質量於受彈性支撐無限長樑上移動時樑的動態反應-幾何非線性效應

隨著科技進步，軌道運輸的速度逐漸提升，而在高速移動後的軌道力學問題，可被模擬成質量於受彈性支撐無限長樑上移動之力學問題。其中動態反應所影響的真正幾何效應是我們深感興趣的，特別是在線性不穩定之臨界移動速度後的單點質量效應我們更加想探討。我們將無限長樑有效地取某一長度，並替換成有限長樑，並用和諧級數以及離散化的技巧使用在偏微分上，這個方法與解析解對比是驗證可行的。藉由特徵值技巧，我們可以找到線性不穩定之臨界速度後對於的指定質量，也就是說質量與速度將會影響線性穩定性。當移動速度超過線性不穩定之臨界速度時，以觀察者跟著施力點移動之移動座標觀看的話，非線性之動態反應會呈現穩定的週期運動，而穩定週期運動後的振幅與非線性參數成反比，也就是說非線性效應小時週期振幅會較大，非線性效應大時週期振幅會較小。這個結果與線性預測的沒有穩定振動是相反的。在週期運動中，樑上的每個點的相都是不同的，施力點會在週期振動的最大高度與最小高度中來回振動。在能量探討中，總能量包含了動能與位能，而能量會在週期運動中呈現週期變化，這些能量變化與水平施力做功抵消基底阻尼功後相同。

關鍵字

質量點在彈性基底之樑上等速移動；集中力在彈性基底之樑上等速移動

並列摘要

We investigate the effect of the nonlinear terms arising from exact geometry on the dynamic response of the mass-beam-foundation system. In particular, we are interested in the case when the moving speed of the point mass exceeds the so-called critical speed. We replace the infinitely long beam with a sufficiently long finite beam and use a harmonic expansion method to discretize the partial differential equations of motion. The feasibility of this technique is verified by comparing our results with existing analytical solutions. By solving the eigenvalues of the linear problem, one can find the critical speed for a specified mass. When the moving speed of the mass is greater than the critical speed, the dynamic response eventually settles to a steady state of periodic motion as seen by an observer travelling along with the mass. This is contrary to the unbounded vibration predicted by the linear theory. During the periodic vibration, the phase of every point on the beam is different from each other. As a result, the location of maximum amplitude of vibration moves back and forth near the loading point. The total energy of the mass-beam-foundation, which includes all the kinetic and potential energy, fluctuates with time during the periodic vibration. The fluctuation of the total energy is balanced by the energy input from the horizontal pushing force and the energy dissipated by the damping in the foundation. The amplitude of the periodic vibration decreases as the geometric nonlinearity parameter increases.

並列關鍵字

mass-beam-foundation ； force-beam-foundation

參考文獻

Achenbach, J.D., Sun, C.T., 1965. Moving load on a flexibly supported Timoshenko beam. International Journal of Solids and Structures 1, 353-370.

Google Scholar

Ansari, M., Esmailzadeh, E., Younesian,D., 2010. Internal-external resonance of beams on non-linear viscoelastic foundation traversed by moving load. Nonlinear Dynamics 61, 163–182

Google Scholar

Bogacz, R., 1983. On dynamics and stability of continuous systems subject to a distributed moving load. Ingenieur Archiv 53, 243-255.

Google Scholar

Chonan, S., 1975. The elastically supported Timoshenko beam subjected to an axial force and a moving load. International Journal of Mechanical Sciences 17, 573-581.

Google Scholar

Den Hartog, J.P., 1952. Advanced Strength of Materials. Dover publication, New York.

Google Scholar

國際替代計量

質量於受彈性支撐無限長樑上移動時樑的動態反應-幾何非線性效應

全文下載

主題瀏覽