受無張力黏彈性基底支撐的無限長樑在過臨界速度移動負載下的週期解

Abstract

隨著高速鐵路的快速發展，軌道在列車經過後產生的變形與振動開始受到人們的關注。列車在軌道上行進的動力學屬於經典的“移動載荷問題”，過去在此領域的研究多為探討移動載荷速度低於臨界速度的反應，本論文主要研究受無張力黏彈性基底支撐的彈性樑，受到過臨界速度的移動集中力所產生的變形。無張力基底的重要特性是樑和基底可能分離，因此我們先定義接觸條件，再以分段匹配法求得平衡解。非線性系統所得的平衡解可能不唯一，在此以特徵值分析與動態分析檢查平衡解的穩定性。當集中力的移動速度大於一臨界值時，系統沒有穩定的平衡解，這個臨界值稱為臨界速度。臨界速度會隨著集中力、阻尼比等因素改變大小。本論文得到阻尼比為0.3時，臨界速度和集中力大小的關係曲線。當集中力移動的速度超過臨界速度時，樑的穩態行為可能為週期解。我們詳細討論了可能發生週期解的條件和週期解的特性。

With the rapid development of high-speed railways, the deformation and vibration of tracks caused by passing trains have become a matter of concern. The dynamic behavior of trains traveling on tracks belongs to the classical "moving load problem." Previous research in this field mainly focused on studying the response when the speed of the moving load is below the critical speed. In this paper, we primarily investigate the deformation of an elastic beam supported by a tension-free viscoelastic foundation subjected to a moving concentrated force exceeding the critical speed.A significant characteristic of the tension-free foundation is the potential separation between the beam and the foundation. Therefore, we first define the contact conditions and then obtain the equilibrium solution using the segmented matching method. The equilibrium solution of a nonlinear system may not be unique, so we analyze the stability of the equilibrium solution through eigenvalue analysis and dynamic analysis.When the speed of the moving force exceeds a critical value, the system does not have a stable equilibrium solution, and this critical value is referred to as the critical speed. The critical speed is influenced by factors such as the magnitude of the concentrated force and the damping ratio. In this study, we obtain the relationship curve between the critical speed and the magnitude of the concentrated force with a damping ratio of 0.3. When the speed of the concentrated force surpasses the critical speed, the beam's steady-state behavior may become a periodic solution. We extensively discuss the conditions and characteristics of the potential periodic solutions.