two spokes 5 l on both sides of the vertical centerline are also prone to stress concentration, and the stress value on the side near the centerline is greater.The stress value of the shear band below the two spokes 5 l on both sides of the wheel centerline is also relatively high.Spoke undergoes a certain degree of bending in all structures, and the deformation is significant when the thickness of spoke 2 l is smaller.

Fig. 6 Fig. 7
Fig. 6 Stress distribution of NPTs under a load of 5000 N

Fig. 8
Reference point diagram for spoke deformation under 5000N load:

Table 6
Specific deformation data of NPTs honeycomb spoke structure (Unit: mm) In order to quantitatively analyze the deformation of the various parts of the honeycomb spoke structure, reference points on the spokes were selected as shown in Fig. 8.The change in distance between node 1 and node 2 represents the deformation of spoke 2 l , the change in distance between node 2 and node 3 represents the deformation of spoke 3 l , the change in distance between node 3 and node 4 represents the deformation of spoke 4 l , and the change in distance L between node 1 and node 4 represents the deformation of the entire honeycomb structure.For comparison, the distances of each spoke are projected onto vertical dashed lines.The specific parameters detailing the deformation of each spoke are presented in Table

Fig. 9
Fig.9 The contact pressure of NPT-3 under different loads Fig.10 The maximum contact pressure of NPTs varies with the reaction force on the road surface Also taken into account is the contact pressure between the tire and the ground.Fig.9shows a diagram of the contact pressure for the NPT-3, where the contact area between the road surface and the tread is approximately rectangular.As the reaction force increases, the contact area gradually expands.Moreover, the stress near the contact center is smaller than those close to the shoulder regions.It is speculated that more weight is concentrated on the shoulder regions, and the tire will bear greater pressure at these positions, resulting in greater contact pressure and relatively larger contact area on both sides.This indicates that the tire shoulders are more vulnerable to wear.Fig.10shows the contact pressure versus the reaction force for all NPTs.For all NPTs, the magnitude of contact pressure increases with increasing reaction force.These NPTs, however, show similar maximum stress responses under the loads considered.This is due to the fact that the contact structure on the radial stiffness and contact properties of NPTs, stiffness testing was conducted in the laboratory using a tire stiffness tester, as shown in Fig. 11.Based on the results of finite element analysis and considering cost issues, 3D printing technology was used to jointly produce experimental sample A with a structure of NPT-1 and experimental sample B with a structure of NPT-7.

Fig. 14
Fig. 14 shows the spoke deformation of sample A in the experiment and the spoke deformation of NPT-1 in the finite element simulation results.Through comparison, it can be found that the deformation of the spoke in the experiment is relatively similar to the simulation results, with the deformation of spoke 5 l indicating the most obvious similarity.The results in Fig. 14 reveal satisfactory agreement between the measured and the predicted deformations of the spokes (NPT-1).Comparable similarities were also observed from those of NPT-7 (not shown).These could establish the validities of the FE models developed for the NPTs.
An angular velocity of 26 rad/s (i.e., 32.28 km/h) was assigned to the NPTs under 5000 N load to analyze their dynamic response characteristics in terms of the reaction force of the tire against the displacement load as well as the stress developed in the spokes.It should be mentioned that only NPT-3 and NPT-4 are selected for these dynamic analyses owing to NPT-3 with a thickness-graded honeycomb structure has strong load-bearing capacity and low stress in the spokes, its overall performance is superior, while NPT-4 is a traditional thickness uniform honeycomb structure.The step time for the rolling simulation was set to 1 second, and the tire rolls approximately 4 revolutions.The comparison of spoke deformation of NPT-3 under the dynamic and static state is shown in Fig. 15.It is evident that the spoke deformation is much more severe in the rolling state than that of the static state.Meanwhile, the deformations of the spokes are symmetric about the center line in the static state.However, during dynamic rolling, the spokes become asymmetric.This maybe attributed to the different deformation modes of the toe and heel regions of the travelling tread.Moreover, Honeycomb spokes mainly undergo cyclic tensile-compression deformations.

Fig. 15
Fig. 15 Deformations of NPT-3 under static and dynamic states Fig. 16 shows the time-varying reaction force of NPT-3 and NPT-4 under steady-state rolling conditions.These normal forces were observed to fluctuate around 4873.5N and 4972.1N,respectively, for NPT-3 and NPT-4, which are smaller than the static load of 5000N.The overall reaction force in rolling state is smaller than that in stationary state.

Fig. 16
Fig. 16 Comparison of reaction force variations between NPT-3 and NPT-4The stress developed in the spokes varies as the NPT rolls.

Fig. 17
Fig. 17 shows the stress responses of the spokes of NPT-3 and NPT-4 under 5000N load and 32.08km/h speed.It can be clearly seen that the maximum stress value occurs periodically, with a total of 4 peaks, which corresponds to the rotation period of the NPT in the simulation time.When entering the compression zone, the stress extraction points experience a significant surge and reaches its peak value.Significant stress decreases, however, were observed as the spoke moves away from the contact region.The average spoke stresses of NPT-3 and NPT-4 are 0.2373MPa and 0.4314MPa, respectively.The average stress in NPT-3 spokes under the rolling state is smaller than that in NPT-4 spokes.

Table 1
Components dimensions of the NPT excluding the honeycomb spoke layer

Table 2
Specific parameters of honeycomb structure

Table 3
Spoke thickness schemes and their equivalent modulus and mass parameters