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Influence Analysis of Machining and Installation Errors on the Radial Stiffness of a NonPneumatic Mechanical Elastic Wheel
Chinese Journal of Mechanical Engineering volumeÂ 31, ArticleÂ number:Â 68 (2018)
Abstract
Machining and installation errors are unavoidable in mechanical structures. However, the effect of errors on radial stiffness of the mechanical elastic wheel (MEWheel) is not considered in previous studies. To this end, the interval mathematical model and interval finite element model of the MEWheel were both established and compared with bench test results. The intercomparison of the influence of the machining and installation errors on the MEWheel radial stiffness revealed good consistency among the interval mathematical analysis, interval finite element simulation, and bench test results. Within the interval range of the MEWheel machining and installation errors, parametric analysis of the combined elastic rings was performed at different initial radial rigidity values. The results showed that the initial radial stiffness of the flexible tire body significantly influenced the MEWheel radial stiffness, and the inverse relationship between the hinge unit length or suspension hub and the radial stiffness was nonlinear. The radial stiffness of the MEWheel is predicted by using the interval algorithm for the first time, and the regularity of the radial stiffness between the error and the load on the MEWheel is studied, which will lay the foundation for the exact study of the MEWheel dynamic characteristics in the future.
1 Introduction
Tires carry the weight of the entire vehicle and are the only contact media with the ground [1]. However, traditional pneumatic tires are punctured easily in poor road conditions, which could lead to vehicle failure as well as traffic accidents. In addition, the pressure of pneumatic tires must be kept stable at all times, thus increasing the time and maintenance costs [2, 3]. Therefore, various new structure safety tires have been proposed. Nonpneumatic tires are airproof and maintenancefree compared to conventional pneumatic tires, thus improving vehicle safety while providing good comfort.
Rhyne introduced the Tweel wheel; the bead and hub of the wheel were connected by dozens of soft deformable polyurethane spokes [4,5,6]. As the Tweel wheel is an airless tire, it cannot lead to a blowout situation in case of a puncture. When crossing obstacles, polyurethane spokes can produce elastic deformation, thus absorbing the impact from the pavement and reducing the vibrations of the road. Fadel et al. [7] and Summers et al. [8] developed the honeycomb wheel, which mainly comprises a bead, honeycomb spoke, and hub, at the Cooper Tire Company and Wisconsin Madison Polymer Research Center. Based on the bionics principle, the hexagonal structure of each spoke supports another to form a honeycomb structure. The vibration reduction performance of the honeycomb wheel increases its strength. Mun et al. [9] developed the nonpneumatic tire, including a cylindrical tread in contact with the ground; the airless tire has good buffering and damping functions. Lee et al. [10] examined the dynamic characteristics of flexible hexagonal lattice spokes using a finite element simulation and compared the vibration characteristics with those of pneumatic tires. Veeramurthy et al. [11] used the finite element model (FEM) to study the effect on the vertical stiffness and rolling resistance response considering two design variables of a nonpneumatic tire.
In recent years, Zhao et al. [12] developed a novel nonpneumatic wheel called the mechanical elastic wheel (MEWheel). The MEWheel consists of three parts, the flexible tire body, hinge unit and suspension hub. In order to improve the adaptability of military vehicles in complex environment, the MEWheel is designed with a statically indeterminate structure. Compared with the conventional pneumatic tire, the MEWheel will not have potential risk factors such as flat tire or leakage. Moreover, the double buffered damping structure adopted by the MEWheel gives it excellent comfort. The MEWheelâ€™s structure and vehicle ride comfort was analyzed by Wang et al. [13]. Wang et al. [14] analyzed the relationship among the excitation frequency, radial deformation, bending stiffness of combined elastic rings, and combined elastic ringsâ€™ laminated structure parameters. Zang et al. [15,16,17] analyzed the influence of conditions on the radial stiffness of the MEWheel. Li et al. [18] studied the mechanical properties of the MEWheels with Laplace transform and obtained the relationship between the tangential deformation and bending angle of the combined elastic rings. Du et al. [19] established a nonlinear threedimensional finite element wheelâ€“soil interaction model, and simulations with different rotational speeds of the MEWheel were conducted.
The radial stiffness of the wheel directly influences the vehicle ride comfort, driving safety, steering stability, and passing ability [20]. Therefore, researchers should thoroughly understand the radial stiffness characteristics of the wheel. In reality, machining and installation errors are inevitable [21, 22]. Owing to the design of the MEWheel, the machining and installation errors significantly influence the radial stiffness of the wheel. Therefore, it is of theoretical and practical significance to investigate the effects of machining and installation errors on the radial stiffness of the MEWheel.
2 Structure and Load Bearing Mode of the MEWheel
2.1 MEWheel Structure
Figure 1 shows the structure of the MEWheel, which comprises the flexible tire body, hinge unit, pin, clamping ring, combined elastic rings, and suspension hub components. The assembly relationship is as follows:

(1)
The five combined elastic rings are juxtaposed. Twelve sets of clamping rings are uniformly installed to lock the elastic rings together to form the MEWheel frame, as shown in Figure 2.

(2)
The suspension hub is installed at the center of the combined elastic rings, and the clamping ring and the suspension hub are connected by the hinge unit.

(3)
The combined elastic rings are embedded in the flexible tire body.
2.2 Load Bearing Mode of the MEWheel
The wheel has two types of load bearings: bottom load bearing and top load bearing [23, 24]. As no force is exerted on the bottom of the MEWheel, the MEWheel is top load bearing.
During motion, the MEWheel bears not only the uneven road shock excitations but also the vehicle weight and torque from the axle shaft (engineâ€“transmissionâ€“axle shaftâ€“wheel). When the MEWheel bears the vertical load, the upper part of the wheel hinge unit is under tensile stress from the suspension hub, the lower portion of the wheel hinge unit is in a relaxed state, the lower part of the flexible tire body is deformed, and the hinge unit is gradually bent. The suspension hub is suspended by the hinge unit on the MEWheel. This type of load bearing can guarantee the optimal carrying capacity of the flexible tire body, greater deformation contact between the MEWheel and the road, and enhanced grip and shock absorbing capacity of the MEWheel. When the hinge unit is pulled, it operates as a twoforce bar and transfers the axial force. On the other hand, when the hinge unit is pressed, the hinge unit is bent and deformed so that the hinge unit does not bear the force, as shown in Figure 3(a).
When the MEWheel is subjected to lateral forces, the mechanical connection (hinge unit, clamping ring) of the MEWheel will transmit the lateral force to the flexible tire body, which will lead to a certain degree of elastic deformation of the flexible tire body in the grounding area. Compared to the pneumatic tire, the MEWheel has greater lateral stiffness, which makes the vehicle equipped with this wheel have better handling stability [25], as shown in Figure 3(b).
3 Interval Mathematical Model of the MEWheel Considering Machining and Installation Errors
Interval mathematics is a branch of mathematics that was originally used to solve error problems [26,27,28]. Interval analysis has been widely used in numeral calculations, especially in numerical error analysis.
3.1 Basic Concepts of Interval Mathematics and Interval Arithmetic
Definition: If x_{1}, x_{2â€‰}âˆˆâ€‰R satisfy x_{1â€‰}â‰¤â€‰x_{2}, then the boundary of the set of real numbers R with a closed interval can be expressed as Xâ€‰=â€‰[x_{1}, x_{2}]â€‰=â€‰{xâ€‰âˆˆâ€‰R x_{1}â€‰â‰¤â€‰Xâ€‰â‰¤â€‰x_{2}}, where x_{1} is the lower endpoint of interval X and x_{2} is the upper endpoint of interval X. If x_{1}â€‰=â€‰x_{2} in interval X, then the interval X is defined as a point interval.
For âˆ€Xâ€‰=â€‰[x_{1}, x_{2}], Yâ€‰=â€‰[y_{1}, y_{2}]â€‰âˆˆâ€‰I(R), the four arithmetic I(R) operations are defined as follows:
If interval Xâ€‰=â€‰[x_{1}, x_{2}], Yâ€‰=â€‰[y_{1}, y_{2}]â€‰âˆˆâ€‰I(R) can satisfy y_{1}â€‰â‰¤â€‰x_{1}â€‰â‰¤â€‰x_{2}â€‰â‰¤â€‰y_{2}, then interval Y contains interval X and is denoted as Xâ€‰âŠ†â€‰Y. If intervals X and Y have common components, the two intervals are considered to intersect: Xâˆ©Yâ€‰â‰ â€‰Î¦ and Xâˆ©Y is still an interval.
Note that the interval operation satisfies the associative law (Xâ€‰+â€‰Y)â€‰Â±â€‰Zâ€‰=â€‰Xâ€‰+â€‰(Yâ€‰Â±â€‰Z), commutative law Xâ€‰+â€‰Yâ€‰=â€‰Yâ€‰+â€‰X, and identity law Xâ€‰+â€‰0â€‰=â€‰0â€‰+â€‰Xâ€‰=â€‰X, but it does not satisfy the distributive law X (Yâ€‰+â€‰Z)â€‰â‰ â€‰(XYâ€‰+â€‰XZ); however, its inclusion relationship X (Yâ€‰+â€‰Z)â€‰âŠ†â€‰XYâ€‰+â€‰XZ satisfies the distributive law.
3.2 Interval Mathematical Model of the MEWheel
Figure 4 shows the force and deformation of the wheel perpendicular to the rigid ground without considering the influence of the surface pattern of the MEWheel and volume compression of the rubber structural material.
The radius R of the MEWheel can be expressed as follows:
where L_{1}, L_{2}, and L_{3} are the lengths of hinge units 1, 2, and 3, respectively; R_{C} is the radius of the suspension hub; and âˆ†r is the thickness of the flexible tire body.
As shown in Figure 5, the radial direction of the flexible tire body was selected as the x axis, and then, a polar coordinate system (r, Î¸, z) was established on the basis of the x axis. The displacements at any point in the coordinate system (r, Î¸, z) are denoted by (u, v, w).
Therefore, for any point M on the flexible tire body, the radial, tangential, and axial normal strain components Îµ_{rr}, Îµ_{Î¸Î¸}, Îµ_{zz} are
The shear strain at point M can be expressed as
According to the generalized Hookeâ€™s law, the constitutive equations of the flexible tire body stresses and strains can be expressed as
where G_{A} is the shear modulus of the flexible tire body, E_{C} is the Youngâ€™s modulus, and V is the Poissonâ€™s ratio.
Ignoring the axial deformation of the flexible tire body, the volume of any small cell (r, râ€‰+â€‰dr) can be expressed as
Thus, the normal stress of a plane of length rdÎ¸ and width l can be expressed as
The shear stress is
The radial load of the small cell is
Therefore, the radial load of the wheel can be obtained as follows:
Based on Eq. (12), the radial deformation of the flexible tire body can be expressed as follows:
where F_{x} is the radial force on the flexible tire body, D is the flexible tire body width, r_{1} is the inner diameter, and r_{2} is the outer diameter.
The radial stiffness of the flexible tire body can be expressed as follows:
where S is the shape factor [29], A_{L} is the crosssectional area of the specimen, and A_{F} is the free surface area of the specimen.
The load between the flexible tire body and suspension hub is transmitted by the circumferentially distributed hinge unit. Load analysis of the MEWheel shows that the load exerted on the MEWheel axle is carried by the extension of all hinge units except the grounded contact area. The MEWheel bearing the hinge unit is only subjected to tensile force. Therefore, the hinge unit is simplified (Figure 6) for ease of model analysis.
In Figure 6, Î±â€‰=â€‰2Ï€/n, where n is the number of hinge units and t is the thickness of the hinge unit.
In the hinge unit model, the tensile force per unit width of the hinge unit at angle Î± is
where E_{D} is the Youngâ€™s modulus of the hinge unit.
The tension of the MEWheelâ€™s hinge unit can be expressed as
where R is the radius of the MEWheel, n is the number of hinge units, Î±â€‰=â€‰2Ï€/n, b is the width of the hinges, L is the length of the hinge unit, t is the thickness of the hinge unit, E_{D} is the Youngâ€™s modulus, and u_{r}(R) is the radial displacement of the MEWheel. The specific data of the MEWheel material properties in the formula are presented in Table 1.
Analysis of the MEWheel force shows that the radial stiffness of the MEWheel is mainly determined by the structural and mechanical properties of the flexible tire body and the hinge unit. The greater the radial stiffness of the flexible tire body, the greater the stiffness of the MEWheel. In addition, the rigidity of the flexible tire body depends on the stiffness of the combined elastic rings, number of clamping rings, and size of flexible tire body structure. On the hinge unit, its stiffness also directly affects the overall stiffness of the MEWheel.
Since the hinge unit material has the same Youngâ€™s modulus during the machining process, in the interval mathematics calculations, one can consider the hinge unit lengths L_{1}, L_{2}, L_{3}, suspension hub radius R_{C}, hinge unit thickness t, and hinge unit width b as the interval number. The main geometric parameters of the MEWheel are shown in Table 2.
Considering that the machining errors are approximately Â±â€‰1 mm and the installation errors are approximately Â±â€‰0.5 mm less than the machining errors, one can determine the error interval as [âˆ’â€‰1.5, +â€‰1.5] mm. Therefore, the interval length of the hinge unit is [Lâ€‰âˆ’â€‰4.5, Lâ€‰+â€‰4.5] mm, the interval radius of the suspension hub is [R_{C}â€‰âˆ’â€‰1.5, R_{C}â€‰+â€‰1.5] mm, the interval thickness of the hinge unit is [tâ€‰âˆ’â€‰1.5, tâ€‰+â€‰1.5] mm, and the interval width of the hinge unit is [bâ€‰âˆ’â€‰1.5, bâ€‰+â€‰1.5] mm.
When the machining and installation errors of the MEWheel are in a certain range, the load characteristics of the MEWheel are determined by the interval mathematical model, as shown in Figure 7. One can find that the load characteristic curve is weakly nonlinear and the influence of the machining and installation errors on the radial stiffness significantly increases with the wheel load.
4 Interval Finite Element Model of the MEWheel
4.1 Finite Element Model of the MEWheel
To avoid obtaining an abnormal unit by comprehensive consideration, one can obtain a neat grid by using the hexahedral and sweep method to mesh the model to enhance the convergence and computational accuracy. The stress distribution and deformation of the hinge unit and combined elastic rings are significant in this test; therefore, the mesh of this part is fine and the other parts of the mesh are relatively coarse. At the same time, the tetrahedral mesh method is adopted for the irregular geometry and the size control method is adopted for some important details. Figure 8 shows the FEM of the MEWheel with 43424 units and 185474 nodes. In the finite element analysis, material properties must be defined to ensure the finite element analysis results. The material properties of the MEWheel in the finite element analysis are given in Table 3.
4.2 Interval Finite Element Model of the MEWheel
The interval FEM can be established when the uncertainty factors of the radial stiffness of the MEWheel can be expressed by the related parameters whose interval boundary can be defined. The interval finite element calculation formula with an uncertain quantity can be written as follows:
or
where the column vector p is represented as the interval quantity.
The FEM of the MEWheel considering the machining and installation errors can be established by the interval finite element method. In contrast to the FEM, the interval FEM is assembled with a shrink fit, as shown in Figure 9. The weak nonlinear relationship shown in Figure 10 is similar to the results in Figure 7.
4.3 Experimental Validation
The radial stiffness experiment of the MEWheel was performed on the tire characteristic test bench. The setup and components of the tire characteristic test bench is shown in Figure 11.
The loading device of the tire characteristic test bench can provide any radial force to the wheel and maintain a certain load. The loading mode of the platform was designed so that the radial stiffness of the MEWheel can be measured accurately.
Data processing for the tire characteristic test bench was performed using Eq. (21),
where F is the elastic force exerted on the wheel, A is the crosssectional area of the piston plate, \(A\, = \,3.115\, \times \,10^{  3}\) m^{2}, and G_{b}â€‰=â€‰1032.43 N is the weight of the platen and the side panels.
For accurate measurements, each radial load was exerted six times on the MEWheel with the hinge unit in the MEWheelâ€™s six oâ€™clock position and the average value was recorded as the result.
The results of the interval finite element calculation (FEM), the interval mathematical model (analytical model) for the same interval as well as the experimental results are shown in Figure 12.
Figure 12 shows that the load characteristic curve of the wheel is weakly nonlinear, and the influence of the machining and installation errors of the MEwheel on the radial stiffness increases with the load. In addition, the experimental results for the radial stiffness of the MEwheel are within the range of the interval mathematical model and interval FEM results. Therefore, the interval mathematical model, interval finite element model, and experimental results for the MEWheel have good consistency.
4.4 Influence of the Machining and Installation Errors on the MEWheel Radial Stiffness
Radial stiffness is an important factor in wheel vibration. This study mainly focused on the influence of the machining and installation errors on the MEWheel radial stiffness, and the length of the hinge unit was used in the parametric analysis.
Four initial radial stiffness values of the flexible tire bodyâ€”80 N/mm, 100 N/mm, 120 N/mm and 140 N/mmâ€”were calculated by the interval FEM. The interval length of the hinge unit and suspension hub was
In the experiment, the hinge unit length was 192 mm and the suspension hub radius was 220 mm. The MEWheel machining and installation errors are shown in Table 4. Figure 13 shows the influence of the machining and installation errors on the MEWheel radial stiffness
Figure 13(a) shows that the radial stiffness of the flexible tire body significantly influences the MEWheel radial stiffness. The maximum difference of the MEWheel radial stiffness at the same hinge unit length was more than 100 N/mm. Assuming that the other parameters are unchanged in the given range, the MEWheel radial stiffness nonlinearly decreased with increasing hinge unit length. The experimental results were similar to the FEM simulation results on the same horizontal axis value, and both results were within the range of radial stiffness calculated by the interval mathematical model.
Figure 13(b) shows the influence of the suspension hub radius on the radial stiffness of the MEWheel. It can be seen that the influence of the hub radius on the radial stiffness is similar to that of the hinge unit length; however, the curves in Figure 13(b) are gentler owing to the smaller interval number range of the suspension hub. The suspension hub radius in the experiment was 220 mm and the experimental value was slightly greater than the simulated value in the same abscissa, and both results satisfied the analysis of the interval mathematical model.
5 Conclusions

(1)
The interval mathematical model and interval FEM of the MEWheel are both established and validated by bench test, which can provide a reference for the subsequent radial stiffness analysis of the MEWheel.

(2)
The interval FEM simulation and theoretical calculation can be obtained, the load characteristic curve of the MEWheel is weakly nonlinear in a certain range, and the influence of the machining and installation errors on the MEWheel radial stiffness increased with the load.

(3)
A parametric analysis is conducted at different initial stiffness values of the flexible tire body. The MEWheel radial stiffness is found to be inversely proportional to the suspension hub and hinge unit length and the reduction rate of the radial stiffness is greater than the increase in the suspension hub and hinge unit length. In the future, the machining and installation errors can be properly controlled by the regularity of this study, thus the required radial stiffness of the MEWheel is obtained.
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Authorsâ€™ Contributions
ZX and FL was in charge of the whole trial; YQZ and ZX wrote the manuscript; MMZ and YJD assisted with sampling and laboratory analyses. All authors read and approved the final manuscript.
Authorsâ€™ Information
YouQun Zhao, born in 1968, is a professor at Nanjing University of Aeronautics and Astronautics, China. His research interests include vehicle dynamics control and automotive design theory and test methods, etc.
Zhen Xiao, born in 1989, is a PhD candidate at Nanjing University of Aeronautics and Astronautics, China. His research interests include vehicle dynamics control and tire dynamics.
Fen Lin, born in 1980, is currently an associate professor at Nanjing University of Aeronautics and Astronautics, China. He received his PhD degree from Nanjing University of Aeronautics and Astronautics, China, in 2008. His research interest is vehicle system dynamics.
MingMin Zhu, born in 1986, is a PhD candidate at Nanjing University of Aeronautics and Astronautics, China. Her research interests include vehicle dynamics control and tire dynamics.
YaoJi Deng, born in 1991, is a PhD candidate at Nanjing University of Aeronautics and Astronautics, China. His research interests include vehicle dynamics control and tire dynamics.
Competing Interests
The authors declare that they have no competing interests.
Funding
Supported by National Natural Science Foundation of China (Grant No. 11672127), Major Exploration Project of the General Armaments Department of China (Grant No. NHA13002), Fundamental Research Funds for the Central Universities of China (Grant No. NP2016412, NP2018403,Â NT2018002), and Jiangsu Provincial Innovation Program for Graduate Education and the Fundamental Research Funds for the Central Universities of China (Grant No. KYLX16_0330).
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Zhao, YQ., Xiao, Z., Lin, F. et al. Influence Analysis of Machining and Installation Errors on the Radial Stiffness of a NonPneumatic Mechanical Elastic Wheel. Chin. J. Mech. Eng. 31, 68 (2018). https://doi.org/10.1186/s100330180273y
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DOI: https://doi.org/10.1186/s100330180273y
Keywords
 Mechanical elastic wheel
 Stiffness characteristic
 Errors
 Interval algorithm
 Interval finite element