臺灣博碩士論文加值系統

為了能有效地應用有限元素分析方法(Finite Element Analysis ,FEA)，進
行木琴琴鍵精確調音之模擬，須確認影響琴鍵彎曲第一、二、三振態頻率
之關鍵區域，以及不同的質量減損率，對琴鍵各振態頻率之影響程度等資
訊。
本研究選擇木琴琴鍵常用之非洲花梨木(Pterocarpus soyauxii)，另外挑選
花旗松(Pseudotsuga menziesii)、鐵杉(Tsuga heterophylla)作為實驗材料，探
討不同的挖削位置及質量減損對彎曲各振態頻率之影響。以長、中、短三
種長度的試材代表低、中、高三個音域，並以試材全長的5%-45%為距離，
將9 個位置分別設置在不同試材上；每支試材只挖削一個位置。每個位置
以直徑5 ㎜、7 ㎜、9 ㎜的鑽頭及兩種鑽除深度(1 ㎝、2 ㎝)，組合出4 種程
度的質量減損，觀察處理前後的頻率變化率。實驗結果顯示，影響第一振
態頻率最明顯的區域，位於試材全長40%～50%之間，影響第二振態頻率
最明顯的區域，位於試材全長25%～35%之間，影響第三振態頻率最明顯
的區域，位於試材全長15%～25%之間，不論試材的樹種及長度均有相似
的結果。此資訊可供有限元素分析方法模擬琴鍵調音之用，亦可歸納出木
琴琴鍵實務調音對應處理方法。各樹種在處理前後的頻率變化率，經分析
之後的結果顯示，非洲花梨木與鐵杉之間並無顯著差異，而花旗松與其他
兩種樹種具有顯著差異。
為了模擬琴鍵到達目標頻率時之輪廓外形，而使用有限元素分析方法模
擬木琴之調音過程。首先，必須驗證未調音試材的有限元素模型，是否能
夠代表試材結構之動態特性。材料選用非洲花梨木，分為長、中、短三種
長度代表低、中、高三個音域，以各柱狀試材經模態分析之各振態頻率，
做為修正有限元素模型之依據。本實驗之誤差範圍設定為目標頻率之±
10%，藉由各項參數漸次修改使有限元素分析模型之各振態頻率達到誤差範
II
圍內，始為有效模型。模型確定有效後，將低音域琴鍵之彎曲振態第1、2、
3 振態頻率的比值，設定為1：4：10，中音域為時之第1、2 振態頻率比值
設定為1：4，高音域則著重基頻；依照各振態之關鍵區域採用最恰當之挖
除比例，利用Solidworks 將琴鍵輪廓漸次修改到ANSYS 模擬結果達到目標
頻率之設定比值為止。依照完成後的琴鍵輪廓對實際材料進行粗調，粗調
後的琴鍵與琴鍵模型之各振態頻率誤差均能達達到在±10%誤差以內之要
求。
本研究在最後以挖削位置與質量減損的實驗，歸納出調音狀況處理方
法，將容許誤差設定在±5 音分，針對粗調後琴鍵之各種狀況採用適當對應
做法進行細調，以驗證處理方法之可行性。琴鍵經細調後的結果顯示，各
振態頻率誤差都在±5 音分內。
在證明前兩項實驗結果可行後，運用相同方法模擬61 鍵全組木琴中各
鍵之外形輪廓及建構出木琴琴鍵設計及調音製作業原則，配合調音對應處
理方法，可提供木琴業界或業餘製作者參考。

Finite element analysis (FEA) can be used on xylophone key tuning
simulation. In order to have an accurate tuning simulation, the crucial area of
each modal frequency, the mass loss of each drilling location, and the various
modal frequency changes of the keys should be verified.
This research selected Pterocarpus soyauxii as the test materials, because
it’s the common materials for making a wooden xylophone, and Pseudotsuga
menziesii, and Tsuga heterophylla were chose for comparison. The effects of
different drilling locations were discussed, and how the mass loss could affect a
modal frequency was analyzed. The test materials were categorized into low,
medium, and high ranges, as well as long, medium, and short three lengths.
The drilling depth could be 5%- 45% of the test material length. Each test
material was given nine drilling locations, but only one would be chosen. The
5 ㎜, 7 ㎜, and 9 ㎜ drills were used to create two drilling depths. There are a
total of four degrees of mass losses, and the frequency changes before and after
the drilling were recorded. The experiments revealed that all the test materials
had similar results regardless of tree species or material length. The most
significant areas that would affect the first, second, and third modal frequencies
were located at the 40-45%, 25-35%, and 15-25% of the material length,
respectively. This information is very useful for FEA tuning simulation, and the
practical xylophone tuning. The frequency change analysis showed that there
was no significant difference between Pterocarpus soyauxii and Tsuga
heterophylla, but there was a significant difference between Pseudotsuga
menziesii and the other two materials.
Before utilizing FEA on tuning simulation, it is necessary to confirm the
applicability of the model parameters. Pterocarpus soyauxii was selected as the
material, and there were three ranges and three lengths. The frequency ratio
was set at 1：4：10, and this ratio represented the frequency goal for the low
IV
range. The ratio for the medium range was set at 1：4. Regarding the high
range, the fundamental frequency was emphasized. Modal analysis was
conducted for each columnar material, and the finite element model
modification was based on the modal frequency data. The error range was set
at ±10%. Only when the parameters were modified and within the error range,
it could be a valid model. After the valid model was established, there was an
appropriate drilling depth in the crucial area of each modal frequency. The
xylophone keys were adjusted until the target frequency ratio was achieved.
Then, a coarse adjustment was taken in order to make all the modal frequency
errors to be within the range of ±10%. After analyzing the different drilling
locations and mass losses, this research proposed a practical tuning approach.
A fine adjustment was conducted after the coarse adjustment so that the
feasibility of the tuning method could be verified. The tolerance error was set
at ±5 cents. The fine-tuning process and the frequency changes were recorded.
The results revealed that all the error values of modal frequencies were within
the permissible range. By using the same method, a 60-key xylophone with its
own appearance could be made. The tuning method proposed by this research
can served as a reference for the xylophone industry and anyone who is
interested in xylophone tuning.

摘 要.........I
Abstract........III
目錄............ V
圖目錄........... VII
表目錄........... IX
壹、前 言........11
貳、文獻回顧.......13
一、音名、音階、音程與音分及和諧音.... 13
二、木琴的分類與發展................................13
三、木琴的調音與製作................................16
四、琴鍵之粗調與細調................................19
五、影響木琴音準的因子...... ........................20
(一)含水率....... ................................21
(二)塗裝......... ................................21
(三)局部質量改變...................................22
(四)溫度......... .................................23
六、有限元素分析(FEA)於振動模擬之應用..................23
参、材料與方法.....................................25
一、試驗材料.......................................26
(一)不同位置及質量減損對柱狀試材彎曲各振態頻率之影響..... .26
(二)電腦輔助工程(CAE)於琴鍵調音之應用.................28
二、試驗方法.......................................28
(一)不同位置及不同質量減損對柱狀試材彎曲各振態頻率之影響...28
(二)電腦輔助工程(CAE)於琴鍵調音之應用..................31
肆、結果與討論......................................37
一、不同位置及質量減損對柱狀試材彎曲各振態頻率之影響.......37
(一)試材彎曲振動第一、二、三振態之確認..................37
(一)相同的質量減損，挖削位置的改變對各振態頻率的影響.......39
(二)相同的挖削位置，不同的質量減損對各振態頻率的影響.......47
(三)調音狀況之對應處理................................51
(四)質量減損對樹種間影響差異之分析......................53
二、電腦輔助工程( CAE)於木琴琴鍵調音之應用...............59
(一)試材有限元素模型之有效性驗證........................59
(二)琴鍵粗調模擬與驗證................................62
(三)模式適用範圍與限制................................67
(四)琴鍵細調原則之應用結果.............................72
三、木琴琴鍵設計與調音之參考步驟........................76
(一)琴鍵尺寸之設計原則................................76
(二)調音實務作法.....................................77
(三)決定穿繩孔位置....................................79
伍、結 論...........................................80
參考文獻.............................................82
附 錄..............................................86

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