Solving the equations of motion for a cylinder with elliptical cross section rolling down an inclined plane constitutes a rather challenging problem. Using Lagrangian mechanics, we have derived the equations of motion analytically and the resulting differential equations are solved numerically. Using a CNC milling machine, we manufactured a solid elliptic cylinder and recorded its dynamics using a high-speed video camera for various inclinations of the plane. We compared theory and experiment by extracting the center of mass coordinates and the angular velocity from the video recording. Limits imposed by the support force reaching zero and insufficient friction to prevent slipping were examined both experimentally and theoretically.

中文翻译：

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滚动椭圆圆柱

对于具有沿倾斜平面滚动的椭圆形横截面的圆柱体，求解其运动方程是一个颇具挑战性的问题。使用拉格朗日力学，我们已经解析地导出了运动方程，并且所得到的微分方程在数值上得到了求解。我们使用CNC铣床制造了一个椭圆形的圆柱体，并使用高速摄像机记录了飞机的各种倾斜度，并记录了其动态。我们通过从录像中提取质心坐标和角速度来比较理论和实验。在实验和理论上都检查了由支撑力达到零而产生的极限以及不足以防止打滑的摩擦力。