| 摘要: | 行星齒輪傳動機構在設計上具有緊湊設計、高速比、高能量密度與輸入與輸出軸的同軸配置等優點。而多階形式的設計更進一步增加行星齒輪機構的多元性，使其在汽車傳動、風力發電、各式產業設備等多有應用。行星齒輪機構具有負載分流之特性，在重負載的條件下，較其他齒輪機構更具優勢，但在開發與設計上，除了各項機構設計上的細節問題外，還需在決定齒輪基本參數時，掌握各項加工、組裝誤差對齒輪承載能力的影響。多階行星齒輪機構會因為不同邊界條件的給定，例如離合器與制動器的作動，而出現不同的扭力分配與功率流向，而每一次的邊界條件變動都需花費時間重新建立分析模型。
在本論文中以單階一般型行星齒輪組分析模型為基礎，向上擴展成通用型多階2K-H型行星齒輪組分析模型，可根據不同的元件連接關係以及邊界條件的給定，由程式自動快速建立分析模型用以計算。一般型行星齒輪組是由太陽齒輪、行星齒輪、環齒輪以及托架所構成，而2K-H型行星齒輪組是由兩個中央齒輪、行星齒輪以及托架所構成。中央齒輪可以為太陽-太陽、環-環、太陽-環三種類型，而行星齒輪則分為單行星型、雙行星型、複合行星型三種類型，總共可以構成九種不同的2K-H型行星齒輪組，其中常見到的如2S-C型行星齒輪組、複合階梯式行星齒輪組等。
為了將單階擴展至多階行星齒輪組，本研究利用矩陣來記錄階與階之間的連接關係以及各元件的邊界條件，可供後續用以建立多階行星齒輪組之齒對嚙合分析模型以及受載齒面接觸分析模型。在齒對嚙合分析中，齒輪的齒面採用標準漸開線配合雙隆起修整以及螺旋修整之數學模型。在加工及組裝誤差中，考慮了各齒輪軸與傳動軸的偏心及偏斜誤差、行星銷軸位置誤差、托架傳動軸偏心及偏斜誤差等。階與階之間的連接件僅考慮轉動角度關係，連接件中各元件的誤差不考慮其相互影響。
由單階擴展至多階行星齒輪組的受載齒面接觸分析模型是以影響係數法為核心所建立，在分析模型中考慮的變形影響包含輪齒齒面接觸、彎曲、剪力變形，托架與行星銷軸變形，太陽齒輪軸扭轉變形等。為了對機構效率進行分析，本研究在力平衡關係中加入了嚙合過程之平均摩擦係數，其會受到接觸負載、齒面相對滑動速度、接觸面曲率、潤滑情況以及表面粗糙度的影響。所分析之結果為在準靜態條件下，各階行星齒輪組之嚙合效率、單階機構效率以及整體機構效率。
為了說明本研究所建立之分析模型的應用廣泛性，在後續以五種類型的行星齒輪組作為分析案例。單腹板行星齒輪組其主要關注於單側腹板的托架其懸臂行星銷所產生的軸變形對於齒面接觸應力的影響，以及齒面修整對於齒面接觸應力的優化表現進行分析。複合階梯式行星齒輪組主要探究不同齒數組合下對傳動性能表現的影響。2S-C型行星齒輪組分析中，主要關注在摩擦力與齒面修整對傳動的影響。差速分流型三階行星齒輪組分析中，主要關注在托架偏心對於多階行星齒輪組傳動表現的影響。五速自排變速三階行星齒輪組分析中，主要關注該機構在各檔位條件下的傳動表現。
本論文所提出之通用化多階2K-H型行星齒輪組分析模型，可提供設計者快速建立各式多階行星齒輪機構進行分析，其中考慮各式加工、組裝誤差以及各元件變形的影響，並可對準靜態下的傳動效率進行分析。
;Planetary gear transmission systems offer advantages such as compact design, high speed ratios, high power density, and coaxial input and output shafts. Multi-stage configurations further enhance the versatility of planetary gear mechanisms, leading to their widespread application in automotive transmissions, wind power generation, and various industrial equipment. Planetary gear mechanisms feature load-sharing characteristics and thus exhibit superior performance under heavy-load conditions compared with other gear systems. However, during development and design, in addition to addressing detailed issues related to mechanism configuration, it is also necessary to understand the effects of manufacturing and assembly errors on gear load-carrying capacity when determining basic gear parameters. In multi-stage planetary gear mechanisms, different torque distributions and power flow paths arise due to variations in boundary conditions, such as the engagement of clutches and brakes, and each change in boundary conditions traditionally requires time to re-establish the corresponding analysis model.
In this study, a generalized multi-stage 2K-H type planetary gear analysis model is developed by extending a single-stage general planetary gear set analysis model. Based on different component connection relationships and prescribed boundary conditions, the analysis model can be automatically and efficiently established through programming for subsequent calculations. A general planetary gear set consists of a sun gear, planet gears, a ring gear, and a carrier, whereas a 2K-H type planetary gear set consists of two central gears, planet gears, and a carrier. The central gears can be of sun–sun, ring–ring, or sun–ring types, and the planet gears can be classified as single-planet, double-planet, or compound-planet types. In total, nine different configurations of 2K-H type planetary gear sets can be formed, including commonly used arrangements such as the 2S-C type planetary gear set and compound stepped planetary gear sets.
To extend the analysis from single-stage to multi-stage planetary gear systems, this research employs matrix representations to record the inter-stage connection relationships and the boundary conditions of each component. These matrices are subsequently used to establish meshing analysis models and loaded tooth contact analysis models for multi-stage planetary gear systems. In the gear meshing analysis, the tooth surfaces are modeled using standard involute profiles combined with double-crowning modifications and lead crowning mathematical models. Manufacturing and assembly errors considered in the analysis include eccentricity and misalignment errors of gear shafts and transmission shafts, positioning errors of planet pin shafts, and eccentricity and misalignment errors of the carrier transmission shaft. For inter-stage connecting components, only rotational angle relationships are considered, and the mutual influence of errors among components within the connecting elements is neglected.
The loaded tooth contact analysis model for extending from single-stage to multi-stage planetary gear systems is established based on the influence coefficient method. The deformation effects considered in the model include tooth surface contact deformation, bending deformation, shear deformation, deformation of the carrier and planet pin shafts, and torsional deformation of the sun gear shaft. To analyze transmission efficiency, an average friction coefficient during the meshing process is incorporated into the force equilibrium equations. This coefficient is influenced by contact load, relative sliding velocity of the tooth surfaces, contact curvature, lubrication conditions, and surface roughness. The analysis results include meshing efficiency of each planetary gear stage, single-stage mechanism efficiency, and overall system efficiency under quasi-static conditions.
To demonstrate the broad applicability of the proposed analysis model, five types of planetary gear systems are investigated as case studies. For the single-web planetary gear system, the analysis focuses on the effects of shaft deformation induced by cantilevered planet pins on one-sided carriers on tooth contact stress, as well as the optimization effects of tooth surface modifications. For compound stepped planetary gear systems, the influence of different tooth number combinations on transmission performance is examined. In the analysis of the 2S-C type planetary gear system, particular attention is given to the effects of friction forces and tooth surface modifications on transmission behavior. For the differential power-splitting three-stage planetary gear system, the primary focus is on the influence of carrier eccentricity on the transmission performance of multi-stage planetary gear systems. In the analysis of a three-stage planetary gear system used in a five-speed automatic transmission, the transmission performance under different gear ratio conditions is investigated.
The generalized multi-stage 2K-H type planetary gear analysis model proposed in this thesis enables designers to rapidly establish and analyze various multi-stage planetary gear mechanisms. The model accounts for manufacturing and assembly errors as well as component deformations, and it allows for the evaluation of transmission efficiency under quasi-static operating conditions |
