# Contact Model of Revolute Joint with Clearance Based on Fractal Theory

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## Abstract

The contact stiffness of a mechanical bonding surface is an important parameter in determining the normal and radial contact force. To improve the calculation accuracy of the contact force model, the surface roughness of the bonding surface and the energy loss that necessarily occurs during the impact process should be considered comprehensively. To study the normal contact force of a revolute joint with clearance more accurately in the case of dry friction, a nonlinear stiffness coefficient model considering the surface roughness was established based on fractal theory, which considers the elastic, elastic-plastic, and plastic deformations of the asperities of the contact surface during the contact process. On this basis, a modified nonlinear spring damping model was established based on the Lankarani–Nikravesh contact force model. The laws influencing the surface roughness, recovery coefficient, initial velocity, and clearance size on the impact force were revealed, and were compared with the Lankarani–Nikravesh model and a hybrid model using MATLAB. The maximum impact force was obtained using a modified contact force model under different initial velocities, different clearances, and different degrees of surface roughness, and the calculated results were then compared with the experiment results. This study indicates that the modified model can be used more widely than other models, and is suitable for both large and small clearances. In particular, the modified model is more accurate when calculating the contact force of a revolute joint with a small clearance.

## Keywords

Fractal theory Modified contact model Surface roughness Lankarani–Nikravesh model## 1 Introduction

In recent years, many scholars domestically and abroad have conducted a number of studies on the contact impact problem of dynamics with clearance. A common contact theory is the Hertz contact theory [1]; however, this theory can only describe the elastic deformation of the material during the contact process, and does not consider the energy dissipation. A contact force model with nonlinear damping (Hunt–Crossley model) was thus proposed by Hunt et al. [2], which considers the dissipation of energy during the contact process. Yan et al. [3] introduced the application of a clearance model in the analysis of nonlinear characteristics, a performance evaluation, and a reliability evaluation of the mechanical system dynamics. Han et al. [4] used the joint contact impact force as a parameter to achieve a high-precision lifetime prediction of a mechanism. Lankarani et al. [5] determined the relationship between the damping factor and restitution coefficient by analyzing the energy loss in a system before and after the contact process, and then put forward the Lankarani–Nikravesh continuous contact force model (L–N model). The L–N model not only considers the dissipation of energy during the contact process, but also reflects the influence of the material properties, geometric characteristics, and motion states of the contact body during the contact process. Based on the L–N model and improved elastic foundation contact model [6], a hybrid model was established by Bai et al. [7], through which the nonlinear stiffness coefficient was presented. Wang et al. [8] established an improved nonlinear contact impact force model by considering the axial dimensions, material coefficients, and energy loss comprehensively. However, the previous modeling methods were all established at a macro level, and do not reflect the microscopic characteristics of the contact surface, such as the influence of the micro topography and surface roughness on the contact stiffness.

There are numerous micro-convex bodies on an actual motion pair surface, all of which have a particular degree of roughness. A rough surface morphology has an important influence on the contact stiffness coefficient [9, 10], and affects the contact impact force during an actual contact collision. Therefore, drawing on research methods regarding the contact problems of the surface morphology, a contact model of a revolute joint with clearance when considering the surface morphology can be established, which is significant to a precision analysis and dynamic design of the mechanical system dynamics.

In fact, many asperities appear on a rough contact surface, and a rough surface morphology has an important influence on [11]. Scholars have conducted a number of studies on theoretical contact force models over the years, including the classic GW model [12], WA model [13], and MB model [14]. Furthermore, taking account of the influences of the surface asperities grade, the deformation characteristic, and the contact area, Yuan et al. [15, 16] established and analyzed the fractal contact mode between rough surfaces based on the fractal theory. Pan et al. [17, 18] derived a theoretical three-dimensional mechanical joint surface based on the W-M function of the three-dimensional joint surface topography. Liu et al. [19] established a fractal model of the tangential contact stiffness of two spherical surfaces when considering the friction factors. Wang et al. [20] established a deterministic model for the elastoplastic contact of rough surfaces under combined loads, and obtained the actual contact area, pressure distribution, and plastic deformation of the micro-convex peaks. A series of studies were carried out on the model of the contact stiffness coefficient of the contact surface. However, there have been relatively few studies establishing the organic connections between the macroscopic levels and microscopic states when considering a rough surface.

In this paper, the contact stiffness coefficient was derived when considering the surface roughness of the contact surface, and a modified contact force model was obtained. Based on the fractal theory, a modified normal contact stiffness coefficient model was also established, which considers the contact surface roughness, as well as the elastic, elastic-plastic, and plastic deformations of the asperities during the contact process. Meanwhile, a modified contact force model was established by combining the L–N model expression. Finally, an experimental system was developed, and the correctness of the model was verified through an experimental study.

## 2 Establishment of Modified Contact Force Model

When two surfaces consisting of asperities are close to each other, the contact only occurs at the top of the asperity owing to the asperities on the contact surface of the actual motion joint. During a study on the surface contact, the contact of two surfaces is usually simplified using the contact between a rough surface and a rigid plane [21].

### 2.1 Contact Deformation of Asperity

*D*is the fractal dimension, \(\varphi\) is the domain extension factor for the asperity size distribution, \(a^{\prime}\) is the area of a contact spot, \(a^{\prime}_{l}\) is the area of the largest contact spot, and \(A_{r}\) is the real area of contact.

*G*is a characteristic length of scale of the surface;

*K*is the correlation coefficient [25], which is usually 2.8;

*H*is the hardness of softer materials; and \(\gamma\) is a constant, which is usually 1.5. The composite elastic modulus \(E^{ * }\) is given by

*v*is the Poisson ratio.

Three deformation stages may occur on the asperity of a rough surface during the contact process, namely, elastic deformation, elastic-plastic deformation, and complete plastic deformation stages.

### 2.2 Modified Contact Stiffness Model

*A*. Thus, an area proportion coefficient \(\lambda\) of less than 1 is found, and the relationship between the actual contact area and the nominal contact area can be described as

*ε*and the penetration displacement

*u*can be expressed as follows:

*ε*is the semi-angle contact,

*u*is the penetration displacement, and

*R*

_{1}and

*R*

_{2}refer to the radii of the pin and the hole in the plate, respectively. Here, \(\Delta R = R_{2} - R_{1}\) represents the radial clearance of the joints.

*R*

_{1}and

*R*

_{2}, the relationship between

*l*and \(\varepsilon\) can be expressed as

*l*is the length of the chord.

*b*, the nominal contact area

*A*can be expressed as

*u*can be obtained through formulae (8) and (11) as

*D*and

*G*[22] are given as

### 2.3 Modified Contact Force Model

*K*is the stiffness coefficient, and

*D*

_{n}is the damping coefficient.

*K*with the modified stiffness coefficient

*K*

_{n}, and is written in the following form:

## 3 Numerical Simulation of Modified Contact Force Model

Using the modified model proposed in this paper, a one-shot contact was analyzed.

### 3.1 Simulation Parameters of the Mechanism

The material and size parameters of the shaft and sleeve were taken as follows: *E* = 207 GPa, ν = 0.3, σ_{ѵ} = 335 MPa, *m *= 1 kg, *b* = 0.01 m, and *R*_{2} = 0.01 m.

Because the L–N and hybrid models [6] do not take the roughness of the contact surface into account, roughness Ra in the modified model has a smaller value of 0.05 for unifying the parameters.

With the increase in contact load, the real contact area tends to be a fixed value; however, this value is less than the nominal contact area (approximately 5–10% of the nominal contact area) [30, 31, 32, 33, 34]. Therefore, the area proportion coefficient *λ* in this paper can be valued at 0.05.

### 3.2 Analysis of Contact Process under Different Restitution Coefficients

Figure 2(a) shows the characteristics of damping hysteresis during the contact process. The damping of the material will cause a loss of energy, and the restitution coefficient will be smaller, whereas the loss of energy during the contact process will be larger. The modified contact force model presented in this paper can better describe the loss of energy during the contact process when the restitution coefficient is small. Furthermore, Figure 2(b) and (c) also conform to the change in contact force and deformation during actual contact. Therefore, it is possible to describe the contact process with different restitution coefficients.

### 3.3 Analysis of Contact Process at Different Initial Velocities

It can be seen from Figure 3 that, with an increase in the initial contact velocity, the contact force and deformation increase rapidly, and the contact time is shortened, which is consistent with the actual contact process. In addition, when the contact force is at maximum, the deformation also reaches the maximum value, which is consistent with the theoretical formula.

### 3.4 Analysis of Contact Process for Different Amounts of Roughness

As Figure 4 shows, the roughness of the contact surface is smaller, the contact force is larger, and the time to reach the maximum contact force is shorter. Moreover, the roughness is smaller, as is the maximum deformation; in addition, the time to reach the maximum deformation is also shorter, the reason for which is that a smaller amount of contact surface roughness leads to a larger fractal dimension, and a larger stiffness coefficient. When the initial conditions are the same, the stiffness is larger, the contact force is greater, and the deformation is smaller.

## 4 Comparative Analysis of Models

In this paper, a single contact simulation of the L–N model and the hybrid model established in Ref. [6] was carried out, and the results were compared with those of the modified model.

### 4.1 Comparative Analysis of Models under Different Clearance Sizes

As shown in Figure 5, the modified model shows a slight difference with the L–N and hybrid models with a large clearance, whereas the difference between the modified model and the hybrid model is greater than that of the L–N model when the clearance decreases, the reason for which is that the L–N model is based on the Hertz contact theory, and is suitable for the case of non-coordinated contact with a large clearance. For a small clearance, the error in the calculation result is large. The modified model proposed in this paper and the hybrid model are suitable for both large and small clearances, and their calculation results are more accurate. In addition, when the clearance is small, the contact becomes closely conformal contact. Because the modified model described in this paper is based on the fractal theory and micro-surface topography, it is more accurate when calculating a small-clearance contact.

### 4.2 Comparative Analysis of Models under Different Restitution Coefficients

As shown in Figure 6, when the restitution coefficient is large, the calculated results of the three models are relatively similar, whereas when the restitution coefficient is small, the calculated results of the modified model in this paper are similar to those of the hybrid model, which is quite different from the L–N model. The damping force terms in the modified model and the hybrid model can reflect the loss of energy better during the contact process with a smaller restitution coefficient, thereby improving the calculation results. The modified contact force model has a broader range of application.

## 5 Experiment Verification

A crash test rig was designed, the results of which prove the correctness of the theoretical analysis.

### 5.1 Test Rig

Parameter values of the modified model

Parameter | Values | Parameter | Values |
---|---|---|---|

Elastic modulus | 207 GPa | Equivalent mass | 0.85 kg |

Poisson ratio | 0.3 | Simulation clearance | 0.5 mm |

Surface roughness | 6.3 | Area scale coefficient | 0.05 |

Convex radius | 16.5 mm | Restitution coefficient | 0.46 |

### 5.2 Experimental Results and Analysis

Experimental measurements and numerical results

Experiment no. | Measured initial velocity (m/s) | Measured contact force (N) | Theoretical contact force (N) | Relative error |
---|---|---|---|---|

1 | 0.131 | 258.18 | 234.5 | − 9.63% |

2 | 0.150 | 308.53 | 281.6 | − 9.56% |

3 | 0.241 | 499.00 | 533.4 | 6.38% |

4 | 0.299 | 711.06 | 708.6 | − 0.42% |

5 | 0.302 | 739.00 | 721.7 | − 2.50% |

6 | 0.351 | 935.00 | 881.4 | − 6.08% |

7 | 0.388 | 1089.17 | 1010 | − 7.82% |

8 | 0.400 | 1188.66 | 1087 | − 9.29% |

9 | 0.421 | 1250.00 | 1128 | − 7.30% |

10 | 0.427 | 1183.78 | 1150 | − 2.87% |

11 | 0.577 | 1661.07 | 1718 | 3.32% |

12 | 0.630 | 2035.22 | 1924 | − 5.77% |

13 | 0.735 | 2205.81 | 2365 | 6.77% |

14 | 0.818 | 2435.45 | 2703 | 9.91% |

15 | 0.882 | 2827.45 | 2994 | 5.58% |

16 | 0.970 | 3191.83 | 3391 | 5.90% |

17 | 0.989 | 3347.17 | 3477 | 3.74% |

18 | 1.044 | 3752.15 | 3722 | − 0.80% |

19 | 1.070 | 3812.87 | 3842 | 0.78% |

20 | 1.110 | 3934.63 | 4039 | 2.60% |

Experimental measurements and numerical results for *c* = 0.5 mm and *R*_{a} = 6.3

Experiment no. | Measured initial velocity (m/s) | Measured contact force (N) | Theoretical contact force (N) | Relative error |
---|---|---|---|---|

1 | 0.131 | 258.18 | 242.4 | − 6.1% |

2 | 0.150 | 308.53 | 288.5 | − 6.5% |

3 | 0.241 | 499.00 | 530.3 | 6.3% |

4 | 0.299 | 711.06 | 698.4 | − 1.8% |

5 | 0.302 | 739.00 | 707.2 | − 4.3% |

6 | 0.343 | 847.78 | 832.4 | − 1.8% |

7 | 0.388 | 1059.17 | 972.7 | − 8.2% |

8 | 0.411 | 1112.06 | 1048.0 | − 5.8% |

9 | 0.422 | 1128.54 | 1084.0 | − 3.9% |

10 | 0.427 | 1183.78 | 1100.0 | − 7.1% |

11 | 0.577 | 1661.07 | 1611.0 | − 3.0% |

12 | 0.624 | 1697.08 | 1778.0 | 4.8% |

13 | 0.633 | 1748.96 | 1812.0 | 3.6% |

14 | 0.735 | 2205.81 | 2188.0 | − 0.8% |

15 | 0.818 | 2435.30 | 2503.9 | 2.8% |

16 | 0.882 | 2827.45 | 2755.8 | − 2.5% |

17 | 0.970 | 3191.83 | 3102.5 | − 2.8% |

18 | 0.989 | 3347.17 | 3177.4 | − 5.1% |

19 | 1.044 | 3752.15 | 3399.4 | − 9.4% |

20 | 1.070 | 3812.87 | 3508.7 | − 8.0% |

21 | 1.110 | 3934.63 | 3677.1 | − 6.5% |

Experimental measurements and numerical results for *c *= 1 mm and *R*_{a }= 3.2

Experiment no. | Measured initial velocity (m/s) | Measured contact force (N) | Theoretical contact force (N) | Relative error |
---|---|---|---|---|

1 | 0.101 | 174.56 | 190.6 | 9.2% |

2 | 0.127 | 249.63 | 256.2 | 2.6% |

3 | 0.253 | 652.16 | 623.3 | − 4.4% |

4 | 0.262 | 624.69 | 651.8 | 4.3% |

5 | 0.297 | 746.77 | 766 | 2.6% |

6 | 0.305 | 822.75 | 792.8 | − 3.6% |

7 | 0.354 | 1044.31 | 959.2 | − 8.1% |

8 | 0.378 | 1011.05 | 1044.5 | 3.3% |

9 | 0.383 | 1084.9 | 1062.3 | − 2.1% |

10 | 0.391 | 1130.68 | 1090.9 | − 3.5% |

11 | 0.392 | 1036.99 | 1094.4 | 5.5% |

12 | 0.402 | 1197.2 | 1130.1 | − 5.6% |

13 | 0.419 | 1176.76 | 1190.7 | 1.2% |

14 | 0.423 | 1163.02 | 1204.9 | 3.6% |

15 | 0.448 | 1313.78 | 1297.7 | − 1.2% |

16 | 0.480 | 1448.97 | 1419.2 | − 2.1% |

17 | 0.484 | 1395.57 | 1434.5 | 2.8% |

18 | 0.499 | 1580.81 | 1491.5 | − 5.6% |

19 | 0.534 | 1752.62 | 1624.7 | − 7.3% |

20 | 0.556 | 1744.38 | 1709.5 | − 2.0% |

21 | 0.602 | 1909.18 | 1896.4 | − 0.7% |

22 | 0.619 | 1992.19 | 1965.7 | − 1.3% |

23 | 0.649 | 2171.33 | 2088 | − 3.8% |

24 | 0.735 | 2647.09 | 2443.3 | − 7.7% |

25 | 0.809 | 2629.39 | 2767.3 | 5.2% |

26 | 0.910 | 3074.65 | 3210.9 | 4.4% |

27 | 0.911 | 3275.15 | 3215.3 | − 1.8% |

28 | 0.925 | 3320.92 | 3276.7 | − 1.3% |

29 | 0.926 | 3533.94 | 3281.1 | − 7.2% |

30 | 0.930 | 3009.95 | 3298.7 | 9.6% |

## 6 Conclusions

- (1)
In this paper, the contact stiffness coefficient was optimized based on the fractal theory, and a modified contact force model was proposed.

- (2)
The change law of the contact force of the modified contact force model under different parameters was provided.

- (3)
The modified model can describe the contact process with different restitution coefficients, which is suitable for both large and small clearances, and has a wider range of application. Furthermore, the modified model is more accurate when calculating the impact force of a contact with a small clearance.

- (4)
The agreement between the simulation and experiment results shows that the modified contact force model proposed in this paper can be used to calculate the contact force better when considering the surface roughness, and can therefore provide a reference for the establishment of contact models.

## Notes

### Authors’ Contributions

S-HL was in charge of the whole trial; X-YH wrote the manuscript; J-QW designed and debugged the experiment; JS and F-JL assisted with sampling and laboratory analyses, as well as implementation of the experiment and date processing. All authors read and approved the final manuscript.

### Authors’ Information

Shi-Hua Li, born in 1966, is currently a professor, PhD Tutor at *Parallel Robot and Mechatronic System Laboratory of Hebei Province, China.* He received his PhD degree in *Yanshan University, China,* in 2004. His research interests include nonlinear dynamics and precision of spatial mechanism, the ory and application of parallel robot, micro-operation and micro-electromechanical systems.

Xue-Yan Han, born in 1978, is currently a PhD candidate at *Parallel Robot and Mechatronic System Laboratory of Hebei Province, China.* He received his master degree in *Yanshan University, China,* in 2007.

Jun-Qi Wan, born in 1993, is currently a master candidate at *Parallel Robot and Mechatronic System Laboratory of Hebei Province, China.*

Jing Sun, born in 1989, is currently a PhD candidate at *Parallel Robot and Mechatronic System Laboratory of Hebei Province, China.* He received his bachelor degree in *Yanshan University, China,* in 2014.

Fu-Juan Li, born in 1985, is currently a experimentalist at *Parallel Robot and Mechatronic System Laboratory of Hebei Province, China.* She received her master degree in *Yanshan University, China,* in 2012.

### Acknowledgements

The authors would like to express gratitude to Professor Shi-Hua Li of Yanshan University for his critical discussion and reading during manuscript preparation.

### Competing Interests

The authors declare that they have no competing interests.

### Funding

Supported by National Natural Science Foundation of China (Grant No. 51775475), Hebei Provincial Natural Science Foundation of China (Grant No. E2016203463).

### Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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