非線性懸吊之彈性樑的內共振分析

The Internal Resonance Analysis of A Nonlinear Suspension Elastic Beam

謝易哲(I-Che Hsieh)

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(2011年)

https://doi.org/10.6846/TKU.2011.00687

摘要

本研究係以一彈性樑為主架構，並以三次方非線性彈簧與線性阻尼模擬其懸吊鋼纜，以分析系統之非線性振動行為。其中，吾人採用多尺度法(Method of Multiple Scales)與正交化法(Orthogonality)來解析此系統之運動方程。由本研究的分析，發現此系統存在著3:1內共振特性，吾人也進一步探討其生成的條件及非線性懸吊之彈性樑的相關振動現象。
本研究分別探討常見的四種邊界狀況之系統內共振(Internal Resonance，簡稱I.R.)分析，並分別激擾低模態(Lower Mode)跟高模態(Higher Mode)，且由Fix Points解(穩態頻率響應)及相位圖(Phase Plot)，來判斷系統之穩定性並觀察其非線性內共振現象。
本文發現兩端為鉸接端之彈性樑，在第一模態和第三模態間具有3：1內共振現象，而本文也發現一端為鉸接一端為滾輪之彈性樑，在第一模態和第三模態也一樣具有3：1內共振現象；另外，本文也發現兩端為固定端與兩端為彈簧之彈性樑，在第二和第四模態也具有3：1內共振現象。對於這四種邊界狀況，皆發現存在非線性內共振
現象，因此在學術界或工程應用上，對這四種邊界狀況應注意產生內共振現象，以避免導致系統發散而產生不穩定現象。

關鍵字

非線性 ； 振動 ； 內共振

並列摘要

The internal resonance (I.R.) of a Bernoulli-Euler Beam with nonlinear suspensions (cubic nonlinear springs) and with different end supports was studied in this thesis. The Bernoulli-Euler Beam is suspended by nonlinear springs (similar to Winkler Type Foundation) to the top ceiling. Four different types of boundary conditions are considered in the study, which are hinged-hinged, hinged-roller support, fixed-fixed, and spring-spring support, respectively. A simple Aerodynamic loading is included to simulate this suspension bridge vibration system. This research is to find the I.R. conditions for the boundary conditions aforementioned and subjected to Aerodynamic and simple harmonic loadings. The method of Multiple Scales (MOMS) is applied to get the conditions for I.R. The classical structural Dynamic analysis is also employed for finding mode shapes of the beam for different boundary conditions. By using the orthogonal properties, this problem can be deduced to a to a time domain ordinary differential equation. The I.R. condition for the cases studied can be determined analytically. The analytical predictions are confirmed by the Fixed Point plots and phase plots.
This research found that the 3:1 I.R. occurred in the case of hinged-hinged boundary conditions in the 1 and 3 modes. The 3:1 I.R. also occurred in the case of hinged-roller boundary conditions in the 1 and 3 modes. The 3:1 I.R. occurred in the case of fixed-fixed boundary conditions in the 2 and 4 modes. The 3:1 I.R. occurred in the case of spring-spring boundary conditions in the 2 and 4 modes.

並列關鍵字

Nonlinear ； Vibration ； Internal Resonances

參考文獻

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