均佈質量於受黏彈性支撐無限長樑上水平等速移動的動態穩定性分析

列車於軌道上高速前進時，可將運動模型模擬為，有限寬度的均佈質量於具有阻尼與彈簧基底支撐的長樑上等速移動。本研究主要探討此模型在不同分佈寬度與不同單位質量下的系統穩定性。因於臨界穩定狀態下長樑變形以週期震盪的行形式存在，為求取週期震盪的臨界參數，將長樑拆成三段進行討論，並分別解出各段解析解，再透過邊界連續的條件將相鄰的兩段進行擬合，其中包括在正負無限遠處的邊界條件需收斂。比較均佈質量分佈寬度為有限長與無限長的穩定性邊界曲線，發現兩曲線無法相互逼近，對於這個問題，推斷原因為兩者在無限遠處的邊界條件本質上是不相同的，分佈寬度無限長於無限遠處的邊界條件是沒有收斂的限制。另一方面點質量模型與分佈寬度趨近於零的均佈質量模型是可以相互逼近的，但兩者於同樣臨界總質量時，均佈質量模型所對應的臨界速度是較低的，也就是系統相對不穩定。隨著質量分佈寬度加大，臨界速度也會隨之降低，簡單來說若採用點質量模型簡化分析，系統的穩定性會被高估。比較均佈質量模型與長方形剛體模型的系統穩定邊界，結果顯示均佈質量系統是相對穩定的。在長方形剛體隨著高度參數增加，系統穩定性會隨之下降，這說明長方形剛體的質量慣性矩為系統不穩定因子。最後探討均佈質量模型系統能量變化，在臨界穩定狀態下，系統保守力作功與水平力推力和基底阻尼的非保守力作功是相同的，代表系統處於動態平衡的狀態。

關鍵字

無限長樑；有限寬度的均佈質量分佈；系統穩定性邊界

並列摘要

We consider a distributed mass of finite length travelling uniformly on an infinite beam resting on a visco-elastic foundation. Our focus is on the effects of the length of the distributed mass on the stability of the mass-beam-foundation system. It is argued that on the stability boundary, the mass-beam-foundation system is in the form of a steady state periodic motion. To seek the periodic solution, we divide the infinite beam into three segments, in each of which the differential equation is solved analytically. By comparing the stability boundary curves of the distributed mass of finite length with the one of infinite length, it is found that the stability boundary curves of finite length do not converge to the one of infinite length. It is believed that the course of the failure to converge towards the infinite length model is the differences in the boundary conditions at infinity. On the other hand, the stability boundary curves of finite length do converge to the point mass model when the mass length approaches zero. Compared to the point mass model, the distributed mass model is less stable. More precisely, for a specified system damping and total mass, the critical speed is in general much smaller if distributed model is adopted. The longer the mass length, the smaller the critical speed is. In short, the conventional point mass model overestimates the stability of the mass-beam-foundation system.

並列關鍵字

Infinite beam ； distributed mass of finite length ； stability boundary

參考文獻

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均佈質量於受黏彈性支撐無限長樑上水平等速移動的動態穩定性分析

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