單向黏彈性基底上無限長梁受等速移動質點的變形

Abstract

隨著高速列車軌道技術的快速發展，對於列車在鐵軌上運行時的動力學特性研究變得愈加重要。其中，軌道與車輛運行時產生的振動對乘客的舒適度影響極大，因此對軌道的變形與振動反應進行深入探討顯得尤為必要。 在過去的研究中，大多採用雙向彈性基底模型來模擬列車和鐵軌的交互作用。為了更貼近實際情況，本論文中使用了單向黏彈性基底分析受其支撐的無限長尤拉梁在等速移動質點作用下的變形。 本論文的主要目標在於探討梁的平衡變形和穩定性。利用分段匹配法，求取梁的平衡變形，並獲得變形隨著各參數（例如速度、質量等）變化的結果。運用微擾法和動態分析判別平衡解的穩定性。求出梁存在平衡解的最高速度(臨界速度)和最大質量(臨界質量)，繪製各參數下的臨界曲線。我們發現單向黏彈性基底的臨界速度遠低於雙向黏彈性基底的臨界速度。

With the rapid development of high-speed railway track technology, studying the dynamic characteristics of trains running on rails has become increasingly important. The vibrations generated during track and vehicle operation greatly impact passenger comfort, making a comprehensive investigation into track deformation and vibration responses crucial. In previous research, a bi-directional elastic foundation model was commonly used to simulate the interaction between trains and rail tracks. To better reflect real-world conditions, this paper employs a unidirectional viscoelastic foundation to analyze the deformation of an infinitely long Euler beam subjected to a uniformly moving point load. The main objective of this paper is to explore the equilibrium deformation and stability of the beam. The segmented matching method is used to calculate the equilibrium deformation and obtain the deformation results with varying parameters (such as speed, mass, etc.). Stability of the equilibrium solution is determined using perturbation methods and dynamic analysis. The critical speed (critical velocity) and maximum mass (critical mass) at which the beam has equilibrium solutions are determined, and critical curves for different parameter values are plotted. We found that the critical speed for the unidirectional viscoelastic foundation is significantly lower than that for the tensionless viscoelastic foundation.