论文标题
具有内部液体腔的旋转顶部的非线性稳定性分析
Nonlinear Stability Analysis of a Spinning Top with an Interior Liquid-Filled Cavity
论文作者
论文摘要
考虑耦合系统的运动，$ \ mathscr s $，由（非必要对称）顶部，$ \ mathscr b $，带有内部空腔，$ \ mathscr c $构成，完全填充了Navier-Stokes liquid，$ \ naverscr l $。 $ \ mathscr s $的特定稳态运动$ \ bar {\ sf s} $（例如）是$ \ mathscr l $相对于$ \ mathscr b $而息息（{\ em直立旋转顶部}）。然后，我们通过粗略地显示$ \ bar {\ sf s} $的非线性稳定性的完全表征，大概是当$ \ bar {\ sf s} $稳定时，仅当$ | \ bar {\ vome {\vΩ} | $就足够大，所有其他物理参数均已固定。此外，我们表明，与$ \ Mathscr c $为空的情况不同，在上述稳定条件下，顶部最终将返回到不受干扰的直立配置。
Consider the motion of the the coupled system, $\mathscr S$, constituted by a (non-necessarily symmetric) top, $\mathscr B$, with an interior cavity, $\mathscr C$, completely filled up with a Navier-Stokes liquid, $\mathscr L$. A particular steady-state motion $\bar{\sf s}$ (say) of $\mathscr S$, is when $\mathscr L$ is at rest with respect to $\mathscr B$, and $\mathscr S$, as a whole rigid body, spins with a constant angular velocity $\bar{\Vω}$ around a vertical axis passing through its center of mass $G$ in its highest position ({\em upright spinning top}). We then provide a completely characterization of the nonlinear stability of $\bar{\sf s}$ by showing, roughly speaking, that $\bar{\sf s}$ is stable if and only if $|\bar{\Vω}|$ is sufficiently large, all other physical parameters being fixed. Moreover we show that, unlike the case when $\mathscr C$ is empty, under the above stability conditions, the top will eventually return to the unperturbed upright configuration.
