兩平面彈性樑非對稱線接觸的穩定性分析

Abstract

當兩個相同的彈性樑受到端點軸向推力作用時，彼此之間可能出現點接觸、對稱以及非對稱線接觸的情況。本論文探討非對稱線接觸的平衡解及其穩定性。 過去的研究指出，對稱線接觸下，兩樑只有在接觸段的兩端有集中力，接觸段內部的接觸力為零，因此可以視為兩點接觸。然而，當線接觸形式為非對稱時，接觸段內部存在分佈力，無法視為兩點接觸。本論文將線接觸段的兩樑視為一根合併樑，由此分析平衡解。這種方法的優點在於，線接觸段的分佈力為內力，不會出現在運動方程式。 平衡解的穩定性可由其附近小擾動的頻率來判別。在振動過程中，接觸點的滑動，導致線接觸長度會隨時間改變，這使得問題變得更加複雜。為了克服這個困難，我們採用隨著接觸點移動的座標系，透過這個移動坐標系和原先固定坐標系的轉換關係，推導得平衡解附近小擾動的線性運動方程式，最後利用打靶法得到小擾動的自然頻率。 為了驗證理論結果，我們設計了實驗裝置。在設定的推進量下，輕敲彈性樑施加微小振動，量測適當點的位移，並透過快速傅立葉轉換得到頻譜圖。最後，比對實驗與理論結果中自然頻率隨推進量變化的趨勢，從而驗證理論分析結果。

When two identical elastic beams are subjected to axial thrust at their endpoints, point contact, symmetric line contact, and asymmetric line contact can occur between them. This paper investigates the equilibrium solutions and their stability for asymmetric line contact. Previous studies have shown that in symmetric line contact, the internal forces in the contact region of the beams exist only at the ends as concentrated forces, while the distributed forces in the interior contact region are zero. Therefore, the two beams can be treated as in contact at two points. However, when the line contact is asymmetric, distributed forces exist in the contact region. In this case, the two-point contact model no longer suitable. In this paper, the two beams in the contact region are treated as a merged beam, and the equilibrium solutions are analyzed accordingly. The advantage of this approach is that the distributed forces in the line contact region are internal forces and do not appear in the equations of motion. The stability of equilibrium solutions can be determined by the frequencies of small perturbations around them. During the vibration process, the sliding of the contact point causes the length of the contact segement to change with time, which makes the problem more complex. To overcome this difficulty, we adopt a coordinate system that moves with the contact point. Through the transformation relationship between this moving coordinate system and the original fixed coordinate system, we derive the linear equations of motion for small perturbations near the equilibrium solutions. Finally, we use the shooting method to obtain the natural frequencies of the perturbations. To validate the theoretical results, we designed an experiment. Under a specified amount of thrust, the elastic beam is lightly tapped to induce small vibrations. The displacements at appropriate points are measured, and the Fourier transform is used to obtain the frequency spectrum. Finally, by comparing the trend of the natural frequencies with the amount of thrust in the experimental and theoretical results, the theoretical analysis is validated.