摩擦力對受通道拘束之彈性樑運動的影響

Abstract

本文中我們考慮摩擦力對於受通道拘束之彈性樑(elastica)接觸運動的影響，並分成圓形通道和直線通道兩個部份討論。彈性樑在通道的輸入端及輸出端均受部分挾持(partially clamped)，在輸出端及拘束通道上受到庫倫摩擦作用，本文主要探討彈性樑在輸出端和輸入端之間的相對運動。本文中使用shooting method來求解彈性樑的半靜態(quasi-static)運動，當彈性樑在輸入端進入及退出的過程中，和拘束邊界之間會有四種接觸模式：滾動點接觸、滾滑混合點接觸、滾動線接觸和滾滑混合線接觸。在圓形通道的部份，我們主要討論當樑在輸入端進行正弦運動過程，輸出端會有兩種運動方式：不會運動和完全跟隨輸入端的運動。這兩種運動模式會交互地出現，而輸出端的運動會落後於輸入端，在實驗觀察中也驗證了理論預期的輸出端延遲現象。在直線通道的部份，我們在輸出端多加入彈簧以及預力，探討預力的影響。在通道間隔夠窄的情況下，小變形理論和樑理論的力位移曲線結果相當的接近。

In this paper we consider the effect of friction force on the contact behavior of an elastica inside straight and circular channels with clearance. The elastica is partially clamped at both the input and output ends of the channel. Coulomb friction is assumed to be present at the output end and on the constraining walls. Focused is placed on the relative motion between the input end and the output end of the elastica. Shooting method is adopted to solve for the quasi-static behavior of the elastica movement. Four types of contact patterns between the elastica and the constraining channel walls may occur when the input end moves in and out of the channel; they are rolling point contact, combined rolling and sliding point contact, rolling line contact, and sliding line contact. When the input end undergoes a sinusoidal motion, the output end either ceases to move at all, or follows exactly the input end motion. These two phases of output end motion occur subsequently, one immediately after the other. In general, the output end lags behind the input end. Experimental observations on the lagging phenomenon at the output end confirm the theoretical prediction. In the straight channel model, we add additional springs and preloads at output end, and discuss the effect of preloads. The force-deformation map predicted from small-deformation theory agree with the elastica theory very well when the channel is narrow enough.