耦合渦輪葉片組的動態特性，是設計高性能汽渦輪機時關注的重點，而複雜的曲面導致葉片在分析動態特性上的困難，在大多數渦輪機組中，為增進葉片剛度而特用的多葉片耦合設計，更使得系統動態特性不易求取，在此類週期性元件組合的複雜系統中，往往須依賴有限元素法進行分析，然因其繁雜的曲面與錐度設計，使得在模式建立與計算方面均面臨許多困難與限制，更惶論在進行多參數變化分析時可能面臨之困難，為改善耦合渦輪葉片組設計時的分析效率。本文首先針對單一渦輪葉片進行簡化解析解，再利用模態合成法，藉由邊界條件的限制，對多葉片耦合系統進行合成分析。期望藉由該法對參數變化分析上的便利性，將其應用於耦合渦輪葉片組動態特性設計層級的分析上。
文中利用具側錐度與扭轉之懸臂樑，作為單一渦輪葉片的近似模型，且加入轉速效應考慮，建立單一葉片之簡化解析解，並利用此懸臂樑模型之解析解，使用模態合成法理論，對其進行週期性結構之邊界條件耦合，求解多葉片耦合後之系統動態特性。為證實該模擬在實際分析上的可行性，文中特別引用真實渦輪葉片作為模擬分析對象，文中亦分別利用數學解析與有限元素法，同時對單一葉片及耦合後之葉片組進行動態特性求解，藉由多項參數的變化比較其解之差異，亦驗證該解析方式的可靠度，並探討葉片設計上常使用的各項參數對系統最低自然頻率之影響，文中也分別討論該解析方法的優點與不足之處，為該解析方式用於渦輪葉片分析的可行性進行論證。

The dynamic characteristics of shroud blade group played a significant role in steam turbine design. However, the complex shape and periodical structure of shroud blades make it so hard to find its dynamic characteristics under high speed operation. The complicate shape, periodic structure, and tedious computation limit the application of finite element method in the design analysis of shroud group blades. In order to design the shroud blade group, the component mode synthesis method was employed to derive the system dynamic equation of the grouped periodical blades.
For simplicity, a pre-twisted and tapered cantilever beam is used to derive the approximate analytic solution of a rotating turbo blade. Then the approximated eigen solution of single blade is synthesized in company with the constrain condition by using the component mode synthesis method. In order to confirm the feasibility of the proposed simulation method, a real size turbine blade is used to discuss in the study. Through a comparison between the results solved from the proposed method and finite element method of single blade and shroud blade group to prove the reliability of the proposed method. The effect of blade parameters on the dynamic characteristic of shroud blade group has investigated in this work. Numerical results indicate the proposed method is feasible and effective in dynamic design analyses of the shroud blade group.

目錄 i
圖目錄 iii
表目錄 vi
符號說明 vii
摘要 xii
Abstract xiii
第一章 緒論 1
1.1 前言 1
1.2 研究動機 4
1.3 文獻回顧 7
1.4 組織與章節 11
第二章 葉片數學模型與模態合成理論 12
2.1 側縮懸臂樑之側向振動 12
2.1.1 寬度及厚度固定之懸臂樑 12
2.1.2 寬度固定厚度線性縮小之懸臂樑 15
2.1.3 寬度及厚度等比例線性縮小之懸臂樑 19
2.2 扭轉側縮懸臂樑之側向振動 23
2.3 模態合成法之分件結構表示 30
2.4 模態合成法之系統方程式 35
第三章 渦輪葉片有限元素分析 42
3.1 動力學數值分析理論 42
3.2 建模方法與基本假設 44
3.3 參數規劃 49
3.3.1 單一葉片分析參數規劃 49
3.3.2 多葉片端接分析參數規劃 54
第四章 結果與討論 59
4.1 葉片之數學模型近似及模態合成條件 59
4.2 可靠度分析與參數探討 66
4.2.1 單葉片模型之可靠度分析與參數探討 66
4.2.2 多葉片耦合系統之可靠度分析與參數探討 72
4.3 模態合成法應用於渦輪葉片之探討 80
4.3.1 動態特性完整度分析 80
4.3.2 數學模擬優勢探討 93
第五章 結論與未來展望 99
5.1 結論 99
5.2 未來展望 100
參考文獻 102

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