本文提出一套系統化的方法，對雙自由度行星齒輪系進行自動變速箱之運動合成與評估。此方法包含行星齒輪系之構造合成、檔位設計、離合器與制動器之配置、齒數合成、力量分析與評估。
在行星齒輪系之構造合成方面，先證明所有基本型行星齒輪系的自由度都是二，再以加太陽齒輪、環齒輪與行星齒輪的方式合成基本型行星齒輪系構造，並提出奇偶特性矩陣法判別各基本型行星齒輪系的同構與等效。接著，以桿件連接的方式合成組合型行星齒輪系，桿件連接的方式有三種，分別是共軸桿件與共軸桿件相連接、行星齒輪與行星齒輪相連接、共軸桿件與行星齒輪相連接。其中，基本型行星齒輪系與第一種組合型行星齒輪系較適用於自動變速箱。
在自動變速箱之檔位設計方面，考量作動連續之功能需求，可作為五速至七速自動變速箱，行星齒輪系之共軸桿件個數應大於或等於五，可據此進行行星齒輪系構造之篩選。其次，進行離合順序之合成。先針對每一個行星齒輪機構，進行減速比公式之推導與各操作屬性之決定，將適用於自動變速箱之操作列表，以利於作動連續之低速檔序列與高速檔序列之排列。再將減速比值遞減之低速檔序列與高速檔序列，和適用於倒退檔之操作相組合，即可列出所有可行之離合順序。
在離合器與制動器之配置方面，針對二自由度行星齒輪機構，提出編碼圖之繪製方式，可找出該行星齒輪機構之共軸桿件序列與各共軸桿件之離合器代碼與制動器代碼。根據共軸桿件序列，可列出配置離合器與制動器時會產生桿件干涉之三明治集合，以檢測所合成之離合順序是否能配置離合器與制動器而不產生桿件干涉。根據共軸桿件之離合器代碼與制動器代碼，則可找出該離合順序所有可行之配置圖。
在齒數合成方面，針對整數變數之各齒輪齒數，使用全域搜尋法，令目標函數為各前進檔減速比公式與減速比目標值之差的絕對值之和。拘束式包含：相鄰檔位間減速比之比值關係之限制、各前進檔減速比值與減速比目標值之誤差限制、倒退檔減速比值範圍之限制、行星齒輪系之幾何限制與齒輪齒數之範圍限制。如此即可找出各檔減速比值與減速比目標值最接近之一組齒數組合，但本文保留幾組目標函數值較小之解，以進行後續自動變速箱之評估。
在力量分析方面，考量行星齒輪相嚙合時，軸心線不與鉛垂線重合之複雜狀況，提出修正後之切線力公式、徑向力公式與行星齒輪軸心受力之水平分量與垂直分量公式，此外亦提出有利於撰寫電腦程式之行星齒輪之向心力公式。在自動變速箱之評估方面，考量之性能包含齒數合成之目標函數值、平均嚙合效率、最大切線力、最大扭矩與行星齒輪系體積等五項。評估函數即為將上述性能函數無因次化之後，乘以權重之值再相加之和。針對齒數合成所保留之各組解，計算出評估函數之值，根據其值之大小即可決定各齒數組合之排名。
本文將這套方法運用於六速自動變速箱之運動合成與評估，篩選出一個二自由度行星齒輪系，桿件數目是八且具有五個共軸桿件。此行星齒輪系共合成出六個有效的離合順序。對第二與第五個離合順序進行齒數合成，各保留七組解以進行自動變速箱之評估，其結果可供六速自動變速箱選用之參考。
本文將這套方法運用於七速自動變速箱之運動合成與評估，選用一個具有10個桿件且共軸桿件數是六的基本型行星齒輪系。此行星齒輪系共合成出200個有效的離合順序，其中離合器與制動器總數是六者，共有27個。於齒數合成時，保留10組解以進行自動變速箱之效率評估，其結果可供七速自動變速箱選用之參考。

This study presents a systematic method to synthesize the feasible configurations and evaluate the performances of automatic transmissions with two-degree-of-freedom planetary gear trains. The methodology includes structural synthesis of planetary gear trains, clutching sequences synthesis, clutches equipment, optimal synthesis of the number of teeth on gears, force analysis, and performances evaluation for automatic transmissions.
On the structural synthesis of planetary gear trains, it is proved that the number of degrees-of-freedom of basic-type planetary gear trains is always two. Sun gears, ring gears, and planet gears are added to the kernel unit of a simple planetary gear train in sequence to construct basic-type planetary gear trains. An odd-even property matrix method is proposed to detect isomorphism and equivalences for basic-type planetary gear trains. Compound-type planetary gear trains are synthesized by link-to-link connections between two or more basic-type planetary gear trains. The connections include three types: (1) coaxial links to coaxial links, (2) planet gears to planet gears, and (3) coaxial links to planet gears. The basic-type planetary gear trains and the compound-type planetary gear trains with the first type connections are suitable for automatic transmissions.
On the clutching sequences synthesis, suitable planetary gear trains are detected first. Under the functional requirement for a single clutch-to-clutch shift between speeds, a screening rule is proposed to assure that the number of coaxial links must be greater than or equal to five for a planetary gear train to be suitable for five-speed to seven-speed automatic transmissions. In order to synthesize feasible clutching sequences, based on the requirement for clutch-to-clutch shifts, a systematic algorithm is presented to classify and permute feasible shift operations. The feasible clutching sequences are obtained by combining the feasible underdrive sequences, overdrive sequences, and the operations suitable for reverse speed for automatic transmissions.
In the clutches equipment aspect, coding sketches for planetary gear mechanisms are proposed for determining their coaxial-link sequences and clutch codes. According to the coaxial-link sequences, the invalid clutching sequences, which lead to interference of coaxial links while equipping clutches, are detected. According to the clutch codes, all feasible equipments of clutches for a valid clutching sequence are constructed.
On the optimal synthesis of the number of teeth on gears, an exhaustive searching method is used to find the global optimum solution for the number of teeth on gears. The objective function is defined as the sum of absolute value of the difference between the generated speed ratio and the desired speed ratio of the forward speeds. Five types of constraints are considered. They are (1) constraints on the step ratios, (2) constraints on the permissible errors between the generated speed ratio and the desired speed ratio for the forward speeds, (3) constraints on the speed ratio range for the reverse speed, (4) geometric constraints for a planetary gear train, and (5) constraints on the range of the number of teeth on gears. The best seven to ten solutions are then obtained.
In the force analysis aspect, considering the situation that several planet gears geared together, the modified tangent force equations, modified radial force equations, and modified equations of the x-component and y-component bearing forces for planet gears are proposed. In this study, the centrifugal force equations for planet gears are also proposed for computer programming.
In the performances evaluation aspect, five items are considered. They are the objective function value in the optimal synthesis of the number of teeth on gears, the average meshing efficiency, the maximum tangent force, the maximum torque, and the volume for a planetary gear train. The evaluation function is defined as the sum of the five items normalized and multiplied by the corresponding weightings.
In order to illustrate the applications for the systematic methodology mentioned above, the kinematic synthesis and evaluation for six-speed automatic transmissions is taken as an example. According to the screening rule, a two-degree-of-freedom planetary gear train, which has eight links or five coaxial links, is chosen. Six valid clutching sequences and 16 feasible clutches equipments are synthesized. After the optimal design for the number of teeth on gears, the best seven solutions for each of the two valid clutching sequences are selected for evaluation. The evaluation results from the seven solutions for each clutching sequence can be used as valuable references for designers.
The systematic methodology is also applied to synthesize seven-speed automatic transmissions. A 10-link basic-type planetary gear train, which has six coaxial links, is chosen as the design target. A total of 200 valid clutching sequences are obtained, but only 27 clutching sequences have the least number of clutches, i.e., six clutches. After the optimal design for the number of teeth on gears, the best ten solutions are obtained from the clutching sequence of the planetary gear mechanism with a sun gear as the output link. The results after the efficiency evaluation for the ten solutions provide good choices for designers.

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