This paper addresses the modeling and simulation of a homogeneous disk that undergoes plane motion constrained to be in permanent contact with and in the same plane as a two-dimensional curve described by a parametric equation. Except for the elementary cases in which the curve is a straight line or an arc of circumference, the investigated problem presents significant challenges to both modeling and simulation. In addition to the necessity of using differential geometry methods to calculate local geometrical properties of the motion, this system exhibits typical difficulties of non-smooth dynamics. Different modes of operation require the synthesis of suitable rules for switching systems of differential equations and, due to the discontinuities caused by the exchange of dynamic models, numerical integration methods suitable for solving stiff problems are necessary. It should also be emphasized that the dynamic model here developed shows the Painlevé’s paradox, caused by the application of a simplified law of friction (Coulomb’s law) to a rigid body subjected to an unilateral constraint. Among the results obtained in this work, we would like to highlight the following: (i) the realization that both the geometry of the contact curve and that followed by the center of mass of the disk must be considered to construct a correct dynamic model for the motion; (ii) a detailed procedure to identify the instant the disk stops sliding and initiates a pure rolling motion on the two-dimensional track represented by a \(\mathfrak{C}_{2}\) class curve \(C\) in parametric form; (iii) the realization that, for arbitrary geometries of flat two-dimensional curves of class \(\mathfrak{C}_{2}\) and simplified models of friction, it is not possible to establish an arbitrary initial kinematic condition (a necessity for integrating the equations of motion) without incurring in a paradoxical result, known in the literature as Painlévé’s paradox. We consider that the approach adopted in this work might contribute to building dynamic models of hybrid systems, especially those with non-elementary geometric characteristics.

中文翻译：

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与参数2D曲线和Painlevé悖论接触的磁盘的运动

本文讨论了均质圆盘的建模和仿真，该均质圆盘受到平面运动的约束，该平面运动被约束为与参数方程所描述的二维曲线永久接触并在同一平面内。除了曲线为直线或圆周弧的基本情况外，研究的问题对建模和仿真都提出了重大挑战。除了使用微分几何方法来计算运动的局部几何特性的必要性之外，该系统还表现出非平滑动力学的典型困难。不同的操作模式需要综合适用于微分方程切换系统的规则，并且由于交换动态模型而导致的不连续性，必须采用适合解决刚性问题的数值积分方法。还应强调的是，此处开发的动力学模型显示了Painlevé悖论，这是由于将简化的摩擦定律（库仑定律）应用于受单边约束的刚体而引起的。在这项工作中获得的结果中，我们想强调以下几点：（i）认识到必须考虑接触曲线的几何形状以及紧随盘的质心的几何形状，才能为议案；（ii）识别磁盘停止滑动并在由a表示的二维轨道上发起纯滚动运动的瞬间的详细过程 由对单向约束的刚体应用简化的摩擦定律（库仑定律）引起的。在这项工作中获得的结果中，我们想强调以下几点：（i）认识到必须考虑接触曲线的几何形状以及紧随盘的质心的几何形状，才能为议案；（ii）识别磁盘停止滑动并在由a表示的二维轨道上发起纯滚动运动的瞬间的详细过程 由对单向约束的刚体应用简化的摩擦定律（库仑定律）引起的。在这项工作中获得的结果中，我们想强调以下几点：（i）认识到必须考虑接触曲线的几何形状以及紧随盘的质心的几何形状，才能为议案；（ii）识别磁盘停止滑动并在由a表示的二维轨道上发起纯滚动运动的瞬间的详细过程 （i）认识到必须同时考虑接触曲线的几何形状和紧随其后的圆盘质心，以为运动建立正确的动力学模型；（ii）识别磁盘停止滑动并在由a表示的二维轨道上发起纯滚动运动的瞬间的详细过程 （i）认识到必须同时考虑接触曲线的几何形状和紧随其后的圆盘质心，以为运动建立正确的动力学模型；（ii）识别磁盘停止滑动并在由a表示的二维轨道上发起纯滚动运动的瞬间的详细过程\（\ mathfrak {C} _ {2} \）类曲线\（C \）以参数形式；（iii）认识到，对于类别为\（\ mathfrak {C} _ {2} \）的平坦二维曲线的任意几何形状和简化的摩擦模型，不可能建立任意的初始运动学条件（a在没有引起悖论的结果的情况下必须整合运动方程式，这在文献中称为Painlévé悖论。我们认为这项工作中采用的方法可能有助于建立混合系统的动态模型，尤其是具有非基本几何特征的系统。