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An Investigation on Optimal Tolerance Allocation Based on Lagrange Multipliers
|關鍵字:||optimal tolerance allocation;最佳化公差配置;Lagrange multipliers;Lambert W function;cost-tolerance model;worst-case and statistical tolerancing.;拉格朗日乘數;Lambert W函數;成本公差模式;最惡狀況與統計公差||出版社:||機械工程學系所||引用:||[Ann & Seng, 1996] Ann, B. N. K. and Seng, M. S. M., “Tolerance Synthesis Adopting a Nonlinear Programming Approach,” International Journal of Advanced Manufacturing Technology, No. 11, pp. 387-393, 1996. [Arora, 2004] Arora, J. S., “Optimum Design Concepts,” In: Introduction to Optimum Design, Chapter 3, pp. 93-148, 2004, Elsevier Academic Press, 2nd Edition, Oxford, UK. [Chase et al., 1990] Chase, K. W., Greenwood, W. H., Loosli, B. G. and Hauglund, L. F., “Least Cost Tolerance Allocation for Mechanical Assemblies with Automated Process Selection,” Manufacturing Review, ASME, Vol. 3, No. 1, pp. 49-59, 1990. [Chen, 1996] Chen, M. 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公差配置的目的在於如何分配或指定合適的公差規格於工件或組件上，最佳化配置準則往往建立在製造成本的考量上，本論文探討如何以最佳化理論搭配Lagrange multipliers (LM)方法建構尺寸公差最佳化課題，以最低製造成本為目標，以組件組合公差即尺寸鏈以及製程能力為限制條件，配合不同成本－公差關係式，進行最惡狀況與統計公差的最佳化配置，同時也建立最佳化公差配置完整分析與設計過程。
Tolerance allocation is the process to distribute or to assign suitable tolerance specifications to a part or to an assembly. The distribution criteria are often based on manufacturing costs. As mathematical models are required for optimal tolerance allocation, this dissertation explores a comprehensive analysis method for optimal tolerance allocation using Lagrange multipliers for minimizing manufacturing cost subject to constraints on dimensional chains and machining capabilities. A series of systematic design procedures is also conducted to resolve the tolerance optimization problem in mechanical design.
The general reciprocal power and exponential cost-tolerance models with both equality and inequality constraints are investigated for employing this method. Moreover, worst-case and statistical tolerancing are also applied in this investigation. In particular, we further derive a closed-form expression of the tolerance optimization problem for reciprocal exponential cost-tolerance model by introducing Lambert W function for which the transcendental equation can be solved. As the near-optimal solutions obtained using nontraditional optimization techniques are probably inconsistent, the optimization problem is solved by applying the algorithmic approach. Therefore, for constrained minimization problems with only equality constraints, the optimum design can be obtained by solving simultaneous equations with closed-form solutions in order to reduce complicated computations while keeps a reasonable accuracy. This computing result displays that truncation errors and round-off errors can be decreased greatly.
Several case studies are illustrated to demonstrate the feasibility of this approach. Through comparisons with the regular Lagrange multipliers method and the proposed approach applied on traditional tolerance analysis methods, the result also reveals that tolerance can be allocated quickly, effectively and accurately using this method.
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