When a single-degree-of-freedom underdamped system is subjected to a step function force in order that it eventually acquires a constant displacement, its response shows undershoots and overshoots. Since a step function force is difficult, if not impossible, to apply to many aerospace, civil, and mechanical engineering systems because of their inertias, this paper looks at simple forces that can be generated from a practical standpoint and are not instantaneously applied, that cause an underdamped oscillator to acquire a constant displacement without any overshoots and/or undershoots. These forces are ramped up (or down) over a short duration of time and held constant thereafter. A preliminary approach to the development of such force–time histories is presented by using a force given by a polynomial in time. Open-loop optimal control is next considered, and then closed-loop optimal control. The optimal control problem does not fall within the standard rubric of terminal control and a new approach for doing this is developed. These ideas are then woven into the development of a methodology that allows an undamped/underdamped single degree of freedom system to track a desired piecewise constant displacement time-history using forces that do not need to be instantaneously applied and that generate rippleless responses with no overshoots/undershoots. The methodology is also applicable to classically damped multi-degree-of-freedom systems with underdamped modes of vibration.

中文翻译：

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通过抑制航空航天、民用和机械系统中的过冲和下冲，欠阻尼振荡器的分段恒定响应

当单自由度欠阻尼系统受到阶跃函数力以使其最终获得恒定位移时，其响应会出现下冲和过冲。由于阶跃函数力由于惯性而难以（如果不是不可能）应用于许多航空航天、土木和机械工程系统，因此本文着眼于可以从实际角度生成且不能立即应用的简单力，即导致欠阻尼振荡器获得恒定位移，没有任何过冲和/或下冲。这些力会在很短的时间内上升（或下降）并在此后保持恒定。通过使用由时间多项式给出的力，提出了发展这种力-时间历程的初步方法。接下来考虑开环最优控制，然后考虑闭环最优控制。最优控制问题不属于终端控制的标准范畴，为此开发了一种新方法。然后将这些想法融入到一种方法的开发中，该方法允许无阻尼/欠阻尼单自由度系统使用不需要立即施加的力来跟踪所需的分段恒定位移时间历程，并产生无过冲的无波纹响应/ 不足。该方法也适用于具有欠阻尼振动模式的经典阻尼多自由度系统。然后将这些想法融入到一种方法的开发中，该方法允许无阻尼/欠阻尼单自由度系统使用不需要立即施加的力来跟踪所需的分段恒定位移时间历程，并产生无过冲的无波纹响应/ 不足。该方法也适用于具有欠阻尼振动模式的经典阻尼多自由度系统。然后将这些想法融入到一种方法的开发中，该方法允许无阻尼/欠阻尼单自由度系统使用不需要立即施加的力来跟踪所需的分段恒定位移时间历程，并产生无过冲的无波纹响应/ 不足。该方法也适用于具有欠阻尼振动模式的经典阻尼多自由度系统。