Skip to main content
SpringerLink
Log in

Application study of fractal theory in mechanical transmission

  • Review
  • Published:
Chinese Journal of Mechanical Engineering Submit manuscript

Abstract

Mechanical transmissions are applied widely in various electrical and mechanical products, but some qualities of some high-end products can’t meet people’s demand, and need to be improved with some new methods or theories. The fractal theory is a new mathematic tool, which provides a new approach for the further study in the area of the mechanical transmission, and helps to solve some problems. The basic contents of the fractal theory are introduced firstly, especially the two important concepts, the self-similar fractal and the fractal dimension. Then, the deferent application of the fractal theory in this area are given to display how to further the study and improve some important characteristics of the mechanical transmission, such as contact surfaces, manufacturing precise, friction and wear, stiffness, strength, dynamics, fault diagnosis, etc. Finally, the problems of the fractal theory and its application are discussed, and some weaknesses, such as the calculation capacity of the fractal theory is not strong, are pointed out. Some new solutions are suggested, such as combining the fractal theory with the fuzzy theory, the chaos theory and so on. The new application fields of the fractal theory in the area of the mechanical transmission are proposed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. KENNECH J F. Fractal geometry-mathematical foundations and applications[M]. London: John Wiley & Sons, 1990.

    Google Scholar 

  2. SRINIVASAN R S, WOOD K L. Geometric tolerancing in mechanical design using fractal-based parameters[J]. Journal of Mechanical Design, 1995, 117(1): 203–206.

    Article  Google Scholar 

  3. JIANG Zhuangde, WANG Hairong, FEI Bin. Research into the application of fractal geometry in characterising machined surfaces[J]. International Journal of Machine Tools & Manufacture, 2001, 41(41): 2179–2185.

    Article  Google Scholar 

  4. SONG Senyi, WANG Lihua, XUE Erpen. Fractal theory and the application in mechanical junction surface[J]. Machinery, 2008, 25(1): 73–78. (in Chinese)

    Google Scholar 

  5. LIU Weiping. Study of the detection method of contact area of gear surface based on machine vision technique[J]. Journal of Mechanical Transmission, 2013, 75(5): 18–26. (in Chinese)

    Google Scholar 

  6. MIAO Xiaomei, HUANG Xiaodiao. A complete contact model of a fractal rough surface[J]. Wear, 2014, 309(1–2): 146–151.

    Article  Google Scholar 

  7. LIU Liyan, CHEN Fuqiang, YANG Wei, et al. The model association and verification of the fractal dimension and the milling parameters[J]. Mechanical Design and Manufacturing, 2015(5): 122–126. (in Chinese)

    Google Scholar 

  8. ABUZEID O M, DADO M H F. Fractal model to predict the crack roughness effect on the local bending compliance of circular shafts[J]. International Journal of Mechanical Sciences, 2004, 46(5): 695–702.

    Article  MATH  Google Scholar 

  9. PENG Yunfeng, GUO Yinbiao. An elastic-plastic adhesion model for contacting fractal rough surface and perfectly wetted plane with meniscus[J]. Chinese Journal of Mechanical Engineering, 2009, 22(1): 9–14.

    Article  MathSciNet  Google Scholar 

  10. CIAVARELLA M. Adhesive rough contacts near complete contact[J]. International Journal of Mechanical Sciences, 2015, 104: 104–111.

    Article  Google Scholar 

  11. QIAN Canrong. Study on the bonding characteristics of heavy duty wet clutch based on fractal theory[J]. Mechanical Transmission, 2016, 40(4): 43–47. (in Chinese)

    Google Scholar 

  12. LING F F. The possible role of fractal geometry in tribology[J]. Tribology Transactions, 1989, 32(4): 497–505.

    Article  Google Scholar 

  13. WANG S, KOMVOPOULOS K. A fractal theory of the interfacial temperature distribution in the slow sliding regime Part I—elastic contact and heat transfer analysis[J]. J. Tribol, 1994, 116(4): 812–822.

    Article  Google Scholar 

  14. YOU J M, CHEN T N. A static friction model for the contact of fractal surfaces[J]. Proceedings of the Institution of Mechanical Engineers Part J, 2010, 224(5): 513–518.

    Article  Google Scholar 

  15. JI Cuicui, ZHU Hua, JIANG Wei, et al. Running-in test and fractal methodology for worn surface topography characterization[J]. Chinese Journal of Mechanical Engineering, 2010, 23(5): 600–605.

    Article  Google Scholar 

  16. LIOU Jengluen, SUN Yihsing, LIN Jenfin, et al. Fractal theory applied to evaluate the tribological performances of two greases demonstrated in four-ball tests[J]. J. Tribol,2012, 134(3): 308–314

  17. YU Qiuping, SUN Jianjun, YU Bo, et al. A fractal model of mechanical seal surfaces based on accelerating experiment[OL]. Tribology Transactions, 2016. Taylor & Francis Group, http: //dx.doi.org/10.1080/10402004.2016.1165900.

    Google Scholar 

  18. JIANG Shuwen, JIANG Bin, LI Yan, et al. Calculation of fractal dimension of worn surface[J]. Tribology, 2003, 23(6): 533–536

    Google Scholar 

  19. LIU Zuomin. Coupled effects of fractal roughness and self-lubricating composite porosity on lubrication and wear[J]. Tribology Transactions, 2013, 56(4): 581–591

    Article  Google Scholar 

  20. BIGERELLE M, NIANGA J M, NAJJAR D, et al. Roughness signature of tribological contact calculated by a new method of peaks curvature radius estimation on fractal surfaces[J]. Tribology International, 2013, 65(3): 235–247

    Article  Google Scholar 

  21. CHEN Qi, ZHAO Han. Study on applications of fractal theory in analysis of gear contact strength[J]. China Mechanical Engineering, 2010, 21(9): 1014–1017. (in Chinese)

    Google Scholar 

  22. WEI Zhang, HE Jianing, MA Wuxing, et al, The contact strength analysis of double-circular-arc modified cycloid gears based on fractal theory[C]//IEEE International Conference on Oxide Material. for Electronic Engineering, 2012: 215–218.

  23. LI Xiaopeng, LIANG Yamin, ZHAO Guanghui, et al. Dynamic characteristics of joint surface considering friction and vibration factors based on fractal theory[J]. Journal of Vibroengineering, 2013, 15(2): 872–883.

    Google Scholar 

  24. CHEN Qi, MA Yun, HUANG Shouwu, et al. Research on gears’ dynamic performance influenced by gear backlash based on fractal theory[J]. Applied Surface Science, 2014, 313: 325–332.

    Article  Google Scholar 

  25. LI Zhixiong, PENG Z. Nonlinear dynamic response of a multidegree of freedom gear system dynamic model coupled with tooth surface characters: a case study on coal cutters[J]. Nonlinear Dynamics, 2015, 84(1): 1–16.

    MathSciNet  Google Scholar 

  26. FREDERICO G S F, LAZO M J. Fractional Noether’s theorem with classical and Caputo derivatives: constants of motion for non-conservative systems[J]. Nonlinear Dynamics, 2016, 85(2): 839–851.

    Article  MathSciNet  Google Scholar 

  27. WANG Qiuyan, LIANG Zhiqiang, WANG Xibin, et al. Research on fractal behavior of surface topography in precision grinding of monocrystal sapphire[J]. Journal of Mechanical Engineering, 2015, 51(19): 174–181. (in Chinese)

    Article  Google Scholar 

  28. LUO Huarong, GAO Chenghui. FEA of thermo-mechanical couple of sliding friction between dual fractal surfaces[J]. Machine Building & Automation, 2014(3): 141–145. (in Chinese)

    Google Scholar 

  29. GE Shirong, CHEN Gouan. Fractal prediction models of sliding wear during the running-in process[J]. Wear, 1999, 231(2): 249–255.

    Article  MathSciNet  Google Scholar 

  30. XU Yuxiu, ZHONG Jianjun, WEN Bangchun. Fractal fault diagnosis and classification to model characteritic of rotor system[J]. Chinese Journal of Mechanical Engineering, 2005, 41(12): 186–189.

    Article  Google Scholar 

  31. ZHU Yunbo, FENG Guangbin, SUN Huagang, et al. Study on gearbox fault diagnosis based on multi-fractal and SVM[J]. Journal of Mechanical Transmission, 2012, 36(6): 99–102. (in Chinese)

    Google Scholar 

  32. LIU Zhichuan, TANG Liwei, CAO Lijun. Fault diagnosis of gearbox based on kalman filter and mathematical morphology fractal analysis[J]. Journal of Mechanical Transmission, 2014, 38(7): 141–144. (in Chinese)

    Google Scholar 

  33. CHEN Jianghai, SUN Qishan. Fault diagnosis of rolling bearings based on fractal and wavelet packets theory[J]. Bearing, 2010(2): 48–52. (in chinese)

    Google Scholar 

  34. ZHANG Yang, YANG Yongfa, ZHU Lizong, et al. Fault diagnosis of rolling bearings based on fractal theory[J]. Bearing Technology, 2008(1): 1–2. (in Chinese)

    Google Scholar 

  35. HAO Yan, WANG Taiyong, WAN Jan, et al. Mechanical fault diagnosis based on cascaded bistable stochastic resonance and multi-fractal[J]. Journal of Vibration and Shock, 2012, 31(8): 181–185. (in Chinese)

    Google Scholar 

  36. LI Bing, ZHANG Peilin, REN Guoquan, et al. Application of mathematical-morphology-based generalized fractal dimensions in engine fault diagnosis[J]. Journal of Vibration and Shock, 2011, 30(10): 208–208. (in Chinese)

    Google Scholar 

  37. LI Tiexi, LI Xizhi. Similarity and similar principle[M]. Harbin: Press of Harbin Institute of Technology, 2014. (in Chinese)

    Google Scholar 

  38. ZHOU Meili. Similarity Science[M]. Beijing: Science Press, 2004, (in Chinese)

    Google Scholar 

  39. FAN Fumei, LIANG Pin, WU Genshen. Experiment study of vibration fault diagnosis of turbine rotors based on the fractal box dimension[J]. Nuclear Power Engineering, 2006, 27(1): 85–89. (in Chinese)

    Google Scholar 

  40. DEEPAK L K, RAMAMOORTHY B. Surface topography characterization of automotive cylinder liner surfaces using fractal methods[J]. Applied Surface Science, 2013, 280(9): 332–342.

    Google Scholar 

  41. WEN Shuhua. Modeling and simulation of contact feature theory of joint surfaces[M]. Beijing: National Defense Industry Publishing House, 2012. (in Chinese)

    Google Scholar 

  42. DI Jizheng, Wavelet analysis principle[M]. Beijing: Science Press, 2010. (in Chinese)

  43. MIESOWICZ K, STASZEWSKI W J, KORBIEL T. Analysis of Barkhausen noise using wavelet-based fractal signal processing for fatigue crack detection[J]. International Journal of Fatigue, 2015, 83: 109–116.

    Article  Google Scholar 

  44. XIE Jijian, LIU Chengping. Fuzzy mathematic methods and application[M]. Wuhan: Press of Huazhong University of Science and Technology, 2000. (in Chinese)

    Google Scholar 

  45. YANG Lunbiao, GAO Yinyi. Principle and application of fuzzy mathematics[M]. Changsha: South China University of Technology Press, 2006. (in Chinese)

    Google Scholar 

  46. ZHANG Weiqiang, WANG Yujuan. The fuzzy logic relationships among the fractal dimension of piston pins and the traditional parameters[J]. Journal of Agricultural Machinery, 2000, 31(6): 88–90. (in Chinese)

    Google Scholar 

  47. PEITGEN H O, JURGENS H J, SAUPE D. Chaos and fractals new frontiers of science[M]. New York: Springer, 2004.

    Book  MATH  Google Scholar 

  48. BIGERELLE M, NIANGA J M, IOST A. Decomposition of a tribological system by chaos theory on rough surfaces[J]. Tribology International, 2015, 82(82): 561–576.

    Article  Google Scholar 

  49. ZHU Daqi, SHI Hui. The principle and application of artificial neural network[M]. Beijing: Science Press, 2006. (in Chinese)

    Google Scholar 

  50. CHU Qinqin, XIAO Han, NU Yong, et al. Fault diagnosis of gears based on multi-fractal theory and artificial neural networks[J]. Vibration and Shock, 2015, 34(21): 15–18. (in Chinese)

    Google Scholar 

  51. ZHAO Qidong, LUO Tianhong, LUO Wenjun, et al. The friction model of multi-disc wet brakes based on the fractal theory[J]. Journal of Huazhong University of Science and Technology, 2015, 43(4): 123–127. (in Chinese)

    MathSciNet  Google Scholar 

  52. FENG Yan, YU Xiaoli, LIU Zhentao. The fractal model of the normal load acted on the joint area based on the thermo elastic plastic theory[J]. Journal of Xi’an Medical University, 2015, 49(9): 18–23. (in Chinese)

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mechanical and Automobile Engineering, Hefei University of Technology, Hefei, 230009, China

    Han Zhao & Qilin Wu

Authors
  1. Han Zhao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Qilin Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Han Zhao.

Additional information

Supported by International Co-operation Program of China(Grant No. 2014DFA80440)

ZHAO Han, born in 1957, is currently a professor at School of Mechanical and Automobile Engineering, Hefei University of Technology, China. He received his PhD degree from Aalborg University, Danmark, in 1990. His research interests include mechanical transmissions, electrical vehicles, mechanical design etc.

WU Qilin, born in 1987, is currently a PhD candidate at School of Mechanical and Automobile Engineering, Hefei University of Technology, China. His research interests include mechanical transmissions, mechanical design etc.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhao, H., Wu, Q. Application study of fractal theory in mechanical transmission. Chin. J. Mech. Eng. 29, 871–879 (2016). https://doi.org/10.3901/CJME.2016.0818.094

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.3901/CJME.2016.0818.094

Keywords

Search

Navigation