雙線性彈性基底上無限長樑的行進波

Abstract

無限長樑在各種基底上受到不同形式負載下的平衡解、週期解一直是重要的研究課題，也有許多成果。 然而關於非受力狀態下，樑內行進波的傳遞，除了已知的在雙向線彈性基底上正弦行進波有解析的色散關係外，對其他形式基底上樑內行進波的研究相對較少。

本論文討論在雙線性基底上無限長樑內週期性行進波的傳遞特性。由於基底的雙線性特性，這個問題是非線性問題。我們利用了兩種方法:分段匹配法和打靶法，求取行進波的波形；利用Floquet理論判別行進波的穩定性。

行進波一波長內至少包含各一個位移為正和位移為負的段；位移為正的段和為負的段的交點稱為分段點。行進波的重要特性為速度以及分段點的位置和數目。我們詳細探討了波速和分段點隨基底的彈性特性變化的情形，畫出對應的分歧圖，並標示穩定、不穩定區域。

The equilibrium and periodic solutions of an infinite beam under various types of loading on different foundations have been an important research topic with numerous achievements. However, regarding the propagation of traveling waves in the beam under non-force conditions, apart from the known analytical dispersion relation for sinusoidal traveling waves on an elastic foundation, there has been relatively less research on traveling waves in beams on other types of foundations.

This paper discusses the characteristics of periodic traveling waves in an infinite beam resting on a bilinear foundation. Due to the bilinear nature of the foundation, the problem becomes nonlinear. We employ two methods, namely the piecewise matching method and the shooting method, to obtain the waveforms of traveling waves. The stability of the traveling waves is determined using Floquet theory.

Within one wavelength, the traveling wave contains at least one segment with positive displacement and one segment with negative displacement, and the intersection point of these segments is referred to as a segment point. The key features of the traveling waves are the velocity, positions and numbers of the segment points. We thoroughly investigate the variations of wave velocity and segment points with respect to the elastic properties (stiffness) of the foundation, constructing corresponding bifurcation diagrams and identifying stable and unstable regions.