臺灣博碩士論文加值系統

本文主要目的在探討微結構樑之尺寸效應對材料機械性質之影響。依據力偶應力理論，結合漢米爾頓原理及變分原理，針對微結構樑之自然頻率與微尺度的關係進行分析探討，並應用柏努利-微尤拉樑來找出材料參數，以解決微結構材料對機械性質因製程技術的影響。
其中應用柏努利-尤拉樑來進行應力探討，其過程中運用兩個邊界值解決簡支樑和懸臂樑的問題，針對兩種邊界條件樑的自然頻率大小作用進行評估。將所得自然頻率及振動模態與理論值之結果加以比較，來分析探討內部材料長度參數的影響，藉此確定材料長度參數，塑造另外一個的材料長度的標準參數，以提供微結構下小尺寸所需的機械性質，當樑受自然頻率及特微長度尺寸大小影響時，厚度減小到可以和材料的特徵長度相比較時，其自然頻率將顯著增大。另外大變形理論也在本文中被應用來與力偶應力理論之結果相互比較。
本文所應用之方法具有明顯的優勢，可提供材料在一個簡單的方式下，來簡化微結構樑材料機械性質參數之設定。

The dynamic problems of Bernoulli–Euler beams are solved analytically on the basis of modified couple stress theory. The governing equations of equilibrium, initial conditions and boundary conditions are obtained by a combination of the basic equations of modified couple stress theory and Hamilton’s principle. Two boundary value problems (one for simply supported beam and another for cantilever beam) are solved and the size effect on the beam’s natural frequencies for two kinds of boundary conditions are assessed. It is found that the natural frequencies of the beams predicted by the new model are size-dependent. The difference between the natural frequencies predicted by the newly established model andclassical beam model is very significant when the ratio of characteristic sizes to internal material length scale parameter is approximately equal to one, but is diminishing with the increase of the ratio.
The large deflection theory also is appliod in this study . The resnlts are
comparied with Couple stress theory and some discussion has been provided.

目錄
中文摘要.........................................I
Abstract.......................................II
誌謝...........................................III
目錄............................................IV
表目錄..........................................VI
圖目錄..........................................VI
符號索引........................................Ⅷ
第一章 導論......................................1
1.1 研究背景.....................................1
1.2 文獻回顧.....................................1
1.3 研究目的.....................................4
1.4 本文大綱.....................................4
第二章 運動方程式.................................5
2.1力偶應力理論..................................5
2.2力偶應力理論之運動方程式........................7
2.3大變形理論之運動方程式.........................10
第三章 振動分析..................................12
3.1 力偶應力理論的自然頻率........................12
3.1.1 S-S 簡支樑................................12
3.1.2 C-F 一端固定一端自由的樑(fixed-free)........14
3.2 力偶應力的自然頻率在矩形與圓形截面的簡化........16
3.2.1簡支樑矩形橫截面............................16
3.2.2簡支樑圓形橫截面............................16
3.2.3懸臂樑矩形橫截面............................17
3.2.4懸臂樑圓形橫截面............................18
3.3大變形振動的自然頻率...........................19
3.4力偶應力樑與大變形樑自然頻率之比較...............21
3.4.1由應變能類比................................21
3.4.2由運動方程式類比............................22
3.4.3 以ω/ω0對h/l之關係..........................23
3.4.3.1由運動方程式類比並將大變形理論之T/h轉換成h/l.23
3.4.3.2由應變能類比..............................24
3.4.4 以ω/ω0對T/h之關係..........................24
3.4.4.1由運動方程式類比並將力偶應力理論之h/l轉換成
T/h......................................24
3.4.4.2由應變能類比..............................25
第四章結果與討論.................................26
4.1矩形斷面之簡支樑h/l值變化與ω/ω0比值的關.........26
4.2圓形斷面之簡支樑d/l值變化與ω/ω0比值的關係........28
4.3矩形斷面之懸臂樑h/l值變化與ω/ω0比值的關.........30
4.4圓形斷面之懸臂樑d/l值變化與ω/ω0比值的關係........32
4.5矩形斷面之簡支樑T/h值與(ωNL/ωL)之關係...........34
4.6自然頻率ω/ω0與樑厚度與特徵長度h/l值變化比值的關係.35
4.7自然頻率ω/ω0與大變形T/h值變化比值的關係.........38
第五章結論......................................42
參考文獻........................................44
個人資料........................................45



表目錄
表1矩形斷面h/l值在 0.1~1時所得到的值..............26
表2矩形斷面h/l值在 0.1~1時所得到的值..............27
表3圓形斷面d/l值在 0.1~1時所得到的值..............28
表4圓形斷面d/l值在 0.1~1時所得到的值..............29
表5矩形斷面h/l值在 0.1~1時所得到的值..............30
表6矩形斷面h/l值在 0.1~1時所得到的值..............31
表7圓形斷面d/l值在 0.1~1時所得到的值..............32
表8圓形斷面d/l值在 0.1~1時所得到的值..............33
表9矩形斷面之簡支樑其形狀T/h值設0.1~1時所得到的值...34
表10矩形斷面之簡支樑其形狀,樑厚度與特徵長度h/l值設1~28時所造成之,自然頻率ω/ω0 與樑厚度與特徵長度h/l值變化比值.............................................36
表11矩形斷面之簡支樑其形狀T/h值設0~1.5時所造成之，自然頻率ω/ω0與大變形T/h值變化的比值...................39
圖目錄
圖1 矩形斷面之簡支樑其形狀h/l值在 設0.1~1時所造成之ω/ω0之關係圖h/l值變化與ω/ω0比值的關係說明..........27
圖2 矩形斷面之簡支樑其形狀h/l值在 設0.1~1時所造成之ω/ω0之關係圖h/l值變化與ω/ω0比值的關係說明..........28

圖3圓形斷面之簡支樑其形狀d/l值在 設0.1~1時因素所造成之ω/ω0之關係圖d/l值變化與ω/ω0比值的關係說明..........29
圖4圓形斷面之簡支樑其形狀d/l值在 設0.1~1時因素所造成之
ω/ω0之關係圖d/l值變化與ω/ω0比值的關係說明.......30
圖5矩形斷面之懸臂樑其形狀h/l值在 0.1~1時所造成之ω/ω0之關係圖h/l值變化與ω/ω0比值的關係說明................31
圖6矩形斷面之懸臂樑其形狀h/l值在 0.1~1時所造成之ω/ω0之
關係圖h/l值變化與ω/ω0比值的關係說明.............32
圖7圓形斷面之懸臂樑其形狀d/l值在 0.1~1時所造成之ω/ω0之關係圖d/l值變化與ω/ω0比值的關係說明................33
圖8圓形斷面之懸臂樑其形狀d/l值在 0.1~1時所造成之ω/ω0之
關係圖d/l值變化與ω/ω0比值的關係說明.............34
圖9矩形斷面之簡支樑其形狀T/h值設0.1~1時所造成之(ωNL/ωL)之關係圖................................35
圖10矩形斷面之簡支樑其形狀,樑厚度與特徵長度h/l值1~10時所造成之ω/ω0之關係圖,樑厚度與特徵長度h/l值變化與ω/ω0比值的關係說明....................................37
圖11矩形斷面之簡支樑其形狀T/h值設0~1.5時所造成之ω/ω0之關係圖大變形T值變化與自然頻率ω/ω0比值的關係明.......40

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