A Comparison between Active and Passive Vibration Control of Non-Linear Simple Pendulum. Part II: Longitudinal Tuned Absorber and Negative Gφ and Gφn Feedback

Abstract

In part I [1] we dealt with a tuned absorber, which can move in the transversally direction, where it is added to an externally excited pendulum. Active control is applied to the system via negative velocity feedback or its square or cubic value. The multiple time scale perturbation technique is applied throughout. An approximate solution is derived up to second order approximation. The stability of the system is investigated applying both frequency response equations and phase plane methods. The effects of the absorber on system behavior are studied numerically. Optimum working conditions of the system are obtained applying passive and active control methods. Both control methods are demonstrated numerically. In this paper, a tuned absorber, in the longitudinal direction, is added to an externally excited pendulum. Active control is applied to the system via negative acceleration feedback or via negative angular displacement or its square or cubic value. An approximate solution is derived up to the second order approximation for the system with absorber. The stability of the system is investigated applying both frequency response equations and phase plane methods. The effects of the absorber on system behavior are studied numerically. Optimum working conditions of the system are extracted when applying both passive and active control methods.

Vibration reduction of multi-parametric excited spring pendulum via a transversally tuned absorber

“…represents the complex conjugate of the previous terms. Substituting (12) into (10) and using the response condition equation (4) leads to secular terms. Eliminating these secular terms leads to solvability for the first-order approximation:…”

“…Eissa [9] reported that one of the most effective tools for the control of passive vibration is the dynamic absorber or the damper or the neutralizer. Vibration and dynamic chaos of both simple pendulum and spring pendulum that represent ship's roll motion have been studied by Eissa and Sayed [10][11][12][13][14]. The effects of the transverse and longitudinal tuned absorbers on the vibrations of both systems under single harmonic excitation have been studied numerically to control the oscillations of both systems.…”

Vibration suppression in a twin-tail system to parametric and external excitations

“…Eissa and Sayed [10,11] made a comparison between the active and the passive vibration control of a simple pendulum described by a second order nonlinear differential equation having both quadratic and cubic nonlinearities. They controlled the system applying either nonlinear absorber (passive control) or negative velocity feedback or its square or cubic value (active control).…”

Vibration suppression in ultrasonic machining described by non-linear differential equations

“…They applied saturation values of different parameters as optimum working conditions for vibration suppression of the cantilever. Eissa et al [16,17] presented tuned absorbers in both transverse and longitudinal directions of a simple pendulum which was designed to control one frequency at primary resonance. They demonstrated the effectiveness of the absorber for passive control.…”