陀螺運動的角動量初等分析

我們從剛體轉動力學的知識出發，利用尤拉角的方法去連結固定座標系和隨物轉動座標系之間座標的關係，隨後利用在固定座標系觀察下某物理量的時變率等於在旋轉座標系下該物理量的時變率加上旋轉角速度與該物理量的外積得到一組聯立的微分方程式，最後利用這組微分方程式研究在均勻重力場作用下一個一端點被固定住的對稱陀螺的旋轉運動。我們用matlab數值模擬了一些一般化情形，討論了起始條件對陀螺的進動與章動的影響。此外，在章動效應很小情況下，我們從基本的力學原理出發，解釋了進動的產生所導致豎直方向看似角動量不守恆的問題。這樣的討論對大學物理剛體部分教學提供一定的參考價值以及對尤拉角這個概念有更深一層的認識。

關鍵字

陀螺儀；進動；章動；定軸轉動；尤拉角

並列摘要

We start with the knowledge of rotational dynamics of rigid body, and use the method of Eulerian angle to connect the coordinate relationship between the fixed coordinate system and the rotating coordinate system which moving with the object. Then making use the fact that the time-varying rate of a certain physical quantity in a fixed coordinate system equals to the time-varying rate of that physical quantity in the rotating coordinate system plus the outer product of the rotational angular velocity and that physical quantity to obtain a set of simultaneous differential equations. Finally, this set of differential equations is used to study the rotational motion of a symmetric gyro whose one end is fixed under the action of a uniform gravity field. We numerically simulated some general situations with Matlab and discussed the influence of initial conditions on the precession and nutation of the gyro. In addition, on the premise of the nutation effect being small, starting with the basic mechanics principle, we explains the problem that the vertical direction seems to be non-conservation of angular momentum caused by the precession. This kind of discussion provides a certain reference value for the teaching of the rigid body part of university physics and a deeper understanding of the concept of Eulerian angle.