- Original Article
- Open Access

# Influence Analysis of Machining and Installation Errors on the Radial Stiffness of a Non-Pneumatic Mechanical Elastic Wheel

- You-Qun Zhao
^{1}Email author, - Zhen Xiao
^{1}, - Fen Lin
^{1}, - Ming-Min Zhu
^{1}and - Yao-Ji Deng
^{1}

**31**:68

https://doi.org/10.1186/s10033-018-0273-y

© The Author(s) 2018

**Received:**13 January 2017**Accepted:**9 August 2018**Published:**20 August 2018

## Abstract

Machining and installation errors are unavoidable in mechanical structures. However, the effect of errors on radial stiffness of the mechanical elastic wheel (ME-Wheel) is not considered in previous studies. To this end, the interval mathematical model and interval finite element model of the ME-Wheel were both established and compared with bench test results. The intercomparison of the influence of the machining and installation errors on the ME-Wheel radial stiffness revealed good consistency among the interval mathematical analysis, interval finite element simulation, and bench test results. Within the interval range of the ME-Wheel 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 ME-Wheel 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 ME-Wheel 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 ME-Wheel is studied, which will lay the foundation for the exact study of the ME-Wheel dynamic characteristics in the future.

## Keywords

- Mechanical elastic wheel
- Stiffness characteristic
- Errors
- Interval algorithm
- Interval finite element

## 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. Non-pneumatic tires are air-proof and maintenance-free 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 [4–6]. 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 non-pneumatic 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 non-pneumatic tire.

In recent years, Zhao et al. [12] developed a novel non-pneumatic wheel called the mechanical elastic wheel (ME-Wheel). The ME-Wheel 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 ME-Wheel is designed with a statically indeterminate structure. Compared with the conventional pneumatic tire, the ME-Wheel will not have potential risk factors such as flat tire or leakage. Moreover, the double buffered damping structure adopted by the ME-Wheel gives it excellent comfort. The ME-Wheel’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 rings’ laminated structure parameters. Zang et al. [15–17] analyzed the influence of conditions on the radial stiffness of the ME-Wheel. Li et al. [18] studied the mechanical properties of the ME-Wheels 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 three-dimensional finite element wheel–soil interaction model, and simulations with different rotational speeds of the ME-Wheel 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 ME-Wheel, 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 ME-Wheel.

## 2 Structure and Load Bearing Mode of the ME-Wheel

### 2.1 ME-Wheel Structure

- (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 ME-Wheel 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 ME-Wheel

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 ME-Wheel, the ME-Wheel is top load bearing.

When the ME-Wheel is subjected to lateral forces, the mechanical connection (hinge unit, clamping ring) of the ME-Wheel 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 ME-Wheel 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 ME-Wheel Considering Machining and Installation Errors

Interval mathematics is a branch of mathematics that was originally used to solve error problems [26–28]. 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*_{2 }∈ **R** satisfy *x*_{1 }≤ *x*_{2}, then the boundary of the set of real numbers **R** with a closed interval can be expressed as *X* = [*x*_{1}, *x*_{2}] = {*x* ∈ **R**| *x*_{1} ≤ *X* ≤ *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} = *x*_{2} in interval *X*, then the interval *X* is defined as a point interval.

*X*= [

*x*

_{1},

*x*

_{2}],

*Y*= [

*y*

_{1},

*y*

_{2}] ∈

*I*(

*R*), the four arithmetic

*I*(

*R*) operations are defined as follows:

If interval *X* = [*x*_{1}, *x*_{2}], *Y* = [*y*_{1}, *y*_{2}] ∈ *I*(*R*) can satisfy *y*_{1} ≤ *x*_{1} ≤ *x*_{2} ≤ *y*_{2}, then interval *Y* contains interval *X* and is denoted as *X* ⊆ *Y*. If intervals *X* and *Y* have common components, the two intervals are considered to intersect: *X*∩*Y* ≠ *Φ* and *X*∩*Y* is still an interval.

Note that the interval operation satisfies the associative law (*X* *+* *Y*) ± *Z* = *X* + (*Y* *±* *Z*), commutative law *X* *+* *Y* *=* *Y* *+* *X*, and identity law *X* + 0 = 0 + *X* = *X*, but it does not satisfy the distributive law *X* (*Y* + *Z*) ≠ (*XY* + *XZ*); however, its inclusion relationship *X* (*Y* + *Z*) ⊆ *XY* + *XZ* satisfies the distributive law.

### 3.2 Interval Mathematical Model of the ME-Wheel

*R*of the ME-Wheel can be expressed as follows:

*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 ∆

*r*is the thickness of the flexible tire body.

*x*axis, and then, a polar coordinate system (

*r*,

*θ*,

*z*) was established on the basis of the

*x*axis. The displacements at any point in the coordinate system (

*r*,

*θ*,

*z*) are denoted by (

*u*,

*v*,

*w*).

*M*on the flexible tire body, the radial, tangential, and axial normal strain components

*ε*

_{rr},

*ε*

_{θθ},

*ε*

_{zz}are

*M*can be expressed as

*G*

_{A}is the shear modulus of the flexible tire body,

*E*

_{C}is the Young’s modulus, and

*V*is the Poisson’s ratio.

*r*,

*r*

*+*d

*r*) can be expressed as

*r*d

*θ*and width

*l*can be expressed as

*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.

*S*is the shape factor [29],

*A*

_{L}is the cross-sectional area of the specimen, and

*A*

_{F}is the free surface area of the specimen.

In Figure 6, *α* = 2π/*n*, where *n* is the number of hinge units and *t* is the thickness of the hinge unit.

*α*is

*E*

_{D}is the Young’s modulus of the hinge unit.

*R*is the radius of the ME-Wheel,

*n*is the number of hinge units,

*α*= 2π/

*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 Young’s modulus, and

*u*

_{r}(

*R*) is the radial displacement of the ME-Wheel. The specific data of the ME-Wheel material properties in the formula are presented in Table 1.

Material properties of the ME-Wheel in the formula

Properties | Flexible tire body | Suspension hub | Hinge unit |
---|---|---|---|

Young’s modulus | 9.61 | 110 | 71 |

Shear modulus | 4 | − | − |

Poisson’s ratio | 0.48 | 0.28 | 0.33 |

Analysis of the ME-Wheel force shows that the radial stiffness of the ME-Wheel 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 ME-Wheel. 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 ME-Wheel.

*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 ME-Wheel are shown in Table 2.

Geometric parameters of the ME-Wheel

Parameters | Hinge unit | Flexible tire body | Suspension hub |
---|---|---|---|

Number | 12 | − | − |

Thickness | 20 | − | − |

Width | 45 | − | − |

Length | 80 | − | − |

Length | 70 | − | − |

Length | 40 | − | − |

Inner diameter | − | 390 | − |

Outer diameter | − | 460 | − |

Width | − | 320 | − |

Radius | − | − | 220 |

Considering that the machining errors are approximately ± 1 mm and the installation errors are approximately ± 0.5 mm less than the machining errors, one can determine the error interval as [− 1.5, + 1.5] mm. Therefore, the interval length of the hinge unit is [*L* − 4.5, *L* + 4.5] mm, the interval radius of the suspension hub is [*R*_{C} − 1.5, *R*_{C} + 1.5] mm, the interval thickness of the hinge unit is [*t* − 1.5, *t* + 1.5] mm, and the interval width of the hinge unit is [*b* − 1.5, *b* + 1.5] mm.

## 4 Interval Finite Element Model of the ME-Wheel

### 4.1 Finite Element Model of the ME-Wheel

Material properties of the ME-Wheel in the finite element analysis

Parts | Young’s modulus (MPa) | Poisson’s ratio | Density (kg/m |
---|---|---|---|

Flexible tire body | 9610 | 0.48 | 960 |

Suspension hub | 71000 | 0.31 | 2810 |

Hinge unit | 110000 | 0.28 | 7850 |

Clamping ring | 71000 | 0.33 | 2770 |

### 4.2 Interval Finite Element Model of the ME-Wheel

*p*is represented as the interval quantity.

### 4.3 Experimental Validation

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 ME-Wheel can be measured accurately.

*F*is the elastic force exerted on the wheel,

*A*is the cross-sectional area of the piston plate, \(A\, = \,3.115\, \times \,10^{ - 3}\) m

^{2}, and

*G*

_{b}= 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 ME-Wheel with the hinge unit in the ME-Wheel’s six o’clock position and the average value was recorded as the result.

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 ME-wheel on the radial stiffness increases with the load. In addition, the experimental results for the radial stiffness of the ME-wheel 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 ME-Wheel have good consistency.

### 4.4 Influence of the Machining and Installation Errors on the ME-Wheel 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 ME-Wheel radial stiffness, and the length of the hinge unit was used in the parametric analysis.

ME-Wheel machining and installation error range

Interval number | Error of interval | Experimental condition |
---|---|---|

Length of the hinge unit | [185.5, 194.5] | 192 |

Suspension hub radius | [218.5, 221.5] | 220 |

Figure 13(a) shows that the radial stiffness of the flexible tire body significantly influences the ME-Wheel radial stiffness. The maximum difference of the ME-Wheel 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 ME-Wheel 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 ME-Wheel. 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 ME-Wheel are both established and validated by bench test, which can provide a reference for the subsequent radial stiffness analysis of the ME-Wheel.

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

- (3)
A parametric analysis is conducted at different initial stiffness values of the flexible tire body. The ME-Wheel 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 ME-Wheel is obtained.

## Declarations

### Authors’ Contributions

ZX and FL was in charge of the whole trial; Y-QZ and ZX wrote the manuscript; M-MZ and Y-JD assisted with sampling and laboratory analyses. All authors read and approved the final manuscript.

### Authors’ Information

You-Qun 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.

Ming-Min 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.

Yao-Ji 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, 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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## Authors’ Affiliations

## References

- J Ju, B Ananthasayanam, J D Summers, et al. Design of cellular shear bands of a non-pneumatic tire-investigation of contact pressure.
*Hundred Schools in Arts*, 2010, 3(1): 598-606.Google Scholar - L Chen, G Wang, D F An, et al. Tread wear and footprint geometrical characters of truck bus radial tires.
*Chinese Journal of Mechanical Engineering*, 2013, 26(3): 506-511.View ArticleGoogle Scholar - Q Liu, A Shalaby. Simulation of pavement response to tire pressure and shape of contact area.
*Canadian Journal of Civil Engineering*, 2013, 40(3): 236-242.View ArticleGoogle Scholar - T B Rhyne, R H Thompson, S M Cron, et al. Non-pneumatic tire: US, US 7201194 B2. 2007-04-10.Google Scholar
- T B Rhyne, R H Thompson, S M Cron, et al. Non-pneumatic tire having web spokes: US, US 7650919 B2. 2010-01-26.Google Scholar
- T B Rhyne, K W Demino, S M Cron. Structurally supported resilient tire: US, US6769465. 2004-08-03.Google Scholar
- G M Fadel, J Ju, A Michaelraj, et al. Honeycomb structures for high shear flexure: US, US8651156. 2014-02-18.Google Scholar
- J D Summers, G M Fadel, J Ju, et al. Shear compliant hexagonal meso-structures having high shear strength and high shear strain: US, US8609220. 2013-12-17.Google Scholar
- D Y Mun, H J Kim, S J Choi. Airless tire: US, US20120060991. 2012-03-15.Google Scholar
- C Lee, J Ju, D M Kim. Vibration analysis of non-pneumatic tires with hexagonal lattice spokes. ASME 2012 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Chicago, Illinois, USA, August 12–15, 2012: 483–490.Google Scholar
- M Veeramurthy, J Ju, L L Thompson, et al. Optimization of a non-pneumatic tire for reduced rolling resistance. ASME 2011 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Washington, DC, USA, August 28–31, 2011: 861–868.Google Scholar
- Y Q Zhao, L G Zang, Y Q Chen, et al. Non-pneumatic mechanical elastic wheel natural dynamic characteristics and influencing factors.
*Journal of Central South University*, 2015, 22(5): 1707-1715.View ArticleGoogle Scholar - W Wang, Y Q Zhao, L G Zang, et al. Structure analysis and ride comfort of vehicle on new mechanical elastic tire. Proceedings of the FISITA 2012 World Automotive Congress, Beijing, China, November 27, 2013: 199–209.Google Scholar
- Q Wang, Y Q Zhao, X B Du, et al. Equivalent stiffness and dynamic response of new mechanical elastic wheel.
*Journal of Vibroengineering*, 2016, 18(1): 431-445.Google Scholar - L G Zang, Y Q Zhao, C Jiang, et al. Mechanical elastic wheel’s radial stiffness characteristics and their influencing factors.
*Journal of Vibration and Shock*, 2015, 34(8): 181–186. (in Chinese)Google Scholar - L G Zang, Y Q Zhao, B Li, et al. Static radical stiffness characteristics of non-pneumatic mechanical elastic wheel.
*Acta Armamentarii*, 2015, 36(2): 355-362. (in Chinese)Google Scholar - L G Zang, Y Q Zhao, B Li, et al. An experimental study on the ground contact characteristics of non-pneumatic mechanical elastic wheel.
*Automotive Engineering*, 2016, 38(3): 350-355. (in Chinese)Google Scholar - B Li, Y Q Zhao, L G Zang. Closed-form solution of curved beam model of elastic mechanical wheel.
*Journal of Vibroengineering*, 2014, 16(8): 3951-3962.Google Scholar - X B Du, Y Q Zhao, Q Wang, et al. Numerical analysis of the dynamic interaction between a non-pneumatic mechanical elastic wheel and soil containing an obstacle.
*Part D: Journal of Automobile Engineering*, 2017, 231(6): 731-742.Google Scholar - A N Gent, J D Walter, Y T Wei, et al.
*The pneumatic tire*. Beijing: Tsinghua University Press, 2013. (in Chinese)Google Scholar - Z W Wang, Q J Yang, G Bao, et al. Effect of manufacturing errors on static characteristics of externally pressurized spherical air bearings.
*Chinese Journal of Mechanical Engineering*, 2009, 22(6): 896-902.View ArticleGoogle Scholar - X H Han, L Hua, D Song, et al. Influence of alignment errors on contact pressure during straight bevel gear meshing process.
*Chinese Journal of Mechanical Engineering*, 2015, 28(6): 1089-1099.View ArticleGoogle Scholar - T B Rhyne. Influence of rim run-out on the nonuniformity of tire-wheel assemblies.
*Tire Science and Technology*, 1994, 22(2): 99-120.View ArticleGoogle Scholar - T B Rhyne, S M Cron. Development of a non-pneumatic wheel.
*Tire Science and Technology*, 2006, 34(3): 222-225.View ArticleGoogle Scholar - H X Fu, Y Q Zhao, F Lin, et al. Theoretical and experimental analysis on steady-state cornering properties of mechanical elastic wheel.
*Journal of Zhejiang University (Engineering Science)*, 2017, 51(2): 1-6. (in Chinese)Google Scholar - G Alefeld, J Herzberger.
*Introduction to interval computation*. New York: Academic Press, 1983.MATHGoogle Scholar - R E Moore.
*Methods and applications of interval analysis*. Philadelphia: Society for Industrial and Applied Mathematics, 1995.Google Scholar - P Thieler. Technical calculations by means of interval mathematics. Interval Mathemantics 1985: Proceedings of the International Symposium, Freiburg I. Br. Federal Republic of Germany, September, 1986: 197–208Google Scholar
- H Tohara, C W Mu.
*Anti-vibration rubber and its application*. Beijing: China Railway Publishing House, 1982. (in Chinese)Google Scholar