基於四元數之演算法在侷限顆粒流動之粒子旋轉研究

Abstract

本論文自行搭建位移平台，使用線性滑軌驅動裝有固定重量聚甲醛乾顆粒的圓管，使顆粒依控制速度運動，並在平台下方裝設有高速攝影機，同時觀察容器移動時之底部顆粒的運動狀態。透過霍夫轉換與粒子追蹤法處理運動過程中之照片得出顆粒與標記之空間狀態。此外我們應用四元數與共軛梯度法開發一新轉動演算法，結合顆粒與標記之空間資訊，可算出當下顆粒轉動速度。針對此新開發演算法，我們使用MATLAB軟體進行轉動模擬，考慮不同情境下之空間誤差，也比較不同演算法參數與起始值之誤差影響。為探討顆粒流中的物理量，我們使用空間平均分析，比較不同控制速度下之速度、轉動速度、顆粒溫度等分布。我們發現顆粒轉動比例隨控制速度上升而遞減，且顆粒角速度與顆粒溫度有負相關趨勢。我們亦發現轉動因子與移動速度除以顆粒溫度的根號有冪定律關係，與文獻中的實驗結果相似。

This study constructs a displacement platform that uses a linear stage to drive a circular tube filled with fixed-weight polyoxymethylene particles. The tube moves in a controlled-velocity and a high-speed camera positioned beneath the platform observes the motion of particles at the bottom of the tube. We perform Hough transformation and Particle tracking on the captured images to obtain the spatial information of particles and markers location. In addition, we employ quaternions and conjugate gradient method to develop a new rotational algorithm. Combining rotation algorithm with particles’ and markers’ location, we can calculate the instantaneous particle rotation speed. Through rotation simulations in MATLAB, we evaluate the limitation and usages of the newly developed rotation algorithm in terms of location errors, algorithm parameters and initialization. We use averaging box analysis in the granular flow field to determine the distribution of velocity, rotation velocity, and particle temperature. We find a negative relation between the rotation data percentage and controlled velocity, and a negative relation between the rotation speed and granular temperature. We also find that the dimensionless rotation index has a power-law correlation with the ratio of translational velocity to the square root of granular temperature, which is similar to the experimental result in the literature.