臺灣博碩士論文加值系統

本文探討影響係數法之轉子動平衡，藉由不同的量測點/平衡面之數目及位置的選擇，所建構的影響係數矩陣，基於線性系統理論，探討條件數對於轉子動平衡校正質量計算的影響。運用的最佳化方法，包括遺傳演算法及禁忌搜尋法，以最小化影響係數矩陣之條件數為目標函數，預先在平衡程序前決定量測點/平衡面之位置或數目，可提高動平衡效率及精度，降低成本，避免因矩陣病態而導致平衡失敗，甚至造成機組損傷。
對於病態嚴重的影響係數矩陣，其條件數過大，微小的量測擾動，將導致傳統的最小平方法所求得之校正量大幅偏離精確值，使得動平衡完全失敗，因此，本文運用求解反問題的Tikhonov正則法，並且使用L曲線準則決定正則參數，使得病態影響係數矩陣之動平衡方程式求解，得到校正質量的精度，可有效改善。並且考量量測誤差之影響，比較最小平方法及正則法之動平衡效果，分析其校正量、平衡精度、條件數以及量測誤差之關係，以提昇平衡精度及抗誤差能力。
本文使用有限元素法建立轉子-軸承系統之數值模型，分析其穩態響應，以實驗及數值分析方法驗證轉子之動平衡，包括單轉速以及雙轉速多平衡面影響係數法之動平衡程序。

In this dissertation, the rotor balancing is investigated by using the influence coefficient method. An influence coefficient (IC) matrix is constructed by various selections of numbers and locations of measurement sensors and/or balancing planes. The influences of condition number on the unbalance determinations are studied based on the linear system theorem. The minimization of condition number of influence coefficient matrix is served as an objective function for the reduction of computation and measurement errors in balancing procedure by using genetic algorithms and tabu search. Thus, credible locations and numbers of measurement sensors and/or balancing planes can be determined in advance. The balancing efficiency and accuracy will be improved as fulfilling the optimization methods. Moreover, the failure of rotor balancing could be avoided due to the wellness of influence coefficient matrix.
For the illness of influence coefficient matrix, its condition number is large and the correction mass determined by least squares algorithm (LSA) will be away from the true value due to slight measurement errors. For this reason, the Tikhonov regularization is utilized to solve influence coefficient equations for the unbalances determination. Additionally, the L-curve criterion is adopted to search appropriate value of regularization parameter. The balancing accuracy and robustness against disturbance can be improved by using Tikhonov regularization (TR). Furthermore, the influences of measurement errors on the unbalances determination are considered in this dissertation. The capability of against perturbation is required for determination of correction mass. The relationships among balancing accuracies, correction mass, condition number, and measurement error are analyzed by adopting both LSA and TR. Also, the balancing results using TR are compared with those using LSA.
The rotor-bearing systems are modelled based on finite element method for the analysis of steady-state responses. The balancing procedures are simulated and experimented by using a rotor kit, which include rigid and flexible rotor balancing with multi-planes.

目 錄

摘要 I
Abstract II
致謝 IV
目錄 V
表目錄 VIII
圖目錄 X
符號說明 XIII
第一章 導論 1
1.1 研究動機 1
1.2 文獻回顧 1
1.3 研究目的 4
1.4 本文大綱 5
第二章 影響係數矩陣之條件數 6
2.1 條件數 6
2.1.1 方陣線性系統 6
2.1.2 非方陣線性系統 7
2.1.3 矩陣維度特徵之條件數證明 8
2.2 影響係數法 14
2.3 校正量之誤差分析 15
2.4 數值例 17
2.4.1 條件數之算例 17
2.4.2 條件數與矩陣維度之關係 22
第三章 條件數與平衡精度 26
3.1 轉子-軸承系統之有限元模型 26
3.2 剛性、撓性轉子之平衡選擇及平衡精度 30
3.3 動平衡分析 30
3.4 IC矩陣2×2之條件數與平衡精度不相關之證明 38
3.5 實驗 39
3.6 小結 41
第四章 最佳化動平衡位置 43
4.1 遺傳演算法 43
4.2 禁忌搜尋法 45
4.3 方法比較 48
4.4 數值模擬 49
4.4.1 轉子-軸承系統 49
4.4.2 最佳化 52
4.5 實驗比較 57
4.5.1 模態試驗 58
4.5.2 動平衡實驗 61
4-6 小結 66
第五章 轉子動平衡反運算 68
5.1 影響係數方程之Tikhonov正則解 68
5.2 正則解之誤差估計 69
5.3 決定正則參數之方法：L曲線準則 71
5.4 平衡精度 72
5.5 條件數對於LSA與ITR之影響 72
5.5.1 分析例：剛性轉子平衡 73
5.5.2 分析例：撓性轉子平衡 74
5.6 小結 75
第六章 量測誤差之影響 83
6.1 求解影響係數方程組 83
6.2 求解加權影響係數方程組 94
6.3 小結 104
第七章 結論與未來展望 105
7.1 結論 105
7.1.1 最小化影響係數矩陣之條件數及最佳化 105
7.1.2 正則法之轉子動平衡 106
7.1.3 量測誤差之影響 106
7.2 實際工業運用 107
7.3 未來展望 107

參考文獻 109
論文發表 114
個人資料 117

表 目 錄

表2-1 影響係數矩陣及其條件數 21
表2-2 量測點及平衡面數目相等時之條件數 23
表2-3 量測點數目大於平衡面時之條件數 23
表2-4 量測點數目小於平衡面時之條件數 24
表2-5 五平衡面及不同量測點數目時之條件數 24
表2-6 五量測點及不同平衡面數目時之條件數 25
表3-1 轉子-軸承系統1號模型之轉盤參數及不平衡量 29
表3-2 轉子-軸承系統2號模型之轉盤參數及不平衡量 29
表3-3 轉子-軸承系統3號模型之轉盤參數及不平衡量 29
表3-4 單轉速3×3平衡結果 32
表3-5 雙轉速(3×2)×3平衡結果 32
表3-6 轉盤之物理參數 40
表3-7 固定量測點時，實驗所求得之校正量 40
表3-8 固定平衡面時，實驗所求得之校正量 40
表4-1 遺傳演算法與禁忌搜尋法之特性比較 48
表4-2 有限元素模型之物理參數 49
表4-3 遺傳演算法之參數設定 52
表4-4 禁忌搜尋法之參數設定 52
表4-5 影響係數矩陣之優化結果 53
表4-6 影響係數矩陣(2×2)之條件數 54
表4-7 平衡結果及條件數之比較 55
表4-8 實驗轉盤之物理參數 58
表4-9 校正量及條件數之比較 62
表5-1 LSA所得之校正量及平衡後之殘振倒數 76
表5-2 ITR所得之校正量及平衡後之殘振倒數 76
表6-1 動平衡方案之量測點、平衡面位置及其條件數 84

圖 目 錄

圖3-1 轉子座標系統 27
圖3-2 轉子-軸承系統之有限元素模型 28
圖3-3 轉子-軸承系統1號模型之模態分析 30
圖3-4 單轉速影響係數之平衡精度、總校正質量與條件數之關係 33
圖3-5 雙轉速影響係數(3×2)×3之平衡精度、總校正質量與條件數之關係 34
圖3-6 單轉速影響係數3×3之平衡前後振幅，平衡轉速1000 rpm 35
圖3-7 雙轉速影響係數(3×2)×3之平衡前後振幅 36
圖3-8 三量測點，三平衡面之校正質量落點分佈 37
圖3-9 2×2平衡精度、總校正質量與條件數之關係 38
圖3-10 實驗設備及rotor kit 模組 39
圖3-11 比較不同平衡面動平衡前後之旋振幅值 36
圖4-1 遺傳演算法之演算流程 44
圖4-2 禁忌搜尋法之演算流程 46
圖4-3 轉子-軸承系統之有限元素模型 50
圖4-4 Campbell Diagram 50
圖4-5 簡諧響應分析之各轉速振形 51
圖4-6 運算200次遺傳演算法之最佳解世代序號落點分佈 54
圖4-7 轉速1000 rpm時，平衡前後之旋振幅值 56
圖4-8 動平衡前後之旋振響應 57
圖4-9 量測點及平衡面之規劃 58
圖4-10 模態試驗之轉移函數頻譜及共振頻率 59
圖4-11 加速規量測軸承座位置之頻瀑圖 59
圖4-12 平衡前，轉速為30Hz、54Hz及70Hz時，位移計所量測之旋振響應頻譜 60
圖4-13 不平衡響應之量測方式 61
圖4-14 平衡轉速30Hz時，不平衡響應之時域訊號 62
圖4-15 校正量之分佈 64
圖4-16 原始不平衡響應之極座標圖 65
圖4-17 影響係數矩陣為2×2，平衡前後各量測點之振幅及相位 66
圖5-1 量測點(S3, S5, S8)、平衡面(P8, P9, P10)時，ITR校正量之收斂情形 77
圖5-2 量測點固定為(S1, S10)、(S3, S5, S8)時，2×2(左行)、3×3(右行)動平衡前後旋振響應 78
圖5-3 LSA與ITR之平衡精度(2×2)及校正量幅值總和隨條件數之變化情形 79
圖5-4 LSA與ITR之平衡精度(3×3)及校正量幅值總和隨條件數之變化情形 80
圖5-5 量測點固定為(S1, S2, S3)時，(3×2)×3動平衡前後旋振響應 81
圖5-6 LSA與ITR之平衡精度(3×2)×3及校正量幅值總和隨條件數之變化情形 82
圖6-1 以LSA求解(3×2)×3 之IC方程組時，平衡精度隨條件數之變化情形，量測位置為S1、S2及S3 87
圖6-2 以LSA求解不平衡量時，平衡精度隨量測誤差之變化情形 88
圖6-3 以L-曲線決定正則參數 89
圖6-4 以TR求解不平衡量時，平衡精度隨量測誤差之變化情形 90
圖6-5 以TR求解Case 1-1時，比較正則參數 對平衡精度之影響 91
圖6-6 以TR求解Case 1-3時，比較正則參數對平衡精度之影響 92
圖6-7 比較以LSA與TR求解Case 1-3所得之不平衡量隨量測誤差之變化情形 93
圖6-8 以TR求解影響係數(IC)方程組時，正則參數由L曲線圖決定 96
圖6-9 以TR求解加權影響係數(WIC)方程組時，正則參數由L曲線圖決定 97
圖6-10 Case 2-1之平衡精度隨量測誤差之變化情形 98
圖6-11 Case 2-2之平衡精度隨量測誤差之變化情形 99
圖6-12 Case 3-1之平衡精度隨量測誤差之變化情形 100
圖6-13 Case 3-2之平衡精度隨量測誤差之變化情形 101
圖6-14 Case 4-1之平衡精度隨量測誤差之變化情形 102
圖6-15 Case 4-2之平衡精度隨量測誤差之變化情形 103

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