本研究考慮一彈性樑，其下方以非線性彈簧支撐主體，以模擬彈性樑置於 Winkler type 彈性基座(Elastic Foundation)的振動行為。在本研究中，彈性樑的一端為扭矩彈簧搭配鉸接邊界支撐之，另一端則是自由端，而在彈性樑上掛載動態減振器(Dynamic Vibration Absorber (DVA))，利用DVA達到避開內共振及減振之效果。由於動態減振器放置於自由端邊界時，此邊界條件改變為時變邊界條件。因此吾人採用 Mindlin-Goodman 法分析此問題，並藉由多項式移位函數 (Shifting Polynomial Function) 將非齊次性邊界條件轉換為齊次性邊界條件。本文並使用多尺度法 (Method of Multiple Scales (MOMS) ) 解析此非線性系統，發現在彈性基座某彈性係數的環境下，系統中之第一模態(Mode)及第二模態存在一對三(1：3) 的內共振情形。本研究並繪製各模態之穩態固定點圖 (Fixed Point Plots)，藉由振幅及振動模態觀察其非線性內共振現象(Internal Resonance)，並藉由此解釋非線性幾何形變(nonlinear geometry)與非線性慣量(nonlinear inertia)兩者之間的關係，以及加裝DVA後，DVA之最佳彈性係數與質量比的組合，使得此系統避開內共振及達到最佳減振效果。本研究最後以數值法及簡易實驗模型驗證結果之正確性。 This study considered a slender hinged-free nonlinear beam embedded in a Winkler type elastic foundation simulated using nonlinear cubic springs. This also created multiple possibilities for mode coupling and internal resonance. The objective of this study was to avoid internal resonance within this system and achieve effective vibration damping. We added a time-dependent boundary dynamic vibration absorber (TDB DVA) that was suspended at the free end of the beam to reduce vibration and prevent internal resonance. The Mindlin-Goodman method was used to analyze this time dependent boundary condition problem. The internal resonance condition based on the ratio of the elastic foundation frequency to the beam frequency of the main structure was obtained. The vibration reduction effects of other positions of the DVA were also studied. The influence of shortening effect (nonlinear inertia) and nonlinear geometry of this beam were taken into account as well. We employed the method of multiple scales (MOMS) to analyze this nonlinear problem. The Fixed point plots (steady state frequency response) were obtained. DVAs with various locations and spring constants were considered and the optimal mass range for the DVA to reduce vibration in the main structure was also proposed. The results of this study were verified using numerical simulation, which, in addition to confirming the accuracy by through comparison, established the applicability in this study.
