非線性強制型懸臂樑理論與實驗研究

The theoretical and experimental studies of forced vibration nonlinear cantilever beam

王嘉成(Jia Cheng Wang)

指導教授 ： 林依恩

國立虎尾科技大學/工程學院/機械與機電工程研究所/碩士(2010年)

摘要

本文目的是藉由不同振動源，探討一垂直式懸臂樑在基礎端受一諧波激振之振動特性。壓電片當振動源實驗部份：藉由ANSYS軟體，提出一懸臂樑FEM模型，來輔以實驗驗證。實驗為一頻率283KHz給壓電片，再透過雷射振動儀進行金屬片實際振動的量測，經實驗與模擬比對後結果相近。激振器當振動源實驗部份：應用Timoshenko樑理論在垂直式懸臂樑上，探討結構參數(楊氏係數、阻尼係數、慣性矩)對FFT的影響。實驗為頻率17.547Hz與加速度1g給激振器，再透過雷射振動儀進行樑實際振動的量測，經理論與實驗驗證後在頻率0.732Hz之峰值大小比對誤差較大(<28%)外，其他頻率比對誤差均小於4%情況下，進一步探討結構參數對FFT的影響。研究顯示，在楊氏係數變為原來+5.15%(204GPa)時，自然頻率會受到影響(0.732Hz),(5.85Hz),(16.35Hz),(17.55Hz),(32.35Hz)，變為(0.86Hz),(6.59Hz),(17.21Hz),(17.55Hz),(33.20Hz)，對峰值大小影響也明顯(-26.68%),(-14.78%),(-10.12%),(-3.42%),(-11.36%)。阻尼係數 (×10)時，對自然頻率不造成影響，在峰值大小則影響較大(-31.57%),(-17.35%),(-21.47%),(-5.98%),(-29.36%)。慣性矩變為原來1.46e-13(-5.51%)時，不僅自然頻率會受到影響(0.64Hz), (5.17Hz),(15.71Hz),(17.55Hz),(31.37Hz)，在峰值大小也會受到影響(+15.47%),(+9.68%),(+7.14%),(+4.28%),(+10.38%)。

關鍵字：非線性振動、懸臂樑、有限元素分析、快速傅立葉轉換

關鍵字

非線性振動 ； 懸臂樑 ； 有限元素分析 ； 快速傅立葉轉換

並列摘要

The vibration of a highly flexible cantilever beam is investigated. The order three equations of motion, developed by Crespo da Silva and Glyn (1978), for the nonlinear flexural-flexural-torsional vibration of inextensional beams, are used to investigate the time response of the beam subjected to harmonic excitation at the base. The equation for the planar flexural vibration of the beam is solved using the finite element method. The finite element model developed in this work employs Galerkin's weighted residuals method, combined with the Newmark technique, and an iterative process. We further apply 0.3g, 1g, 2.5g, 2.97g accelerations in our experimental work and observe the FFT variations. The results show reasonable agreement between theoretical and experimental data at resonant frequencies. The computations are also extended to study the impacts on FFT resulting from material properties, damping, geometries of the excited beam.

Keywords：Nonlinear Vibration、Cantilever Beam、FEM、FFT

並列關鍵字

Nonlinear Vibration ； Cantilever Beam ； FEM ； FFT

參考文獻

[1]林峪嶙，液流阻尼器對非線性結構之減震分析，博士論文，國立

[3]R. R. Craig, Jr. 1985.Structural Dynamics：An

Introduction to Computer Methods.台北:北門出版社。

uniform beams carrying a concentrated mass., Journal of

Sound and Vibration, 57(3), pp.357-365, 1978