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Tolerance analysis of assemblies with asymmetric tolerances by unified uncertainty–accumulation model based on fuzzy logic

Abstract

In mechanical assemblies, individual components are placed together to deliver a certain function. The performance, quality, and cost of the mechanical assembly are significantly affected by its tolerances. Toleranced dimensions inherently generate an uncertain environment in a mechanical assembly. This paper presents a proper method for tolerance analysis of mechanical assemblies with asymmetric tolerances based on an uncertainty model. This mathematical approach is based on fuzzy logic and tolerance accumulation models such as worst-case and root-sum-square methods. A fuzzy-based tolerance representation is developed to model uncertainty of tolerance components in the mechanical assemblies. According to this scheme, toleranced components are described as fuzzy numbers with their membership functions constructed using the statistical distributions of manufactured variables. In this way, the uncertainty of assembly requirements and accumulation of tolerances are represented in the form of fuzzy number. In this paper, a new factor, the fuzzy factor, is introduced that helps converting the membership functions into fuzzy intervals that can be used for modal interval analysis. Equations for estimation of percent contributions of individual tolerances are introduced in terms of uncertainty parameter. These equations yield percent contributions of upper and lower bounds of independent variables (manufactured dimensions) on the upper and lower bounds of dependent variables (assembly dimensions). The proposed method is applied to an example, and its results are discussed.

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References

  1. 1.

    Ngoi BKA, Ong CT (1998) Product and process dimensioning and tolerancing techniques. A state-of-the-art review. Int J Adv Manuf Technol 14:910–917

    Article  Google Scholar 

  2. 2.

    Scholz F (1995) Tolerance stack analysis methods. Research and Technology Boeing Information & Support Services, Boeing, Seattle, pp 1–44

    Google Scholar 

  3. 3.

    Fortini ET (1967) Dimensioning for interchangeable manufacture. Industrial Press, New York

    Google Scholar 

  4. 4.

    Evans DH (1975) Statistical tolerancing: the state of art. Part II: Methods for estimating moments. J Qual Technol 7(1):1–12

    Google Scholar 

  5. 5.

    Roy U, Liu CR, Woo TC (1991) Review of dimensioning and tolerancing: representation and processing. Comput Aided Des 23(7):466–483

    MATH  Article  Google Scholar 

  6. 6.

    Greenwood WH, Chase KW (1988) Worst case tolerance analysis with nonlinear problems. J Eng Industry ASME 110:232–235

    Article  Google Scholar 

  7. 7.

    Greenwood WH, Chase KW (1990) Root sum squares tolerance analysis with nonlinear problems. J Eng Industry ASME 112:382–384

    Article  Google Scholar 

  8. 8.

    Movahhedy MR, Khodaygan S (2007) Tolerance analysis of mechanical assemblies with asymmetric tolerances. SAE 2007 Transactions J Mater Manuf 116:44

    Google Scholar 

  9. 9.

    Wu W, Rao SS (2006) Fuzzy analysis of geometric tolerances using interval method. Proc IME C: J Mech Eng Sci 220(4):489–497

    Article  Google Scholar 

  10. 10.

    Celikyilmaz A, Türksen IB (2009) Modeling uncertainty with fuzzy logic with recent theory and applications. Springer, Berlin

    MATH  Book  Google Scholar 

  11. 11.

    Li HX, Yen VC (1995) Fuzzy sets and fuzzy decision-making. CRC Press, Boca Raton, FL

    MATH  Google Scholar 

  12. 12.

    Moore RE (1966) Interval analysis. Prentice-Hall, Englewood Cliffs

    MATH  Google Scholar 

  13. 13.

    Moore RE, Kearfott RB, Cloud MJ (2009) Introduction to interval analysis. Society for Industrial and Applied Mathematics, Philadelphia

    MATH  Book  Google Scholar 

  14. 14.

    Gardenes E, Sainz MA, Jorba L, Calm R, Estela R, Mielgo H, Trepat A (2001) Modal intervals. Reliab Comput 7(2):77–111

    MathSciNet  MATH  Article  Google Scholar 

  15. 15.

    Popova ED (2001) Multiplication distributivity of proper and improper intervals. Reliab Comput 7(2):129–140

    MathSciNet  MATH  Article  Google Scholar 

  16. 16.

    Wang Y (2008) Semantic tolerance modeling based on modal interval analysis. In: Proceedings of the international workshop on reliable engineering computing REC'08, Savannah, Georgia, pp 46–59, 20–22 February 2008

  17. 17.

    Chase KW, Gao J, Magleby SP (1997) Tolerance analysis of 2-D and 3-D mechanical assemblies with small kinematic adjustments. In: Zhang H-C (ed) Advanced tolerancing techniques. Wiley, New York, pp 103–137

    Google Scholar 

  18. 18.

    Creveling CM, Wesley A (1997) Tolerance design: a handbook for developing optimal specifications. Longman, White Plains, NY

    Google Scholar 

  19. 19.

    Civanlar MR, Trussell HJ (1986) Constructing membership functions using statistical data. Fuzzy Sets Syst 18(1):1–13

    MathSciNet  MATH  Article  Google Scholar 

  20. 20.

    Rice J (1995) Mathematical statistics and data analysis, 2nd edn. Duxbury Press, North Scituate, MA

    MATH  Google Scholar 

  21. 21.

    Mansoor EM (1963) The application of probability to tolerances used in engineering designs. Proc Inst Mech Eng 178(1):29–51

    Article  Google Scholar 

  22. 22.

    Bjørke Ø (1978) Computer aided tolerancing. Tapir Publishers, Norway

    Google Scholar 

  23. 23.

    Henzold G (2006) Geometrical dimensioning and tolerancing for design, manufacturing and inspection: a handbook for geometrical product specification using ISO and ASME standards, 2nd edn. Elsevier, Amsterdam

    Google Scholar 

  24. 24.

    Sleeper AD (2006) Design for six Sigma Statistics-59 tools for diagnosing and solving problems in DFSS initiatives. McGraw-Hill, USA

    Google Scholar 

  25. 25.

    Hoyle MH (1973) Transformations: an introduction and a bibliography. Int Stat Rev 41(2):203–223

    MathSciNet  Article  Google Scholar 

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Correspondence to M. R. Movahhedy.

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Khodaygan, S., Movahhedy, M.R. Tolerance analysis of assemblies with asymmetric tolerances by unified uncertainty–accumulation model based on fuzzy logic. Int J Adv Manuf Technol 53, 777–788 (2011). https://doi.org/10.1007/s00170-010-2855-4

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Keywords

  • Tolerance analysis
  • Tolerance accumulation
  • Fuzzy logic
  • Modal interval analysis
  • Asymmetric tolerances