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研究生： 徐毓宏
Yu-hung Hsu
論文名稱： 大垂度索彎曲行為與小垂度索振動分析
The Analysis on the Bending Behavior of Cables with Large Sag and on the Vibration of Cables with Small Sag
指導教授： 潘誠平
Chan-ping Pan
口試委員: 黃震興
Jenn-Shin Hwang
鍾立來
Lap-Loi Chung
陳瑞華
Rwey-Hua Chen
姚忠達
Jong-dar Yau
郭世榮
S. R. Kuo
學位類別： 博士
Doctor
系所名稱： 工程學院 - 營建工程系
Department of Civil and Construction Engineering
論文出版年： 2014
畢業學年度： 103
語文別： 中文
論文頁數： 298
中文關鍵詞： Bessel方程 、Irvine方程 、傾斜索 、變索力 、大垂度 、邊界層 、懸鏈線 、WKB法 、彎曲剛度 、索 、自由振動 、平均弦向拉力 、振動控制 、風雨激振 、模態阻尼比 、黏滯性阻尼器 、黏彈性阻尼器 、通用設計曲線 、廣義頻率近似方程式
外文關鍵詞： Free vibration, Inclined cable, Variable tension, Large sag, Boundary layers, Catenary, WKB method, Bending stiffness, Cable, Irvine equation, Bessel equations, Average chordwise tension, Vibration control, Rain–wind induced vibrations, Modal damping ratio, Viscous dampers, Viscoelastic dampers, Universal curve relating modal damping ratio, Generalized frequency approximate equation
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    • 外形細長且承拉為主的構件被廣泛用於土木與海洋工程中，如滿足交通需求的斜拉橋與懸索橋的索、及滿足深水開採石油與天然氣需求的錨泊系統，這類結構可歸類於廣義索問題。廣義索結構其疲勞與振動所衍生的斷裂與銹蝕問題造成龐大的金錢損失，為了保障索結構的安全營運和使用壽命，須對索的疲勞與振動問題給予足夠的關注。
    進行疲勞分析必須了解大垂度索彎曲行為，其難點為微分方程複雜無解析解，若使用數值分析又有邊界層問題，接近索支承的微短區域內有急速彎矩變化，在此一般有限差分法或有限元素法等數值方法會失效，一直是學理上難以突破的問題。邊界層問題同樣影響到傳統解決此類問題的攝動法，讓匹配過程與解答都複雜。為解決傳統攝動法缺點，本文提出一個新的彎矩方程式，避開傳統攝動法須四階微分方程式開始求解的難點，直接由彎矩方程切入並用WKB 法(Wentzel-Kramers-Brillouin method)得解。
    小垂度傾斜索振動分析方面，先考慮索力變化的影響建立靜力與運動方程式，其中拉力可用高處、低處或是平均支承水平力表示，共建立三組方程式，數值使用Galerkin method驗證正確性。發現靜力方程式與面內運動方程式中使用的水平力要一致，亦即若是採對應平均弦向索力的靜力方程式，則運動方程式在考慮索中弦向拉力的變化時，也要以平均弦向索力為基準來推導。倘若錯置將會得到不合理的振態與頻率比結果。
    再針對傾斜索含彎曲剛度的振動行為進行研究，在小垂度的假設下，分別考慮鉸接與固端的邊界條件求得水平索固定索力與弦向座標傾斜索變化索力各自含彎曲剛度的拋物線解，並運用到後續以平均支承弦向拉力為基準的運動方程式其靜力垂度項中，得到振態與頻率方程式。在同時考慮小垂度、線性振動、彎曲剛度、索力變化與邊界條件下，使用五種組合：鉸接（水平固定力）－鉸接、鉸接（傾斜變力）－鉸接、鉸接（水平固定力）－固端、鉸接（傾斜變力）－固端、固端（水平固定力）－固端，去確定垂度、彎曲剛度、索力變化與邊界條件對索的振動行為的影響程度。
    最後進行斜拉索振動控制研究，先由近似漸進法建立理想黏滯性阻尼器的設計通用曲線。以其為基礎提出阻尼器的廣義頻率近似方程式，可適用各種形式的阻尼器。理想黏滯性阻尼器假定內部液體不可壓縮，阻尼器內部勁度為無窮大。但實際上液體仍具可壓縮性，故內部勁度為有限值。據此阻尼器的內部變形就會增加。而黏彈性阻尼器內部勁度是其固有特性不可忽略。此外阻尼器必由支座連接至橋面板上，理想阻尼器亦假定支座勁度為無窮大，但事實不然。本章分成黏滯性的Maxwell串聯模型與黏彈性的Kelvin並聯模型分別討論阻尼器內部勁度與支座勁度對最大模態阻尼比的影響。
    黏滯性最大模態阻尼比與支座勁度和內部勁度成正比。即支座勁度和內部勁度越大，最大模態阻尼比越大。黏彈性最大模態阻尼比與支座勁度成正比但和內部勁度成反比，即支座勁度越大或是內部勁度越小，最大模態阻尼比越大。在實際工程中應用阻尼器裝置時，需考慮內部勁度與支座勁度對最佳模態阻尼比的影響。
    實務設計要讓自然頻率0.3~3Hz的索必須要達最小阻尼比的要求，不論是黏滯性或黏彈性阻尼器，因為內部彈性變形與支座變形的增加都將使通用設計曲線趨向扁平，會造成在結構要求的最小阻尼比其對應的阻尼比參數較大根變小，進而造成設計所需的阻尼值下降。換言之用較小的阻尼值即可達到預計的設計成果，反而使設計優化。在黏滯性阻尼器中，內部勁度跟阻尼器內裝液體性質與機械運作方式有關。同理，在黏彈性阻尼器中，內部勁度跟阻尼器內裝彈性介質性質與機械運作方式有關。設計者必須注意內部變形與勁度數值對最大模態阻尼比的影響，在滿足最小阻尼比前提下，適度調低支座勁度可降低設計阻尼值。

    Slim tension members have been comprehensibly applied to civil and ocean engineering to create cables of cable-stayed and suspension bridges that satisfy traffic demands, as well as mooring systems that satisfy the demands of mining deep-sea petrol and natural gas. Such structural problems could be understood as general cable problems. The rupture and corrosion caused by the fatigue and vibrations of cable structures have resulted in billions of dollars of financial loss. To ensure safe operations and lengthen lifetime, more attention should be devoted to the fatigue and vibration of cables.
    It is necessary to understand the bending behavior of cables with large sag before conducting fatigue analysis. However, of incorporating the bending stiffness effect in cables results from the complex differential equations. bending stiffness is the cause of boundary layers in anchorages, and rapid variations of bending moments occur near the differential regions of the cable anchorage. Moreover, the finite difference method and the finite element method in numerical analysis can often be ineffective because of inappropriate parameter configuration and the drastic variation of functions in the boundary layers.
    Previous studies have tackled this problem with the perturbation method; yet, due to the complexity of the matching process and solution finding, the method might not be an ideal solution for engineering applications. To correct the weaknesses of a conventional perturbation method, this study proposed a novel catenary bending moment equation. By winding around the difficulty of a conventional perturbation method, which requires a fourth order differential equation for finding the solution, the proposed equation could directly identify cable sag with the help of bending moment equations. A solution was found by applying the WKB method (Wentzel-Kramers-Brillouin method) to overcome the complex problem of boundary layers.
    This study established equations that describe the static equilibrium and motion resulted from cable tension. The established equations in motion are represented in the form of zero-order Bessel functions. Since the tension in the equations can be represented by the chordwise tension occurring in the top bearing and in the bottom bearing, as well as by the average chordwise tension, three equations were respectively developed for structures in static equilibrium and in motion. Moreover, the equation based on the average chordwise bearing tension was selected to obtain explicit results of corresponding modes and frequency ratios. The solutions generated by this formula were verified using Galerkin methods. Moreover, the results show that the horizontal tension in the equations of static equilibrium should be equal to those of in-plane motion; otherwise incorrect modes and frequency ratios may be obtained.
    In this study, by considering small sag, linear vibration, bending stiffness, cable force change, and boundary condition, five combinations of different conditions were applied to identify the influence of cable vibration.
    In the end, the vibration control of stayed cables was researched and a generalized frequency approximate equation suitable for various damper applications. Moreover, the influence of damper internal stiffness and support stiffness on the maximum modal damping ratio, which of viscous dampers is directly proportionate to the support stiffness and the internal stiffness. Therefore, the larger the support stiffness and internal stiffness is, the larger the maximum modal damping ratio is. The maximum modal damping ratio of viscoelastic dampers is directly proportionate to the support stiffness but inversely proportionate to the internal stiffness. Consequently, the larger the support stiffness or the smaller the internal stiffness is, the larger the maximum modal damping ratio.

    第1章 緒論 1 1.1 研究動機 1 1.2 研究目的 8 1.3 文獻回顧 8 1.3.1 索靜力行為的研究 8 1.3.1.1 索的靜力解析解研究 8 1.3.1.2 索的靜力數值解研究 12 1.3.2 索動力行為的研究 14 1.3.2.1 索的線性動力研究 14 1.3.2.2 索的非線性動力研究 17 1.3.2.3 索的動勁度矩陣研究 22 1.3.2.4 索的拉力測定研究 22 1.3.2.5 拉索的振動控制 26 1.4 研究內容 35 第2章 大垂度含彎曲剛度索的靜力行為 37 2.1 前言 37 2.2 懸索靜力理論 38 2.2.1 懸鏈線解 38 2.2.2 拋物線解 40 2.2.3 彈性懸鏈線(elastic catenary cable) 45 2.2.4 準彈性懸鏈線(associate elastic catenary cable) 49 2.3 懸鏈線索元素的柔度與切線勁度矩陣 51 2.4 傾斜懸索的弦向座標靜力理論 52 2.4.1 弦向與水平座標間的轉換關係 52 2.4.2 弦向座標的靜力微分方程 53 2.4.3 弦向座標的拋物線解 54 2.4.4 弦向支承拉力的靜力微分方程 56 2.4.5 較低支承弦向拉力的靜力解 57 2.4.6 較高支承弦向拉力的靜力解 57 2.4.7 平均支承弦向拉力的靜力解 58 2.5 含彎曲剛度懸鏈線與拋物線傳統靜力解 61 2.5.1 含彎曲剛度懸鏈線微分方程 61 2.5.2 含彎曲剛度拋物線解 63 2.6 含彎曲剛度新式懸鏈線靜力模型 65 2.6.1 新懸鏈線彎矩方程式 65 2.6.2 拋物線解驗證新彎矩方程式正確性 66 2.6.3 新的懸鏈線微分方程式 66 2.7 含彎曲剛度的WKB懸鏈線解 68 2.8 數值比較 70 2.9 本章結論 91 第3章 索力變化之小垂度傾斜索自由振動分析 93 3.1 前言 93 3.2 最簡單的弦振動理論 94 3.3 傾斜索計垂度但不計拉力變化的振動理論 96 3.3.1 有垂度但不計拉力變化的運動方程式 96 3.3.2 彈性索的面外振態與頻率方程式 98 3.3.3 彈性索的Irvine面內振態與頻率方程式 99 3.3.4 Wu et al.的面內修正Irvine振態與頻率方程式 102 3.4 傾斜索考慮垂度與拉力變化的振動理論 105 3.5 平均支承索力 的運動方程解 107 3.6 較低支承弦向索力 的運動方程解 111 3.7 較高支承弦向索力 的運動方程解 114 3.8 用Galerkin method進行數值驗證 118 3.8.1 懸索振動微分方程之Galerkin method解 118 3.8.2 平均索力 的Galerkin method解 123 3.8.3 較低支承弦向拉力 的Galerkin method解 124 3.8.4 較高支承弦向拉力 的Galerkin method解 126 3.9 數值驗證與討論 128 3.10 本章結論 144 第4章 小垂度含彎曲剛度索自由振動分析 145 4.1 前言 145 4.1 弦向座標含彎矩傾斜索的垂度靜力解 148 4.1.1 弦向座標下含彎曲剛度的垂度微分方程 148 4.1.2 含彎曲剛度固定索力拋物線解 150 4.1.3 含彎曲剛度變索力拋物線解 151 4.2 梁振動理論 153 4.2.1 梁理論運動方程式 153 4.2.2 無軸拉力梁振態與頻率方程式 155 4.2.3 有軸拉力鉸支承梁振態與頻率方程式 156 4.2.4 有軸拉力固端梁振態與頻率方程式 157 4.2.5 Zui et al.的簡化索力計算公式 161 4.3 傾斜索計垂度與彎曲勁度的振動理論 162 4.4 含彎曲剛度之鉸接－鉸接索的振態與頻率方程式 164 4.4.1 鉸接－鉸接水平索對應的面內振態與頻率方程式 164 4.4.2 鉸接－鉸接傾斜索對應的面內振態與頻率方程式 167 4.4.3 鉸接－鉸接索對應的面外振態與頻率方程式 168 4.5 含彎曲剛度之鉸接－固端索的振態與頻率方程式 168 4.5.1 鉸接－固端水平索對應的面內振態與頻率方程式 168 4.5.2 鉸接－固端傾斜索對應的面內振態與頻率方程式 170 4.5.3 鉸接－固端索對應的面外振態與頻率方程式 171 4.6 含彎曲剛度之固端－固端索的振態與頻率方程式 172 4.6.1 固端－固端水平索對應的振態與頻率方程式 172 4.7 數值比較與討論 174 4.8 本章結論 192 第5章 斜拉索振動控制 193 5.1 前言 193 5.2 斜拉橋索大敵－風雨激振 198 5.3 含阻尼的弦振動理論 201 5.4 含黏滯性阻尼器的弦振動理論 202 5.4.1 黏滯性阻尼器的頻率方程式 202 5.4.2 黏滯性阻尼器的振態等效阻尼比近似解 204 5.4.3 黏滯性阻尼器的通用設計曲線 207 5.5 阻尼器的廣義弦振動理論 212 5.5.1 阻尼器的廣義頻率方程式 212 5.5.2 阻尼器的廣義振態等效阻尼比近似解 213 5.6 非線性黏滯性阻尼器的弦振動理論 216 5.6.1 非線性黏滯性阻尼器的頻率方程式 216 5.6.2 非線性黏滯性阻尼器的模態等效阻尼比近似解 218 5.7 非理想黏滯性阻尼器的弦振動理論 219 5.7.1 非理想黏滯性阻尼器的頻率方程式 219 5.7.2 非理想黏滯性阻尼器的模態等效阻尼比近似解 221 5.7.3 非理想黏滯性阻尼器無內部彈簧變形的通用設計曲線 223 5.7.4 非理想黏滯性阻尼器無支座變形的廣義通用設計曲線 224 5.7.5 非理想黏滯性阻尼器的廣義通用設計曲線 225 5.7.6 非理想黏滯性阻尼器的設計參數分析 227 5.7.6.1 內部勁度對模態阻尼比的影響 227 5.7.6.2 支座勁度對設計參數的影響 231 5.7.6.3 最佳模態阻尼比和內部勁度與支座勁度的互制 234 5.7.7 非理想黏滯性阻尼器的通用設計範例 237 5.8 黏彈性阻尼器的弦振動理論 241 5.8.1 黏彈性阻尼器的頻率方程式 241 5.8.2 黏彈性阻尼器的振態等效阻尼比近似解 242 5.8.3 黏彈性阻尼器無內部勁度的廣義通用設計曲線 245 5.8.4 黏彈性阻尼器無支座變形的廣義通用設計曲線 245 5.8.5 黏彈性阻尼器的廣義通用設計曲線 246 5.8.6 黏彈性阻尼器的設計參數分析 249 5.8.6.1 內部勁度對模態阻尼比的影響 249 5.8.6.2 支座勁度對設計參數的影響 252 5.8.6.3 最佳模態阻尼比和內部勁度與支座勁度的互制 256 5.8.7 黏彈性阻尼器的通用設計範例 257 5.9 本章結論 261 第6章 結論與展望 263 6.1 結論 263 6.2 展望 267 參考文獻 269

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