非線性減振器之於非線性懸臂樑之減振效益

本研究以一非線性彈性樑為主體，其一端為固定端(fixed end)，另一端為自由端(free end)，以模擬機翼、一般飛行器或是海上鑽油平台的吊臂。首先，吾人以牛頓第二運動定律為基礎，推導出此非線性彈性樑之運動方程，再利用時間多尺度法(Method of Multiple Scales (MOMS))將非線性運動方程式，分成兩個不同的時間尺度，找出系統之振動頻率，並探討是否有內共振之現象。此外，為了達到減振效益，吾人在此彈性樑下方分別掛載線性及非線性調質減振器(Tuned Mass Damper (TMD))分析此懸臂樑系統，在不同的TMD質量、彈性係數、阻尼係數以及擺放位置對於整體振動之影響，並利用本系統最大振福的3D圖及投影之3D Maximum Amplitude Contour Plot (3D MACP)觀察TMD的最佳組合，達到本系統之最佳減振目的。最後，吾人以一簡單的空氣動力函數模擬氣流對於本彈性樑系統之阻尼的影響，藉由改變風速的大小，利用Floquet Theory 搭配 Floquet Multipliers (F.M.)判定法則來分析此系統之穩定性，吾人分別以附加線性及非線性TMD之系統之最佳減振組合與各種風速影響之下的Basin of Attration (BOA)圖形，觀察此系統在不同狀態下之穩定性，以獲得最後結論。

關鍵字

非線性之振動；振動模態；調質減振器；減振

並列摘要

This study considered a slender fixed-free nonlinear beam subjected to distributed loads and unsteady aerodynamic forces. The objective of this study was to find if there is any internal resonance in this system and achieve effective vibration damping. We added a tuned mass damper (TMD) that was suspended under the beam to reduce vibration and prevent internal resonance. The effects of linear and nonlinear TMDs were 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. TMDs with various locations and spring constants were considered and the optimal mass range for the TMD to reduce vibration in the main structure was also proposed. The Poincaré Map was also utilized to identify the system instability frequency region of the jump phenomenon. The parameters of the added TMD were studied. The internal resonance can be avoid for the existence of the TMD. The optimal TMD mass and the spring constant were provided for best beam vibration reduction. Finally, the wind speeds and aerodynamic loads were included to investigate the stability of this system. The system stability was analyzed by Floquet theory and Floquet multipliers. The basin of attraction charts were made to verify the effects of the combinations of TMD’s mass and the spring constant at diverge speed.

並列關鍵字

Nonlinear Vibration ； Mode Shape ； Vibration Reduction ； Tuned Mass Damper (TMD)

參考文獻

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[9] Y.M. Fu, J.W. Hong and X.Q. Wang, “Analysis of nonlinear vibration for embedded carbon nanotubes,” Journal of Sound and Vibration,Vol. 296, (2006), pp.746-756.

國際替代計量

非線性減振器之於非線性懸臂樑之減振效益

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