Optimum Design of a Nonlinear Vibration Absorber Coupled to a Resonant Oscillator: A Case Study

Abstract

This paper presents the optimal design of a passive autoparametric cantilever beam vibration absorber for a linear mass-spring-damper system subject to harmonic external force. The design of the autoparametric vibration absorber is obtained by using an approximation of the nonlinear frequency response function, computed via the multiple scales method. Based on the solution given by the perturbation method mentioned above, a static optimization problem is formulated in order to determine the optimum parameters (mass and length) of the nonlinear absorber which minimizes the steady state amplitude of the primary mass under resonant conditions; then, a PZT actuator is cemented to the base of the beam, so the nonlinear absorber is made active, thus enabling the possibility of controlling the effective stiffness associated with the passive absorber and, as a consequence, the implementation of an active vibration control scheme able to preserve, as possible, the autoparametric interaction as well as to compensate varying excitation frequencies and parametric uncertainty. Finally, some simulations and experimental results are included to validate and illustrate the dynamic performance of the overall system.

Free and forced vibrations of elastically restrained cantilever with lumped oscillator

“…Impact of secondary oscillators upon dynamic responses of vibration suppressors was conducted by Ma et al [14]. Nonlinear effects of an oscillator upon dynamic responses of a vibration absorber has been published by Abundis et al [15]. Paunovic et al [16] studied the vibration analysis of viscoelastic microbeams with attached tip mass under base excitations.…”

Oscillation Attenuation in a Building-like Structure by Using a Flexible Vibration Absorber

“…On the other hand, configuration (c) corresponds to an autoparametric system, where the dynamic model is represented by a system of nonlinear differential equations that cannot be linearized due to the decoupling between the degree of freedom of the primary system and the dynamics of the absorber to be implemented [22,23]. This kind of configuration is not considered in the present work due to it having been widely studied in the nonlinear-vibrations literature, addressing passive and active control aspects for different hosting structures [24][25][26][27][28][29][30][31][32][33][34][35][36][37].…”

On-Line Modal Parameter Identification Applied to Linear and Nonlinear Vibration Absorbers

“…It is important to note that ( 5) is clearly a parametric type equation whereẍ acts as a coefficient of the coordinate y. However, according to the tuning condition between an autoparametric vibration absorber and a mechanical oscillator, the following expressions must be satisfied (for more details see [16])…”

“…Hui and Ng [15] presented the implementation of autoparametric phenomena to reduce symmetrical vibration of a curved beam/panel under external harmonic excitation showing that internal energy transfer of a first symmetric mode into first anti-symmetric mode in a curved panel is one example of autoparametric vibration absorber effect. Abundis-Fong et al [16] developed an optimum design of an autoparametric absorber (cantilever beam configuration) coupled to a resonant oscillator where the implementation of the nonlinear absorber was obtained by using an approximation of the nonlinear frequency response function, computed via a perturbation method. Recently, Ting Tan et al [17] used the nonlinear saturation principle and 1:2 internal resonance in the design of the piezoelectric autoparametric vibration absorber for vibration suppression and energy harvesting.…”