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研究生： 王立民
Wang, Li-Min.
論文名稱： 扭轉圓筒的摺痕與幾何機制
Geometric Mechanism of Crease Formation on Twisting Cylindrical Shell
指導教授： 洪在明
Hong, Tzay-Ming
口試委員: 陳宣毅
CHEN, Hsuan-Yi
欒丕綱
Luan, Pi-Gang
王道維
Wang, Daw-Wei
周亞謙
Chou, Ya-Chang
學位類別： 碩士
Master
系所名稱： 理學院 - 物理學系
Department of Physics
論文出版年： 2018
畢業學年度： 106
語文別： 中文
論文頁數： 112
中文關鍵詞： 摺痕 、扭轉 、彈塑性力學 、摺紙 、斷詞 、語言學
外文關鍵詞： crease, twisting, origami, Föppl–von Kármán equations, text segmentation, linguistics
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    • 本論文包含了兩個工作：（一）扭轉圓筒的摺痕與幾何機制（二）語言統計學的縮
    放結構。兩者皆是跨領域的研究，前者包含了摺紙數學還有彈塑性力學，後者涉及
    統計學、語言學、自然語言處理與物理。
    論文的第一部分，討論在不同幾何限制下，薄膜圓筒被扭轉時因挫屈（buckling）
    產生的規律摺痕。發現圓筒上的摺痕數量N只要透過半徑R與寬度w的比例
    就能被唯一決定，即N = N(R/w)，是一個無關厚度的二維參數。從簡單的幾何關
    係、能量最小化與一對可以直接描述摺痕形成的新材料參數，我們理論推導出的解
    析解，可以成功擬合實驗事實，在摺痕力學的描述上比舊有的彈塑性力學參數表
    現更加簡單。除了摺痕數量，R/w還能用來描述摺痕的型態：斷裂摺痕、規律摺痕
    與不規律摺痕三種型態。本文還會探討如何從這一簡單、可解析的結果出發，建
    立一座串聯小形變與大形變問題的橋樑，不需要透過複雜的von Kármán–Donnell
    equations求數值解。除了最簡單的圓筒，我們還將問題拓展到截錐、多邊形管、
    斜切管、圓球等不同幾何結構，試圖在材料和幾何上統一這類問題。根據尺度縮放
    不影響摺痕形態的這一特性，我們將此結果應用在解釋地球板塊裂痕與山脈形成的
    機制上，提出因扭轉產生板塊形變的機制。
    論文的第二部分，延續了前任學長蔡孟學在詞（word）頻和字/音節（character/
    syllable）頻排名的統計關係研究，深入探討「縮放結構」（scaling structure）
    的來源：將詞的出現次數排名x與其組成字的出現次數排名y畫成y 􀀀 x圖，會
    得到一層一層、階梯狀的規律結構。此一結構在不同的自然語言文本中皆會出現，
    無關文本的類型與語言，而且和語言統計學中知名的齊夫法則（Zipf’s law），
    即單詞的出現次數N會和其次數排名x成反比，是互補的定律。也就是說，這一定
    律可以獨立於Zipf’s law出現。我們進一步發現描述這一結構完整度的指標：SP
    vaule、搭配函數（Collocation function）以及連結函數（Link function），可以
    11
    用來判斷切詞（word segmentation）的準確度，而不需依賴大量數據或者對該語
    言的先備知識。我們根據此結果提出一個新演算法來切詞，並且可以藉由自我評分
    來改善切詞的能力。因為這一新演算法具有不需要先備知識的特性，它可以被用來
    研究未知語言與語料庫不夠完整的語言，比如動物語言和嬰兒語言，或者更進一步
    用來研究一個領域的統計結構與統計單位組成。此外，我們還透過人造文本來研究
    縮放結構產生與破壞的機制，證明這一結構並非路邊隨便找來的文本就有，而是具
    有特定、待更深入研究的統計意義。

    This thesis contain two work: 1. Geometric Mechanism of Crease Formation on
    Twisting Cylindrical Shell, and 2. Scaling Pattern: A Fundamental Structure in
    Statistical Linguistics. Both of them are interdisciplinary research. The former
    contain mathematics of paper folding and elastic-plastic mechanics, the latter
    involved statistics, linguistics, natural language processing and physics.
    The first part of thesis discuss the mechanism of regular creases’ pattern
    formation due to twisting buckling under different geometry constraints. Discovering
    the numbr of creases N can unique determined by the ratio of radius
    R and width w of cylindrical shell, a 2D parameter, N = N(R/w). We get analytic
    solutions that fits the experimental data from simple geometry, energy
    minimization and two new material properties. These new properties display
    more concise than ordinary material properties used in elastic-plastic mechanics.
    In addition to N, R/w can be used to depict three types of creases: fracture,
    regular and irregular. This thesis also allow us to build a bridge between
    small and large deformation problem form simple and analytic results instead
    of numerical solutions of von Kármán–Donnell equations which are complicated.
    Beside the simplest case, cylindrical shell, we extend this problem to
    truncated cone, polygon cylindrical shell, beveled end pipe and ball, trying to
    unify the influence of material and geometry. According to the scaling property
    of types of creases, we apply the result to explain of creases of earth’s plates,
    the formation of mountains, and propose the twisting buckling mechanism of
    deformation of earth’s plates.
    The second part of this thesis continues, the work of Meng-Xue Tsai in our
    13
    lab, the reaserch of statistical relationship between frequency rank of words
    and that of characters/syllables and investigate the origin of “scaling structure”:
    ploting word rank x and character/syllables ranky, you will get a stepped
    layer structure. This structure shown in different natrual language corpus, writing
    styles and languages. That is to say, we discover a stastistics law which is
    complementary to Zipf’s law. Furthermore we describe the completeness of
    scaling structure with three quantities: SP vaule, Collocation function and Link
    function. These index can be used to judge accuracy of word segmentation,
    and does not dependent on large data or prior knowledge of the language.
    We popose a new word segmentation algorithm which can improve its ability
    by self-scoring mechanism, Due to its feature, this algorithm can be applied to
    unknown language or language with small data base such as animal language
    or baby language, or finding out the basic statistics unit and structure in a field.
    On top of that, by finding out conditions ruining scaling structure, we prove this
    structure is not arbitrary , but has statistically significant.

    0.1 摘要. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 0.2 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 0.3 致謝. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 I 扭轉圓筒的摺痕與幾何機制 Geometric Mechanism of Crease Formation on Twisting Cylindrical Shell 16 1 簡介17 1.1 摺紙. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.2 扭轉的大形變問題. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2 實驗22 2.1 實驗目的. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.2 儀器配置. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.3 流程. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.3.1 實驗一：探討扭圓筒的摺痕分佈. . . . . . . . . . . . . . . . . . 25 2.3.2 實驗二：記錄扭轉過程之力矩及扭轉角度. . . . . . . . . . . . 27 2.3.3 實驗三：研究截錐之摺痕分布. . . . . . . . . . . . . . . . . . . 27 1 2.3.4 實驗四：檢視正m角柱(m=3,4,5,6)摺痕數與半徑長度比關係. 30 2.3.5 實驗五：研究斜切管之摺痕分佈. . . . . . . . . . . . . . . . . . 32 2.3.6 實驗六：探討球殼之摺痕數及型態. . . . . . . . . . . . . . . . 33 2.4 分析工具. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3 實驗結果與理論解釋35 3.1 實驗一：探討扭圓筒的摺痕分佈. . . . . . . . . . . . . . . . . . . . . . 35 3.1.1 檢視不同實驗參數與摺痕數之關係. . . . . . . . . . . . . . . . 37 3.1.2 分析不同實驗參數與凹痕偏斜角 之關係. . . . . . . . . . . . . 41 3.1.3 建構理論描述圓筒實驗參數與摺痕數、凹痕偏斜角α的關係. . 42 3.2 實驗二：紀錄扭轉過程之力矩及扭轉角度. . . . . . . . . . . . . . . . . 45 3.3 實驗三：研究截錐之摺痕分布. . . . . . . . . . . . . . . . . . . . . . . . 48 3.3.1 截錐幾何(半徑與長度比，Rav/w)與摺痕數之關係. . . . . . . . 48 3.3.2 推廣圓筒理論至截錐. . . . . . . . . . . . . . . . . . . . . . . . . 49 3.4 實驗四：檢視正m角柱(m=3,4,5,6)摺痕數與半徑長度比關係. . . . . 54 3.4.1 檢視m造成的影響. . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.4.2 推廣理論至多角柱. . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.5 實驗五：研究斜切管之摺痕分佈. . . . . . . . . . . . . . . . . . . . . . 59 3.5.1 斜切管的摺痕產生時間. . . . . . . . . . . . . . . . . . . . . . . 59 3.5.2 摺痕數N與管半徑、平均管長wav關係. . . . . . . . . . . . . . 60 3.6 實驗六：探討球殼之摺痕數及型態. . . . . . . . . . . . . . . . . . . . . 61 4 形態變換的幾何理論63 4.1 圓筒的幾何理論. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 4.2 截錐的幾何理論. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 2 4.3 多角柱的幾何理論. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 5 結論與未來展望70 .1 圓筒的情況，證明式3.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 .2 截錐的情況，證明式3.22 . . . . . . . . . . . . . . . . . . . . . . . . . . 79 II 附錄：統計語言學的縮放結構 Appendix: Scaling Pattern: A Fundamental Structure in Statistical Linguistics 81 A Introduction 82 A.1 Power law in complex system . . . . . . . . . . . . . . . . . . . . . . . 82 A.2 Law of statistical linguistics . . . . . . . . . . . . . . . . . . . . . . . . 83 B Statistical method 86 B.1 Fitting Zipf’s law (discrete) . . . . . . . . . . . . . . . . . . . . . . . . 86 C Scaling structure 88 C.1 Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 C.2 Logical reasoning of layer structure . . . . . . . . . . . . . . . . . . . 91 C.3 Theory of scaling function . . . . . . . . . . . . . . . . . . . . . . . . . 92 D Quantification and application 95 D.1 Who is the main role? . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 D.2 Link and Collcation function . . . . . . . . . . . . . . . . . . . . . . . 99 D.3 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 D.4 Clues of scaling sturcture . . . . . . . . . . . . . . . . . . . . . . . . . 104 3 E Conclusion and future development 107

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