Relation between posicast-shaped function and vibration manipulation function of undamped 1-DOF linear oscillator for one-time rest-to-rest motion in a natural period

Abstract

Although more than 55 years have been passed since its first development, it remains unknown whether a posicast-shaped function (PSF) is a unique intermittent input command for a one-time rapid rest-to-rest motion in one natural period. In addition, it has not been determined whether more generalized expressions exist. We recently deduced an intermittent input command of a feedforward formula, called a vibration manipulation function for one natural period (oVMF), to manipulate an undamped 1-DOF linear floating oscillator in an arbitral operational period based on the dynamics of three vibro-impact oscillators under a Grover algorithm in the field of quantum information. As a special case of oVMF, we can define the formula of an input command for a restto-rest motion in one natural period, called roVMF. In this study, we proved that any PSF for an undamped 1-DOF oscillator (uPSF) can be expressed as an infinite series of roVMF. Moreover, because uPSF has a stricter symmetrical property than roVMF, we have concluded that the set of uPSF, Uup, is contained in the set of roVMF, UroV. Hence roVMF is a more general expression than uPSF.