在本研究中，吾人將研究彈性樑在兩端點為非線性支撐的條件下，承受簡諧外力時的頻率響應分析。其中，彈性樑假設為Bernoullis-Euler beam，兩端以三次方彈簧的支撐以模擬非線性邊界條件，再利用傅立葉展開式(Fourier expansion)描述樑的運動行為。如此，以建立吾人之偽非線性振動模型。在運動方程建立完成後，利用貝索函數(Bessel function)來模擬解析函數的形式，其次再以漢克轉換(Hankel transform)來求其特解。觀察研究結果可發現，在非線性頻率響應分析當中會有跳躍現象(jump phenomenon)產生，其現象不只在單一頻率時發生，而是在一頻率域之中發生。由此現象可得知物體振動時的共振頻率及不穩定頻率域，而線性邊界條件假設則無法捕捉此現象。因為線性邊界條件狀況下只能觀察到物體的共振頻率。此一模型及本文所提之新的分析方式可運用在廣泛的工程問題當中，如吊橋、高速鐵路以及各種機械裝置。根據研究結果，在探討振動問題時，除了要關心共振頻率外，在非線性運動狀況下，不穩定頻率域的觀察也是極其重要的。 In this research, the beam vibration with nonlinear boundary conditions sustaining simple harmonic loads is studied. The elastic beam is modeled by the Bernoullis-Euler beam theory. The two ends are supported by cubic springs allowing nonlinear boundaries. The Fourier expansion is applied to the linear motion of the beam alone. However, cubic spring forces give the nonlinear constrains on the boundaries. This quasi-nonlinear analytic model for the beam vibration equation is established. The Bessel function is used to formulate the beam vibration. The Hankel transform is applied to obtain the solution. The results discover the nonlinear jump phenomenon in several frequency ranges. This gives additional information of the unstable behavior for the beam vibration, which cannot be predicted from linear approximations. This model and the new analytic technique can be applied in a wide range of engineering problems, such as suspension bridge, high speed rail road, and other mechanical devices. According to the results, one should not only concern the resonant frequency, but also the unstable frequency ranges for the nonlinear motion.
