标题： 参数转子和摆的稳态 显示英文标题

标题： Steady states of the parametric rotator and pendulum

摘要： 我们讨论了有阻尼的平面参数旋转器和摆的几种稳态旋转和振荡模式。 我们考虑了旋转器和摆的悬挂点的一般椭圆轨迹，对于摆而言，其与重力成任意角度，线性和圆形轨迹作为特殊情况。 我们对参数旋转器和摆的阻尼非线性运动方程进行了微扰处理，适用于小参数激励和阻尼的情况，尽管我们的微扰方法也可以扩展到其他区域。 我们的处理仅涉及常系数的二阶常微分方程，并以基本函数的形式提供了数值精确的微扰解。 这里研究的一些稳态旋转和振荡模式在以前的文献中未被讨论过。 其他众所周知的模式，如参数共振和倒置摆，被扩展到相对于重力倾斜的椭圆参数激励。 本文的结果应可供高年级本科生理解，并对研究生和非线性力学领域的专家具有参考价值。 显示英文摘要

摘要： We discuss several steady-state rotation and oscillation modes of the planar parametric rotator and pendulum with damping. We consider a general elliptic trajectory of the suspension point for both rotator and pendulum, for the latter at an arbitrary angle with gravity, with linear and circular trajectories as particular cases. We treat the damped, non-linear equation of motion of the parametric rotator and pendulum perturbatively for small parametric excitation and damping, although our perturbative approach can be extended to other regimes as well. Our treatment involves only ordinary second-order differential equations with constant coefficients, and provides numerically accurate perturbative solutions in terms of elementary functions. Some of the steady-state rotation and oscillation modes studied here have not been discussed in the previous literature. Other well-known ones, such as parametric resonance and the inverted pendulum, are extended to elliptic parametric excitation tilted with respect to gravity. The results presented here should be accessible to advanced undergraduates, and of interest to graduate students and specialists in the field of non-linear mechanics.