Piecewise constant response of underdamped oscillators through suppression of overshoots and undershoots in aerospace, civil, and mechanical systems

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.