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Phase space structure of spinning disks

机译：旋转盘的相空间结构

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Using a Hamiltonian formalism and a sequence of canonical transformations, we show that the ordinary differential equations associated with the forced oscillations of rotating circular disks admit the first integral of motion. This reduces the phase space dimension of the governing equations from five to three. The phase space flows of the reduced system are then visualized using Poincare maps. Our results show that single mode oscillations of rotating disks are subject to chaotic behavior through the emergence of higher-order resonant islands that surround fundamental periodic cycles. We extend our new formalism to imperfect disks and construct adiabatic invariants near to and far from resonances. For low-speed imperfect disks, we find a new kind of bifurcations of the phase space flows as the system parameters vary. We study the effect of structural damping using Hamilton's principle for non-conservative systems and reveal the existence of asymptotically stable limit cycles for the damped system near the 1:1 resonance. We show that a low-speed disk is eventually flattened due to damping effect.
机译：使用哈密顿形式论和一系列规范变换，我们证明了与旋转圆盘的强迫振动相关的常微分方程允许运动的第一积分。这将控制方程的相空间维从五个减小为三个。然后使用庞加莱图可视化简化系统的相空间流。我们的结果表明，旋转盘的单模振荡会通过围绕基本周期周期的高阶共振岛的出现而受到混沌行为的影响。我们将新的形式主义扩展到不完善的磁盘，并构造接近和远离共振的绝热不变量。对于低速不完美磁盘，随着系统参数的变化，我们发现了一种新型的相空间流分支。我们使用汉密尔顿原理研究非保守系统的结构阻尼效应，并揭示了阻尼系统在1：1共振附近的渐近稳定极限环的存在。我们显示，低速磁盘最终由于阻尼效应而变平。

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