The dynamics of rotating objects is an area of classical mechanics that has many unsolved problems. Among these problems are the gyroscopic effects manifested by the spinning objects of different forms. One of them is the Tippe top designed as the truncated sphere which is fitted with a short, cylindrical rod for rotation. The unexplainable gyroscopic effect of the Tippe top is manifested by its inversion towards the support surface. Researchers tried to describe this gyroscopic effect for two centuries, but all modelings were on the level of assumptions. It is natural because the Tippe top has a more complex design than the simple spinning disc, which gyroscopic effects did not have an analytical solution until recent time. The latest research, in the area of gyroscopic effects, reveals the action of the system of several interrelated inertial torques on any spinning object. The gyroscopic inertial torques are generated by their rotating mass. These inertial torques and the variable ratio of the angular velocities of the spinning object around axes of rotations constitute the fundamental principles of gyroscope theory. These physical principles of dynamics of rotating objects enable to description and compute of any gyroscopic effects and also the Tippe top inversion.

中文翻译：

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Tippe顶部反演的数学模型

旋转物体的动力学是经典力学领域，存在许多未解决的问题。这些问题中有不同形式的旋转物体表现出的陀螺效应。其中之一是Tippe顶，设计为截顶球，装有短圆柱棒进行旋转。Tippe顶部无法解释的陀螺效应是通过其向支撑表面的倒置来体现的。研究人员试图描述这种陀螺效应达两个世纪，但所有模型均处于假设水平。这很自然，因为Tippe顶部的设计比简单的旋转盘要复杂得多，陀螺效应直到最近才有分析解决方案。陀螺效应领域的最新研究，揭示了多个相互关联的惯性转矩对任何旋转物体的作用。陀螺惯性扭矩是由它们的旋转质量产生的。这些惯性转矩和旋转物体绕旋转轴的角速度的可变比率构成了陀螺仪理论的基本原理。这些旋转物体动力学的物理原理可以描述和计算任何陀螺效应以及Tippe顶部反演。