Abstract
As one of key enabling technologies in digital twin-based assembly precision analysis, three dimensional (3D) assembly tolerance analysis technology has increasingly become an important means for predicting the assembly accuracy and verifying the assembly quality of mechanical assemblies. However, current methods exist some deficiencies that (i) the traditional model mostly cannot cover geometric tolerances or form errors in 3D assembly tolerance analysis, and (ii) a loss of assembly accuracy can be caused by ignoring these parallel connections in the assembly deviation propagation. To address these issues, this study proposes a novel assembly tolerance analysis method considering form errors and partial parallel connections with assembly accuracy and reliability guarantees in mechanical assemblies. First of all, through the integration of the unified Jacobian-Torsor model and skin model shapes, the resulting integrated Jacobian-skin model shapes model is presented, which contains the two advantages of easy-to-use tolerance propagation and geometric tolerance representation. Secondly, a novel improved approach combined with progressive contact method and algebraic operation is introduced into the assembly deviation propagation, which can realize the calculation of assembly relative positioning errors in both serial and partial parallel connections. Meanwhile, an overall calculation scheme of the proposed method is elaborated for assembly tolerance analysis with a statistical way, which is used to obtain the final deviation results of the assembly functional requirements (AFRs). At last, a typical mechanical assembly involving three parts is used as a case study to illustrate the effectiveness and feasibility of this solution.
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References
Qi QL, Tao F, Hu TL, Anwer N, Liu A, Wei YL, Wang LH, Nee AYC (2021) Enabling technologies and tools for digital twin. J Manuf Syst 58:3–21
Sun XM, Bao JS, Li J, Zhang YM, Liu SM, Zhou B (2020) A digital twin-driven approach for the assembly-commissioning of high precision products. Rob Comput Integr Manuf 61:101839
Guo FY, Liu JH, Zou F, Zhai YN, Wang ZQ, Li SZ (2019) Research on the state-of-art, connotation and key implementation technology of assembly process planning with digital twin. Chin J Mech Eng 55(17):110–132
Yi Y, Yan YH, Liu XJ, Ni ZH, Feng JD, Liu JS (2021) Digital twin-based smart assembly process design and application framework for complex products and its case study. J Manuf Syst 58:94–107
Liu JH, Sun QC, Cheng H, Liu XK, Ding XY, Liu SL, ** trends of the products assembly technology. Chin J Mech Eng 54(11):2–28
Li H, Zhu HP, Li PG, He F (2014) Tolerance analysis of mechanical assemblies based on small displacement torsor and deviation propagation theories. Int J Adv Manuf Technol 72(1–4):89–99
Guo CY, Liu JH, Jiang K (2016) Efficient statistical analysis of geometric tolerances using unified error distribution and an analytical variation model. Int J Adv Manuf Technol 84(1):347–360
Hong YS, Chang TC (2002) A comprehensive review of tolerancing research. Int J Prod Res 40(11):2425–2459
Chen H, ** S, Li ZM, Lai XM (2014) A comprehensive study of three dimensional tolerance analysis methods. Comput-Aided Des 53:1–13
Requicha AAG (1983) Toward a theory of geometric tolerancing. Int J Rob Res 2(4):45–60
Desrochers A, Rivière A (1997) A matrix approach to the representation of tolerance zones and clearances. Int J Adv Manuf Technol 13(9):630–636
Liu ZY, Zhou SE, Qiu C, Tan JR (2019) Assembly variation analysis of complicated products based on rigid–flexible hybrid vector loop. Proc Inst Mech Eng Part B J Eng Manuf 233(10):2099–2114
Laperrière L, Elmaraghy HA (2000) Tolerance analysis and synthesis using Jacobian transforms. CIRP Ann - Manuf Technol 49(1):359–362
Bourdet P, Mathieu L, Lartigue C, Ballu A (1996) The concept of the small displacement torsor in metrology. Ser Adv Math Appl Sci 40:110–122
Desrochers A, Ghie W, Laperrière L (2003) Application of a unified Jacobian—Torsor model for tolerance analysis. J Compu Inf Sci Eng 3(1):2–14
Zou ZH, Morse EP (2004) A gap-based approach to capture fitting conditions for mechanical assembly. Comput-Aided Des 36(8):691–700
Davidson JK, Mujezinović A, Shah JJ (2002) A new mathematical model for geometric tolerances as applied to round faces. J Mech Des Trans ASME 124(4):609–622
Arroyave-Tobon S, Teissandier D, Delos V (2016) Tolerance analysis with polytopes in HV-description. J Comput Inf Sci Eng 17(4):041011
Corrado A, Polini W (2017) A comprehensive study of tolerance analysis methods for rigid parts with manufacturing signature and operating conditions. J Adv Mech Des Syst 11(2):17–00004
Anwer N, Ballu A, Mathieu L (2013) The skin model, a comprehensive geometric model for engineering design. CIRP Ann - Manuf Technol 62(1):143–146
Anwer N, Schleich B, Mathieu L, Wartzack S (2014) From solid modelling to skin model shapes: shifting paradigms in computer-aided tolerancing. CIRP Ann - Manuf Technol 63(1):137–140
Schleich B, Anwer N, Mathieu L, Wartzack S (2014) Skin model shapes: a new paradigm shift for geometric variations modelling in mechanical engineering. Comput-Aided Des 50:1–15
Corrado A, Polini W (2017) Manufacturing signature in Jacobian and Torsor models for tolerance analysis of rigid parts. Rob Comput Integr Manuf 46:15–24
Corrado A, Polini W (2017) Manufacturing signature in variational and vector-loop models for tolerance analysis of rigid parts. Int J Adv Manuf Technol 88(5–8):2153–2161
Corrado A, Polini W, Moroni G, Petro S (2018) A variational model for 3D tolerance analysis with manufacturing signature and operating conditions. Assem Autom 38(1):10–19
Zhang M, Anwer N, Stockinger A, Mathieu L, Wartzack S (2013) Discrete shape modeling for skin model representation. Proc Inst Mech Eng Part B J Eng Manuf 227(5):672–680
Schleich B, Wartzack S (2015) Evaluation of geometric tolerances and generation of variational part representatives for tolerance analysis. Int J Adv Manuf Technol 79(5–8):959–983
Schleich B, Wartzack S (2018) Novel approaches for the assembly simulation of rigid skin model shapes in tolerance analysis. Comput-Aided Des 101:1–11
Schleich B, Anwer N, Mathieu L, Wartzack S (2015) Contact and mobility simulation for mechanical assemblies based on skin model shapes. J Comput Inf Sci Eng 15(2):979–985
Schleich B, Anwer N, Mathieu L, Wartzack S (2017) Sha** the digital twin for design and production engineering. CIRP Ann - Manuf Technol 66(1):141–144
Yi Y, Liu XJ, Liu TY, Ni ZH (2021) A generic integrated approach of assembly tolerance analysis based on skin model shapes. Proc Inst Mech Eng Part B J Eng Manuf 235(4):689–704
Liu JH, Zhang ZQ, Ding XY, Shao N (2018) Integrating form errors and local surface deformations into tolerance analysis based on skin model shapes and a boundary element method. Comput-Aided Des 104:45–59
Zhang ZQ, Liu JH, Pierre L, Anwer N (2021) Polytope-based tolerance analysis with consideration of form defects and surface deformations. Int J Comput Integ Manuf 34(1):57–75
Homri L, Goka E, Levasseur G, Dantan JY (2017) Tolerance analysis — form defects modeling and simulation by modal decomposition and optimization. Comput-Aided Des 91:46–59
Liu T, Cao YL, Zhao QJ, Yang JX, Cui LJ (2019) Assembly tolerance analysis based on the Jacobian model and skin model shapes. Assem Autom 39(2):245–253
Zuo FC, ** X, Zhang ZJ, Zhang TY (2013) Modeling method for assembly variation propagation taking account of form error. Chin J Mech Eng Engl Ed 04:641–650
Cao YJ, Li X, Zhang ZX, Shang JZ (2015) Dynamic prediction and compensation of aerocraft assembly variation based on state space model. Assem Autom 35(2):183–189
Chen H, ** S, Li ZM, Lai XM (2015) A solution of partial parallel connections for the unified Jacobian-Torsor model. Mech Mach Theory 91:39–49
Mao J, Chen DJ, Zhang LQ (2016) Mechanical assembly quality prediction method based on state space model. Int J Adv Manuf Technol 86(1–4):107–116
Guo FY, Zou F, Liu JH, **ao QD, Wang ZQ (2019) Assembly error propagation modeling and coordination error chain construction for aircraft. Assem Autom 39(2):308–322
Sun QC, Zhao BB, Liu X, Mu XK, Zhang YL (2019) Assembling deviation estimation based on the real mating status of assembly. Comput-Aided Des 115:244–255
Laperrière L, Ghie W, Desrochers A (2002) Statistical and deterministic tolerance analysis and synthesis using a unified Jacobian-Torsor model. CIRP Ann - Manuf Technol 51(1):417–420
Desrochers A, Ghie W, Laperrière L (2003) Application of a unified Jacobian-Torsor model for tolerance analysis. J Comput Inf Sci Eng 3(1):2–14
Clément A, Desrochers A, Rivière A (1991) Theory and practice of 3D tolerancing for assembly. Proceedings of the CIRP Seminar on Computer Aided Tolerancing, Penn State University, USA, pp. 25–55
Yan XY, Ballu A (2019) Review and comparison of form error simulation methods for computer-aided tolerancing. J Comput Inf Sci Eng 19(1):010802
Yi Y, Liu XJ, Feng JD, Liu TY, Liu JS, Ni ZH (2019) Representation and generation method of digital twin-oriented product skin model. Comput Integr Manuf Syst 25(6):1454–1462
Liu JH, Zhang ZQ, **a HX, Gong H, Shao N (2021) Assembly accuracy analysis with consideration of form defects and surface deformations. Chin J Mech Eng 57(3):207–219
Schleich B, Wartzack S (2015) Approaches for the assembly simulation of skin model shapes. Comput-Aided Des 65:18–33
Zhang J, Qiao LH (2018) Research on computing position and orientation deviations caused by mating two non-ideal planes. Procedia CIRP 75:309–313
Hervé JM (1999) The Lie group of rigid body displacements, a fundamental tool for mechanism design. Mech Mach Theory 34(5):719–730
Funding
The authors would like to acknowledge the financial supports from the National Key Research and Development Program of China (grant no. 2018YFB1701301) and the Natural Science Foundation of Jiangsu Province (BK20202007).
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Yi, Y., Liu, T., Yan, Y. et al. A novel assembly tolerance analysis method considering form errors and partial parallel connections. Int J Adv Manuf Technol 131, 5489–5510 (2024). https://doi.org/10.1007/s00170-022-09628-9
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DOI: https://doi.org/10.1007/s00170-022-09628-9