臺灣博碩士論文加值系統

本文主要目的在於探討一攜帶雙自由度彈簧-質量系統樑（簡稱：負載樑）的自由振動特性。與現有文獻不同之處在於：現有文獻忽略每個彈簧-質量系統之螺旋彈簧的慣性效應，而本文將此效應納入考慮。為達上述目的，本文提出等效質量的理論來取代雙自由度彈簧-質量系統，如此，便可從攜帶多個等效質量之樑系統預測出負載樑的自由振動特性。其中，負載樑的運動方程式乃藉由膨脹原理（或重疊原理）結合裸樑的自然頻率與振態來推導。另外，本文亦推導了上述考慮彈簧慣性效應之雙自由度彈簧-質量系統的質量矩陣與勁度矩陣，以方便利用有限元素法進行本研究。由於從等效質量法（EMM）所獲得的所有數值結果，與有限元素法（FEM）所獲得的結果相當接近，因此，本文之方法應該可靠。因每個雙自由度彈簧-質量系統的等效質量取決於其集結質量、彈簧常數與彈簧質量的大小，因此等效質量法提供了一個計算連接樑之雙自由度彈簧-質量系統的整體慣性效應的有效方法。此外，當連接樑之雙自由度彈簧-質量系統的總數很多，則在等效質量法中，負載樑運動方程式之整體特性矩陣的階數會遠小於在有限元素法中的階數。因此，利用等效質量法所需的電腦儲存記憶體亦比有限元素法所需要的少。

This paper investigates the free vibration characteristics of a beam carrying multiple two-degree-of-freedom (two-dof) spring-mass systems (i.e., the loaded beam). Unlike the existing literature to neglect the inertia effect of the helical springs of each spring-mass system, this paper takes the last inertia effect into consideration. To this end, a technique to replace each two-dof spring-mass system by a set of rigidly attached equivalent masses is presented, so that the free vibration characteristics of a loaded beam can be predicted from those of the same beam carrying multiple rigidly attached equivalent masses. In which, the equation of motion of the loaded beam is derived analytically by means of the expansion theorem (or the mode superposition method) incorporated with the natural frequencies and the mode shapes of the bare beam (i.e., the beam carrying nothing). In addition, the mass and stiffness matrices including the inertia effect of the helical springs of a two-dof spring-mass system, required by the conventional finite element method (FEM), are also derived. All the numerical results obtained from the presented equivalent mass method (EMM) are compared with those obtained from FEM and satisfactory agreement is achieved. Because the equivalent masses of each two-dof spring-mass system are dependent on the magnitudes of its lumped mass, spring constant and spring mass, the presented EMM provides an effective technique for evaluating the overall inertia effect of the two-dof spring-mass systems attached to the beam. Furthermore, if the total number of two-dof spring-mass systems attached to the beam is large, then the order of the overall property matrices for the equation of motion of the loaded beam in EMM is much less than that in FEM and the computer storage memory required by the former is also much less than that required by the latter.

摘要………………………………………………………………………I
Abstract……………………………………………………………………. II
謝誌………………….………………………………………………………Ⅳ
目錄………………………………………………………………………….V
表目錄………………………………………………………………...……Ⅶ
圖目錄………………………………………………………………….….VIII
符號說明……………………………………………………………….……Ⅸ
第一章 緒論 1
1.1 研究動機與目的 1
1.2 論文流程架構 3
第二章 雙自由度彈簧-質量系統運動方程式及性質矩陣推導 4
2.1雙自由度彈簧-質量系統運動方程式 4
2.2雙自由度彈簧-質量系統性質矩陣推導 6
第三章 雙自由度彈簧-質量系統的等效質量 9
第四章 負載樑的特徵方程式 14
第五章 求解方法 17
5.1使用等效質量法(EMM) 17
5.2使用有限元素法(FEM) 18
第六章 數值結果及討論 20
6.1本文所提方法的可靠性 20
6.2模態總數( )之影響 24
6.3質量比和彈簧常數之影響 26
6.4攜帶多個相同的雙自由度彈簧-質量系統之樑的自由振動特性 34
6.5攜帶多個相異的雙自由度彈簧-質量系統之樑的自由振動特性 39
第七章 結論 43
參考文獻. 45
表目錄
表1.質量比 對一攜帶雙自由度彈簧-質量系統之兩端固定樑的前五個自然頻率 ( = 1到5)的影響 21
表2.模態總數( )對攜帶一個雙自由度彈簧-質量系統之兩端固定樑的前五個自然頻率 ( = 1到5)的影響 24
表3.質量比 和彈簧常數 對一攜帶雙自由度彈簧-質量系統之兩端固定樑的前五個自然頻率 ( =1到5)的影響 26
表4.相同質量比( = = )對一攜帶三個雙自由度彈簧-質量系統之兩端固定樑的前五個自然頻率 ( =1到5)的影響 35
表5.相異質量比( = 0.01, = 0.05, =0.1)對一攜帶三個雙自由度彈簧-質量系統之兩端固定樑的前五個自然頻率 ( =1到5)的影響 40
圖目錄
圖1.論文流程架構圖 3
圖2. 攜帶一任意雙自由度彈簧-質量系統之均衡樑 4
圖3. 一組等效集結質量相同的攜帶樑 與 13
圖6.1. 攜帶一個雙自由度彈簧-質量系統之兩端固定樑 21
圖6.2. 質量比 = 0.1時，攜帶一個雙自由度彈簧-質量系統之兩端固定樑的前五個振態 23
圖6.3. 質量比 對攜帶一個雙自由度彈簧-質量系統兩端固定樑之五種振態的影響 28
圖6.4.質量比 和彈簧常數 對攜帶一個雙自由度彈簧-質量系統兩端固定樑之五種振態的影響. 31
圖6.5. 一攜帶三個雙自由度彈簧-質量系統之兩端固定樑 34
圖6.6. 相同質量比( = = )對一攜帶三個雙自由度彈簧-質量系統之兩端固定樑前五個振態的影響 36
圖6.7. 相異質量比( = 0.01, =0.05, =0.1)對一攜帶三個雙自由度彈簧-質量系統之兩端固定樑前五個振態的影響 40

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