Please use this identifier to cite or link to this item: http://hdl.handle.net/11455/1610
標題: 以Lagrange Multiplier為基礎之最佳公差配置探討
An Investigation on Optimal Tolerance Allocation Based on Lagrange Multipliers
作者: 鄭國銘
Cheng, Kuo-Ming
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
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摘要: 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。 本研究首先探討倒數次方與倒數指數之一般成本公差模式,配合等式與不等式限制條件推導應用此方法求解最佳化公差配置問題,且進一步研究探討最惡狀況與統計公差模式下的最佳化公差配置。在應用LM法處理倒數指數型之成本公差模式時,因為微分面臨超越函數問題,本研究特別引入Lambert W函數以解決此課題,進而得到一封閉形式之數學表示式。應用文中所推導的表示式,配合聯立方程式求解在等式限制條件下之限制條件最佳化問題,可解決使用非傳統最佳化方法求解所得近似最佳解不一致現象,並且免去傳統LM方法所需要的微分過程複雜計算,更大大地降低數值方法所產生的截去誤差以及肇因於電腦精確度不足所衍生的捨去誤差。 文中同時舉例說明本方法之應用與驗證本方法的效果,並且經由與傳統LM方法所配置之公差與成本作一比較,結果顯示此法可快速、有效以及準確地執行最低成本之公差配置。
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.
URI: http://hdl.handle.net/11455/1610
其他識別: U0005-2105201112505400
文章連結: http://www.airitilibrary.com/Publication/alDetailedMesh1?DocID=U0005-2105201112505400
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