By transforming a 1D second‐order linear oscillator into a 2D first‐order polar motion differential equation, it can be shown that the finite smoothness (i.e., the presence of jump in finite order derivatives) of the applied Newtonian forcing constitutes the sufficient and necessary condition for instantaneous excitation of free eigen‐mode. This condition can be met by forcing functions originated from turbulent and multiphase fluid motions. Sub‐macroscopic transition time associated with astatic elastic deformation limits the physical smoothness of the applied forcing for the Earth's polar motion. Eigen‐modes can also be excited by an infinitely smooth forcing that has a finite domain of non‐zero values. The eigen‐period serves as a macroscopic timescale to characterize the inertia of a linear oscillator. If a zero mean irregular forcing of finite smoothness exhibits a high degree of randomness and the timescale is much shorter than the eigen‐period, then for negligible damping the eigen‐waveform will increase in proportion to the squareroot of time, while the waveform distortion is statistically a constant. As a result, the pattern of distinctive eigen‐oscillation will dominate the forced solution for longer enough duration.

中文翻译：

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关于线性振荡器的本征模激励和地球的极运动

通过将一维二阶线性振荡器转换为二维一阶极移运动微分方程，可以证明，所应用的牛顿力的有限光滑度（即，有限阶导数中存在跳跃）构成了充分必要的条件。自由本征模态瞬时激发的条件 可以通过强制源于湍流和多相流体运动的函数来满足此条件。与静态弹性变形相关的亚宏观过渡时间限制了地球极运动施加的力的物理平滑度。本征模式也可以通过具有非零值的有限域的无限平滑强迫来激发。本征周期用作宏观时间尺度，用以表征线性振荡器的惯性。如果有限平滑度的零均值不规则强迫表现出高度的随机性，并且时间尺度比本征周期短得多，则对于可忽略的阻尼，本征波形将与时间的平方根成比例地增加，而波形失真为统计上是常数。结果，独特的本征振动模式将在足够长的持续时间内主导强迫解决方案。