二維端點支撐之非線性振動系統研究

A Two Dimensional Vibration Analysis with Nonlinear End Struts

林漢翔(Han-Shiang Lin)

指導教授 ： 王怡仁

淡江大學/工學院/航空太空工程學系碩士班/碩士(2010年)

https://doi.org/10.6846/TKU.2010.00071

摘要

本文研究一極細平面剛體剖面振動系統，包含主體及其下方附加減振器之上下位移與俯仰旋轉共四個自由度。並將主體兩端之支撐假設為三次方非線性彈簧與二次方非線性阻尼，以分析系統之非線性振動行為。另外吾人更進一步假設此系統含有3 to 1內共振之特性並考慮系統受一空氣動力之影響，再利用牛頓定律推導出四自由度之運動方程組。而後藉由多尺度法(Method of Multiple Scales)求其穩態之頻率響應並分別激發第一、二自由度，與內共振等值線圖(Internal Resonance Contour Plot)相互比較並觀察內共振對於系統之影響性。而後吾人將利用古典振顫分析的觀念繪製Eigen Plot，找出系統的振顫速度，並與相位圖做比對，再藉由特徵值分析及Floquet Theorem來判斷系統之穩定性。
最終吾人藉由內共振等值線圖與振顫速度等值線圖(Flutter Speed Contour Plot)相互對照，嘗試找出減振器最佳之擺放位置，以達到不變動主體架構原則之下，僅改變減振器位置之最佳減振效果。本文發現，在系統設計時，除了振顫速度是必須注意的項目外，內共振對於系統之影響性也是不容小覷，某些飛行狀況之下，在未達到振顫速度時，可能發生內共振現象，且伴隨極大之振幅及不穩定現象，因此，必須將兩者皆列入設計上考慮的因素，缺一不可，如此一來將能使系統擁有更高的安全性。

關鍵字

振動 ； 非線性 ； 減振器 ； 內共振 ； 振顫速度

並列摘要

This research used a two-dimensional rigid body plate section as the main body, which contains pitch and plunge motion. Both ends of this main body were supported by cubic spring and quadratic damper acted as the support to simulate main body’s nonlinear vibration. A rigid body vibration absorber was suspended underneath the main body, and was mounted with linear spring and damper at both ends. This study also took aerodynamic influence into consideration to form an aeroelastic system. The 3:1 internal resonance of the main body’s plunge and pitch modes was included in this research. The Method of Multiple Scales was employed to analyze the fixed points solutions of this system.
We made an Internal Resonance Contour Plot to get an overview of this 3:1 internal resonance system. This research found that simply adjust the absorber position can significantly reduce internal resonance. We also used the concept of Classical Flutter Analysis to make an eigen-plot to analyze the aerodynamic effects on this system. The phase plots and Flutter Speed Contour Plots were also correlated to verify the system flutter speed.
Finally, the Floquet Theorem was employed to get the Basin of Attraction of this system and the system stability information was concluded. In the case studied, we found that the internal resonance may play an important role in the aeroelastic system. It could result large vibration amplitude before the flutter speed. We also found that in the internal resonance-flutter coupled system, just change the absorber position can also significantly reduce vibration amplitude and increase flutter speed.

並列關鍵字

Vibration ； Nonlinear ； Absorber ； Internal Rensonance ； Flutter Speed

參考文獻

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