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研究生： 林冠廷
Lin, Kuan-Ting
論文名稱： 全尺寸小提琴有限元素模型之建立
Establishment of a Full Size Finite Element Violin Model
指導教授： 褚晴暉
Chue, Ching-Hwei
學位類別： 碩士
Master
系所名稱： 工學院 - 機械工程學系
Department of Mechanical Engineering
論文出版年： 2012
畢業學年度： 100
語文別： 中文
論文頁數： 99
中文關鍵詞： 全尺寸小提琴有限元素模型 、彈性力學 、振動力學 、自然頻率 、振動模態 、驅動點導納值 、弦張力 、琴橋高度
外文關鍵詞： finite element model, full size violin, natural frequencies and modes, elasticity, mechanical vibration, input admittances, string tension, bridge height
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    • 本文建立一全尺寸小提琴有限元素模型，藉由彈性力學與振動力學進行模態分析，求出其自然頻率與振動模態，同時利用頻率響應分析求出驅動點導納。首先比較琴體與全尺寸小提琴有無架設弦之間的頻率響應。結果顯示全尺寸小提琴架設弦與琴體之自然頻率是十分接近的，接近的現象同樣發生在驅動點導納值的比較。因此，預先施加張力之弦在全尺寸小提琴模型中是不可被忽略的。其次討探弦張力對於振動行為所帶來之影響；分配給各弦三組不同的張力，其可能對整體小提琴帶來動態響應之差異。結果指出T1、C3 和C4 模態之自然頻率與驅動點導納幾乎是相同的。考量弦之使用壽命，建議給予較小之弦張力即可。最後討論小提琴琴橋之高度在固定四條弦的自然頻率下所扮演之角色。琴橋高度分別為34mm、33mm 和32mm。依據小提琴製琴師王聖哲老師所提出之建議：小提琴四條弦，G、D、A 與E 弦，四條弦應等長，長度容許差異值為±0.1mm；而本模型之弦長度最大差異為0.9mm，故小提琴琴橋高度所帶來影響之結果，其驅動點導納之結論僅為參考依據。依據本主題數值結果顯示，琴橋高度34mm 與33mm 會呈現較佳之驅動點導納值。

    A finite element model of a full size violin is established in this research. The natural frequencies and modes are computed by performing the modal analysis according based on the theories of elasticity and mechanical vibration. Also, the input admittances are obtained by using the harmonic response analysis.
    Firstly, the difference of the vibration response between the models of the simple violin body and the full size violin with or without the strings attached is studied. It shows that the natural frequency of the full model with attached strings is approximately the same as that of the violin body model. Similar results can be observed from the variations of the input admittances. Therefore, the strings with pretension cannot be ignored when the full model of violin is considered.
    Secondly, the influence of the string tensions on the vibration responses will be investigated. Three tension forces, that may affect the whole structure dynamic response, are assigned to the strings. The results indicate that the natural frequencies and the input admittances of modes T1, C3, and C4 in three cases are almost the same. In order to extend the string life, smaller tension forces assigned to the strings are suggested.
    Lastly, the role of the bridge height will be studied under the condition that the natural frequencies of the four strings are kept to be the same. Three bridge heights chosen in this study are 34mm, 33mm and 32mm, respectively. According to the opinion of a violin repairer-restorer, Baroque Wang, the lengths of four strings G, D, A and E for a good violin should be kept within the tolerance ±0.1mm. In our model, the largest difference between the strings is 0.9mm. Therefore, the conclusion of the bridge height effects on input admittance can be used only as a reference. In our case, the heights with 34mm and 33mm give better input admittance.

    摘要 ............................................. I Abstract.......................................... II 誌謝 ............................................. IV 目錄 ............................................. V 表目錄............................................ VIII 圖目錄............................................ IX 符號表............................................ XIV 第一章 緒論....................................... 1 1.1 前言.......................................... 1 1.2 文獻回顧...................................... 1 1.3 研究動機...................................... 7 1.4 論文架構...................................... 7 第二章 小提琴簡介與理論基礎....................... 9 2.1 小提琴簡介.................................... 9 2.1.1 小提琴的構造................................ 9 2.1.2 琴體的振動.................................. 15 2.2 彈性振動力學.................................. 18 2.2.1 非等向性材料................................ 18 2.2.2 正交性材料.................................. 19 2.2.3 圓柱正交性材料.............................. 21 2.2.4 等向性材料.................................. 23 2.2.5 弦的振動.................................... 24 2.2.6 模態分析.................................... 25 2.2.7 簡諧響應分析................................ 26 2.2.8 驅動點導納.................................. 27 第三章 有限元素模型建立與問題分析................. 29 3.1 三維模型建立.................................. 29 3.1.1 琴體........................................ 29 3.1.2 琴頭、弦軸、琴頸、弦枕與指板等.............. 36 3.1.3 尾板、尾枕與尾端腸弦環...................... 41 3.1.4 組合........................................ 43 3.1.5 弦之架設.................................... 44 3.2 有限元素模型建立.............................. 44 3.2.1 單位系統.................................... 44 3.2.2 有限元素之選擇.............................. 45 3.2.3 MPC（Multi-Point Constraint） .............. 48 3.2.4 有限元素模型................................ 49 3.2.5 材料性質參數................................ 50 3.3 問題分析...................................... 54 3.3.1 邊界設定.................................... 54 3.3.2 討論主題.................................... 56 3.3.3 模擬流程.................................... 57 第四章 結果與討論................................. 59 4.1 琴體與全尺寸小提琴之差異...................... 59 4.2 弦張力大小的影響.............................. 68 4.3 琴橋高度的影響................................ 82 第五章 結論....................................... 95 參考文獻.......................................... 97

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