在市場競爭的需求下，機器的運轉速度不斷提高以降低成本提高產能，但在運轉速度不斷增快的同時，機器所產生之搖撼力及搖撼力矩也隨之增加，而所導致的振動將降低其加工精度；因此常使用最佳化方法進行其平衡設計，更有前人提出使用SOCP搜尋出平衡設計之全域最佳解。而本研究之主要目的為針對平面機構及空間機構分別建立二次規劃模式(Quadratic programming, QP)的最佳平衡設計，以期亦能搜尋出其全域最佳解。
本文所要探討者為針對史提芬生III型平面六連桿添加圓盤配重，及針對RRCS空間四連桿機構添加一圓盤及一圓球配重的最佳平衡設計。首先根據所提出之慣性參數系統，分別建立其運動與動力分析模式，再據以分別建立其最佳平衡設計模式，且探討其設計變數中那些是自由參數(Free variables)，再將其化為具相依變數以滿足非線性等式限制式的二次規劃模式，並分別提出其計算流程，以期搜尋所得之解為全域最佳解。所建立之模式中，配重的密度與厚度均可由設計者根據實務上之需求而給定；而目標函數則採用權重因子將搖撼力、搖撼力矩及輸入扭矩的權重因子加以組合而得兩種模式，而且據以探討其對平衡效果的影響。
在實例中，分別以一組史提芬生III型機構與RRCS機構為例，探討在使用不同模式下之搜尋結果。另外，所有的實例都另建立其非線性規劃模式，且以亂數各產生1000組的起始估計值，再搜尋其結果。在只最小化搖撼力為目標下，兩機構分別以二次規劃模式及其計算流程所得的結果都優於非線性規劃所得者；而在以最小化搖撼力、搖撼力矩及輸入扭矩之和為目標的實例，則以非線性規劃模式所得的結果較佳。因此，使用目前本研究所使用的二次規劃模式及其計算流程所得者不一定是全域最佳解。

Due to the market competition, the operation speeds of machinery need be increased to augment the productiveness and to lower the manufacturing costs. However, higher operating speed leads to greater shaking force and moment, which cause vibration and reduce manufacturing precisions. Therefore, using the optimum method to search for balancing designs is a common way, some researchers even tried to search the global optimum of balancing designs by using second-order cone programming.
The optimal balancing designs of Stephenson-III planar six-bar linkages with adding cylindrical counterweights on the links with fixed pivots, and RRCS spatial four-bar linkages with adding a cylindrical counterweight and a ball counterweight on each link with fixed pivots, is the main concern of this study. The models of kinematic and kinetostatic analyses are built based on the inertial parameter system. Then the design variables of the models are analyzed to check they are free variables or not. Finally their optimization models with modified forms of the quadratic programming which include nonlinear equality constraints but can be satisfied analytically by using dependent variables are proposed. Consequently, their computational algorithms are developed, in order that the solutions can be global optimums. The models allow the designers to specify the density and thickness of the counterweights. By using the weighting factors of shaking force, shaking moment and driving torque functions, two criteria are used as the objective functions, so as to investigate their effects on the balancing designs.
A Stephenson-III six-bar linkage and a RRCS spatial four-bar linkage are used as the examples. Besides, a nonlinear programming model is constructed for each example with 1000 sets of initial estimates generated randomly, then compare each result with the one gotten by using the modified quadratic programming model and algorithm. Based on the results of minimizing the shaking force only, the results of using the modified quadratic programming models and algorithms are better than those of nonlinear programming. But when minimizing the sum of shaking force, shaking moment and driving torque, the results of nonlinear programming are better. So the results of using the proposed modified quadratic programming models and the algorithms might not be the global optimum.

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