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Research Papers

Application of Quasi-Monte Carlo Method Based on Good Point Set in Tolerance Analysis

[+] Author and Article Information
Yanlong Cao

State Key Laboratory of Fluid Power
and Mechatronic Systems,
College of Mechanical Engineering,
Zhejiang University,
Zheda Road 38,
Hangzhou 310027, China;
Key Laboratory of Advanced Manufacturing
Technology of Zhejiang Province,
College of Mechanical Engineering,
Zhejiang University,
Zheda Road 38,
Hangzhou 310027, China
e-mail: sdcaoyl@zju.edu.cn

Huiwen Yan

Key Laboratory of Advanced Manufacturing
Technology of Zhejiang Province,
College of Mechanical Engineering,
Zhejiang University,
Zheda Road 38,
Hangzhou 310027, China
e-mail: 969848190@qq.com

Ting Liu

Key Laboratory of Advanced Manufacturing
Technology of Zhejiang Province,
College of Mechanical Engineering,
Zhejiang University,
Zheda Road 38,
Hangzhou 310027, China
e-mail: 1197857467@qq.com

Jiangxin Yang

Key Laboratory of Advanced Manufacturing
Technology of Zhejiang Province,
College of Mechanical Engineering,
Zhejiang University,
Zheda Road 38,
Hangzhou 310027, China
e-mail: yangjx@zju.edu.cn

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTING AND INFORMATION SCIENCE IN ENGINEERING. Manuscript received April 23, 2015; final manuscript received February 16, 2016; published online May 10, 2016. Editor: Bahram Ravani.

J. Comput. Inf. Sci. Eng 16(2), 021008 (May 10, 2016) (7 pages) Paper No: JCISE-15-1147; doi: 10.1115/1.4032909 History: Received April 23, 2015; Revised February 16, 2016

Tolerance analysis is increasingly becoming an important tool for mechanical design, process planning, manufacturing, and inspection. It provides a quantitative analysis tool for evaluating the effects of manufacturing variations on performance and overall cost of the final assembly. It boosts concurrent engineering by bringing engineering design requirements and manufacturing capabilities together in a common model. It can be either worst-case or statistical. It may involve linear or nonlinear behavior. Monte Carlo simulation is the simplest and the most popular method for nonlinear statistical tolerance analysis. Monte Carlo simulation offers a powerful analytical method for predicting the effects of manufacturing variations on design performance and production cost. However, the main drawbacks of this method are that it is necessary to generate very large samples to assure calculation accuracy, and that the results of analysis contain errors of probability. In this paper, a quasi-Monte Carlo method based on good point (GP) set is proposed. The difference between the method proposed and Monte Carlo simulation lies in that the quasi-random numbers generated by Monte Carlo simulation method are replaced by ones generated by the method proposed in this paper. Compared with Monte Carlo simulation method, the proposed method provides analysis results with less calculation amount and higher precision.

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References

Polini, W. , 2011, “ Geometric Tolerance Analysis,” Geometric Tolerances, Springer, Berlin, pp. 39–68.
Chase, K. W. , and Parkinson, A. R. , 1991, “ A Survey of Research in the Application of Tolerance Analysis to the Design of Mechanical Assemblies,” Res. Eng. Des., 3(1), pp. 23–37. [CrossRef]
Roy, U. , Liu, C. , and Woo, T. , 1991, “ Review of Dimensioning and Tolerancing: Representation and Processing,” Comput. Aided Des., 23(7), pp. 466–483. [CrossRef]
Zhang, H. , and Huq, M. , 1992, “ Tolerancing Techniques: The State-of-the-Art,” Int. J. Prod. Res., 30(9), pp. 2111–2135. [CrossRef]
Nigam, S. D. , and Turner, J. U. , 1995, “ Review of Statistical Approaches to Tolerance Analysis,” Comput. Aided Des., 27(1), pp. 6–15. [CrossRef]
Hong, Y. , and Chang, T. , 2002, “ A Comprehensive Review of Tolerancing Research,” Int. J. Prod. Res., 40(11), pp. 2425–2459. [CrossRef]
Marziale, M. , and Polini, W. , 2009, “ A Review of Two Models for Tolerance Analysis of an Assembly: Vector Loop and Matrix,” Int. J. Adv. Manuf. Technol., 43(11–12), pp. 1106–1123. [CrossRef]
Marziale, M. , and Polini, W. , 2011, “ A Review of Two Models for Tolerance Analysis of an Assembly: Jacobian and Torsor,” Int. J. Comput. Integr. Manuf., 24(1), pp. 74–86. [CrossRef]
Polini, W. , 2012, “ Taxonomy of Models for Tolerance Analysis in Assembling,” Int. J. Prod. Res., 50(7), pp. 2014–2029. [CrossRef]
Chen, H. , Jin, S. , Li, Z. , and Lai, X. , 2014, “ A Comprehensive Study of Three Dimensional Tolerance Analysis Methods,” Comput. Aided Des., 53(2), pp. 1–13. [CrossRef]
Gupta, S. , and Turner, J. U. , 1993, “ Variational Solid Modeling for Tolerance Analysis,” IEEE Comput. Graphics Appl., 13(3), pp. 64–74. [CrossRef]
Li, B. , and Roy, U. , 2001, “ Relative Positioning of Toleranced Polyhedral Parts in an Assembly,” IIE Trans., 33(4), pp. 323–336.
Chase, K. W. , Gao, J. , and Magleby, S. P. , 1995, “ General 2-D Tolerance Analysis of Mechanical Assemblies With Small Kinematic Adjustments,” J. Des. Manuf., 5, pp. 263–274. [CrossRef]
Whitney, D. E. , Gilbert, O. L. , and Jastrzebski, M. , 1994, “ Representation of Geometric Variations Using Matrix Transforms for Statistical Tolerance Analysis in Assemblies,” Res. Eng. Des. Theory Appl. Concurrent Eng., 6(4), pp. 191–210.
Davidson, J. , Mujezinovic, A. , and Shah, J. , 2002, “ A New Mathematical Model for Geometric Tolerances as Applied to Round Faces,” ASME J. Mech. Des., 124(4), pp. 609–622. [CrossRef]
Jaishankar, L. N. , Davidson, J. K. , and Shah, J. J. , eds., 2013, “ Tolerance Analysis of Parallel Assemblies Using Tolerance-Maps® and a Functional Map Derived From Induced Deformations,” ASME Paper No. DETC2013-12394.
Jiang, K. , Davidson, J. K. , Liu, J. , and Shah, J. J. , 2014, “ Using Tolerance Maps to Validate Machining Tolerances for Transfer of Cylindrical Datum in Manufacturing Process,” Int. J. Adv. Manuf. Technol., 73(1–4), pp. 465–478. [CrossRef]
Desrochers, A. , and Clément, A. , 1994, “ A Dimensioning and Tolerancing Assistance Model for CAD/CAM Systems,” Int. J. Adv. Manuf. Technol., 9(6), pp. 352–361. [CrossRef]
Clément, A. , Desrochers, A. , and Riviere, A. , 1991, “ Theory and Practice of 3-D Tolerancing for Assembly,” CIRP International Working Seminar on Computer-Aided Tolerancing, Penn State University, State College, PA, May 16–17, pp. 25–55.
Clément, A. , and Riviere, A. , eds., 1993, “ Tolerancing Versus Nominal Modeling in Next Generation CAD/CAM System,” 3rd CIRP Seminar on Computer Aided Tolerancing-Tolérancement Géométrique, Ecole Normale Supérieure de Cachan, Paris.
Desrochers, A. , Ghie, W. , and Laperrière, L. , 2003, “ Application of a Unified Jacobian—Torsor Model for Tolerance Analysis,” ASME J. Comput. Inf. Sci. Eng., 3(1), pp. 2–14. [CrossRef]
Ghie, W. , Laperriere, L. , and Desrochers, A. , 2007, “ Re-Design of Mechanical Assemblies Using the Unified Jacobian–Torsor Model for Tolerance Analysis,” Models for Computer Aided Tolerancing in Design and Manufacturing, Springer, Berlin, pp. 95–104.
Ghie, W. , Laperrière, L. , and Desrochers, A. , 2010, “ Statistical Tolerance Analysis Using the Unified Jacobian–Torsor Model,” Int. J. Prod. Res., 48(15), pp. 4609–4630. [CrossRef]
Teissandier, D. , Couétard, Y. , and Gérard, A. , 1999, “ A Computer Aided Tolerancing Model: Proportioned Assembly Clearance Volume,” Comput. Aided Des., 31(13), pp 805–817. [CrossRef]
Giordano, M. , Samper, S. , and Petit, J.-P. , 2007, “ Tolerance Analysis and Synthesis by Means of Deviation Domains, Axi-Symmetric Cases,” Models for Computer Aided Tolerancing in Design and Manufacturing,” Springer, Berlin, pp. 85–94.
Dantan, J.-Y. , and Ballu, A. , 2002, “ Assembly Specification by Gauge With Internal Mobilities (GIM)—A Specification Semantics Deduced From Tolerance Synthesis,” J. Manuf. Syst., 21(3), pp. 218–235. [CrossRef]
Dantan, J.-Y. , Mathieu, L. , Ballu, A. , and Martin, P. , 2005, “ Tolerance Synthesis: Quantifier Notion and Virtual Boundary,” Comput. Aided Des., 37(2), pp. 231–240. [CrossRef]
Spotts, M. F. , 1959, An Application of Statistics to the Dimensioning of Machine Parts.
Bender, A. , 1968, “ Statistical Tolerancing as it Relates to Quality Control and the Designer (6 Times 2.5 = 9),” SAE Technical Paper No. 680490.
Chase, K. W. , and Greenwood, W. H. , 1988, “ Design Issues in Mechanical Tolerance Analysis,” Manuf. Rev., 1(1), pp. 50–59.
Evans, D. H. , 1975, “ Statistical Tolerancing: The State of the Art—II: Methods for Estimating Moments,” J. Qual. Technol., 7, pp. 1–12.
Evans, D. H. , 1971, “ An Application of Numerical Integration Techniques to Statistical Tolerancing—II: A Note on the Error,” Technometrics, 13(2), pp. 315–324.
Evans, D. H. , 1972, “ An Application of Numerical Integration Techniques to Statistical Tolerancing—III: General Distributions,” Technometrics, 14(1), pp. 23–35.
Parkinson, D. , 1982, “ The Application of Reliability Methods to Tolerancing,” ASME J. Mech. Des., 104(3), pp. 612–618. [CrossRef]
Lee, W.-J. , and Woo, T. , 1990, “ Tolerances: Their Analysis and Synthesis,” J. Eng. Ind., 112(2), pp. 113–121. [CrossRef]
Taguchi, G. , 1978, “ Performance Analysis Design,” Int. J. Prod. Res., 16(6), pp. 521–530. [CrossRef]
D'Errico, J. R. , and Zaino, N. A. , 1988, “ Statistical Tolerancing Using a Modification of Taguchi's Method,” Technometrics, 30(4), pp. 397–405. [CrossRef]
DeDoncker, D. , and Spencer, A. , 1987, “ Assembly Tolerance Analysis With Simulation and Optimization Techniques,” SAE Technical Paper No. 870263.
Lin, C.-Y. , Huang, W.-H. , Jeng, M.-C. , and Doong, J.-L. , 1997, “ Study of an Assembly Tolerance Allocation Model Based on Monte Carlo Simulation,” J. Mater. Process. Technol., 70(1), pp. 9–16. [CrossRef]
Bruyère, J. , Dantan, J.-Y. , Bigot, R. , and Martin, P. , 2007, “ Statistical Tolerance Analysis of Bevel Gear by Tooth Contact Analysis and Monte Carlo Simulation,” Mech. Mach. Theory, 42(10), pp. 1326–1351. [CrossRef]
Dantan, J.-Y. , and Qureshi, A.-J. , 2009, “ Worst-Case and Statistical Tolerance Analysis Based on Quantified Constraint Satisfaction Problems and Monte Carlo Simulation,” Comput. Aided Des., 41(1), pp. 1–12. [CrossRef]
Wu, F. , Dantan, J.-Y. , Etienne, A. , Siadat, A. , and Martin, P. , 2009, “ Improved Algorithm for Tolerance Allocation Based on Monte Carlo Simulation and Discrete Optimization,” Comput. Ind. Eng., 56(4), pp. 1402–1413. [CrossRef]
Qureshi, A. J. , Dantan, J.-Y. , Sabri, V. , Beaucaire, P. , and Gayton, N. , 2012, “ A Statistical Tolerance Analysis Approach for Over-Constrained Mechanism Based on Optimization and Monte Carlo Simulation,” Comput. Aided Des., 44(2), pp. 132–142. [CrossRef]
Chase, K. W. , Gao, J. , Magleby, S. P. , and Sorenson, C. , 1996, “ Including Geometric Feature Variations in Tolerance Analysis of Mechanical Assemblies,” IIE Trans., 28(10), pp. 795–808.
Gao, J. , Chase, K. W. , and Magleby, S. P. , 1998, “ Generalized 3-D Tolerance Analysis of Mechanical Assemblies With Small Kinematic Adjustments,” IIE Trans., 30(4), pp. 367–377.
Chase, K. W. , 1999, “ Tolerance Analysis of 2-D and 3-D Assemblies With Small Kinematic Adjustments,” ADCATS Report No. 94-4.
Grossman, D. D. , 1976, “ Monte Carlo Simulation of Tolerancing in Discrete Parts Manufacturing and Assembly,” Report No. STAN-CS-76-555.
Hua, L. K. , and Wang, Y. , 1981, Application of Number Theory to Numerical Analysis, Springer, Berlin.
Chung, K.-L. , 1949, “ An Estimate Concerning the Kolmogroff Limit Distribution,” Trans. Am. Math. Soc., 67, pp. 36–50.
Halton, J. H. , 1960, “ On the Efficiency of Certain Quasi-Random Sequences of Points in Evaluating Multi-Dimensional Integrals,” Numerische Math., 2(1), pp. 84–90. [CrossRef]
Armillotta, A. , 2014, “ A Static Analogy for 2D Tolerance Analysis,” Assem. Autom., 34(2), pp. 182–91. [CrossRef]

Figures

Grahic Jump Location
Fig. 1

Three sets of two-dimensional points for size 1597: (a) random numbers, (b) GP set, and (c) GLP set

Grahic Jump Location
Fig. 3

The relationships between the means and sample size obtained by different methods: (a) Y1, (b) Y2, (c) Y3, (d) Y4, (e) Y5, and (f) Y6

Grahic Jump Location
Fig. 2

Flow chart of GP method and GLP method

Grahic Jump Location
Fig. 6

The relationships between mean and standard deviation of function requirement g and sample size n : (a) mean of g versus sample size n and (b) standard deviation of g versus sample size n

Grahic Jump Location
Fig. 4

The relationships between the standard deviations and sample size obtained by different methods: (a) Y1, (b) Y2, (c) Y3, (d) Y4, (e) Y5, and (f) Y6

Grahic Jump Location
Fig. 5

An example assembly applied to the case study

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