- 學位論文
利用力偶應力理論與大變形理論對微形樑自然頻率之探討
Study of natural frequency of micro-beams by using couple stress theory and large deflection theory
摘要
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。
並列摘要
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.