Suppression of motion-induced residual longitudinal vibration of an elastic rod by input shaping

Abstract

Various input-shaping schemes such as the "zero-vibration" (ZV), "zero-vibration-andderivative" (ZVD), "negative ZV" (NZV), and "negative ZVD" (NZVD) schemes have previously been proposed to suppress motion-induced residual vibration of lightly damped structures. In such schemes, the input command of the dynamical system in question is properly administered (i.e., shaped), so that the dominant induced vibration modes are annihilated through destructive interference. Here we are concerned with the effects of system payload on the vibration reduction capabilities of the aforementioned input-shaping schemes, especially when they are applied to continuous systems. By use of the simple structure of a linearly elastic rod as a specific example, it is demonstrated that both the minimum achievable residual vibration amplitude and the tolerance of detuning parameter errors of the inputshaping schemes are sensitive to the amount of payload on the system. It is therefore imperative to take the factor of system payload into account in the design of practical input shapers.

Asymptotic analysis of a two-step input-shaping scheme for suppressing motion-induced residual vibration of nonlinear mechanical systems

“…Moreover, using (10), (11) and (12), we find y = 1 − + 2 + O( 3 ); the exact result is y = 1/(1 + ), as can be readily obtained by solving (9). From (14), (15) and (16), it is calculated that the half-period of the system response to (13) gives the exact expression T y = T 0 A /2. The exact result simply is a reminder of the elementary fact that the intrinsic vibration period of a linear SDOF system is independent of the vibration amplitude.…”

“…The execution times of the second and the third force steps can be calculated accordingly, using (8) with similar expressions for the kinetic energy. It is intuitively clear that an input force having more steps requires a longer task time, but would excite less residual vibration when the potential function used to determine the input force contains an estimation error (see [14] for example). The residual vibration excited by a two-step input force with slightly incorrect force magnitudes and task time (due to the potential function estimation error) is calculated in the next section.…”

“…the last equality is inserted by use of (14). Exploiting the smallness of y − z (and hence of y − x), we then expand the integrand in the first line of (17) in powers of y − x, and evaluate the integral there accordingly, yielding…”

“…During the past two decades or so, extensive research on input shaping has been done in robotics, flexible-arm control, gantry-crane control, microelectromechanical systems, and various other applications (see, for example, [2][3][4][5][6][7][8][9][10][11][12][13][14], and references cited therein). In such studies, the system dynamics generally is modeled by linear differential equations, and a number of input-shaping schemes for suppressing residual vibrations have been proposed.…”

“…Briefly, some schemes are more tolerant of system parameter errors (which are almost inevitable in practice), while some require a shorter task time (a desirable property for input shaping), than others. A comprehensive review of earlier works on input shaping is given in the doctoral dissertation of Singhose [7], and a brief discussion of some common input-shaping schemes can be found in the work of Yang et al [14] on the suppression of residual longitudinal vibration of an elastic rod.…”