Suppression of Limit Cycle Oscillations using the Nonlinear Tuned Vibration Absorber

The objective of the present study is to mitigate, or even completely eliminate, the limit cycle oscillations in mechanical systems using a passive nonlinear absorber, termed the nonlinear tuned vibration absorber (NLTVA). An unconventional aspect of the NLTVA is that the mathematical form of its restoring force is not imposed a priori, as it is the case for most existing nonlinear absorbers. The NLTVA parameters are determined analytically using stability and bifurcation analyses, and the resulting design is validated using numerical continuation. The proposed developments are illustrated using a Van der Pol-Duffing primary system

A principle of similarity for nonlinear vibration absorbers

This paper develops a principle of similarity for the design of a nonlinear absorber, the nonlinear tuned vibration absorber (NLTVA), attached to a nonlinear primary system. Specifically, for effective vibration mitigation, we show that the NLTVA should feature a nonlinearity possessing the same mathematical form as that of the primary system. A compact analytical formula for the nonlinear coefficient of the absorber is then derived. The formula, valid for any polynomial nonlinearity in the primary system, is found to depend only on the mass ratio and on the nonlinear coefficient of the primary system. When the primary system comprises several polynomial nonlinearities, we demonstrate that the NLTVA obeys a principle of additivity, i.e., each nonlinear coefficient can be calculated independently of the other nonlinear coefficients using the proposed formula

Nonlinear Generalization of Den Hartog's Equal-Peak Method

This study addresses the mitigation of a nonlinear resonance of a mechanical system. In view of the narrow bandwidth of the classical linear tuned vibration absorber, a nonlinear absorber, termed the nonlinear tuned vibration absorber (NLTVA), is introduced in this paper. An unconventional aspect of the NLTVA is that the mathematical form of its restoring force is tailored according to the nonlinear restoring force of the primary system. The NLTVA parameters are then determined using a nonlinear generalization of Den Hartog's equal-peak method. The mitigation of the resonant vibrations of a Duffing oscillator is considered to illustrate the proposed developments

Performance, robustness and sensitivity analysis of the nonlinear tuned vibration absorber

The nonlinear tuned vibration absorber (NLTVA) is a recently-developed nonlinear absorber which generalizes Den Hartog's equal peak method to nonlinear systems. If the purposeful introduction of nonlinearity can enhance system performance, it can also give rise to adverse dynamical phenomena, including detached resonance curves and quasiperiodic regimes of motion. Through the combination of numerical continuation of periodic solutions, bifurcation detection and tracking, and global analysis, the present study identifies boundaries in the NLTVA parameter space delimiting safe, unsafe and unacceptable operations. The sensitivity of these boundaries to uncertainty in the NLTVA parameters is also investigated.Comment: Journal pape

Coexisting attractors in floating body dynamics undergoing parametric resonance

This study pertains to analysing the dynamical behaviour of a floating body undergoing parametric resonances. A simple vertical cylinder, representing a classical spar-buoy, is considered, limiting its motion to heave and pitch degrees of freedom. Its geometry and mass distribution are chosen such that a 2:1 ratio of heave to pitch/roll natural frequency makes the spar-buoy prone to parametric resonance. The system is then studied by the shooting method, combined with a pseudo-arclength continuation, and the harmonic balance procedure. Results show that an extensive bistable region exists, where stable parametric resonance coexists with a regular resonance response. The analysis also unveiled the existence of stable quasiperiodic motions existing in correspondence of both pitch and heave resonance. Results are qualitatively validated using a model based on the explicit nonlinear Froude–Krylov force calculation

analytical investigation of single and double neimark sacker bifurcations

The analytical investigation of bifurcations is a very challenging task for many applied scientists and engineers. Often, numerical simulations cannot clarify the complicated dynamics of mechanical systems, in this cases, preprogrammed softwares can be of valid help during the investigation. Also, in the literature, methodology to study bifurcations are presented for most of the cases. However, the presented procedures, are often very hard to be understood from applied scientists with low mathematical background. In this paper we present in details the typical procedure to analyze single and double Neimark-Sacker bifurcations. Especially regarding the double Neimark-Sacker bifurcations of maps, very few sources can be found in the literature, although this kind of bifurcation is very common in many dynamical systems

Passive Linearization of Nonlinear System Resonances

In this work we demonstrate that the addition of properly-tuned nonlinearities to a nonlinear system can increase the range over which a specific resonance responds linearly. Specifically, we seek to enforce two important properties of linear systems, namely the force-displacement proportionality and the invariance of resonance frequencies. Theoretical findings are validated through numerical simulations and experiments