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Dynamics analysis of a nonlinear energy sink for passive suppression of a parametrically excited system

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Abstract

Nonlinear energy sinks (NESs) have been extensively studied to develop passive suppression strategies, with the primary objective of minimizing hazardous oscillatory responses in structures. In this work, we investigate the dynamical regimes of a parametrically excited one-degree-of-freedom system with a rotary NES (RNES) acting as a passive suppressor. By performing numerical pseudo-arclength continuations we determine the comprehensive local bifurcation scenario and illustrate, through locus maps, the impact of various RNES parameters. We identify configurations of the parametric excitation amplitude, mass, and absorber radius that result in stable vibration ranges. The dynamic scenario necessitates a precise adjustment of the RNES characteristics, tailored for either passive suppression or energy harvesting applications. Finally, we assess the resilience of the suitable vibration regions by examining the global dynamics. Basins of attraction display a fractal form, indicating a high sensitivity of the response to initial conditions.

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The authors may provide the numerical codes after the analysis of requests.

Abbreviations

RNES:

Rotary nonlinear energy sink

BPC:

Branching point cycles

PD:

Period-doubling

LPC:

Limit-point cycles

NS:

Neimark–Sacker

M :

Mass of the main structure

D :

Characteristic length of the main structure cross-section

c :

Damping constant associated with the main structure

\(\bar{k}\) :

Time-average value of the main structure’s stiffness

\(\Delta k\) and \(\Omega\) :

Amplitude and frequency of the stiffness variation, respectively

Y :

Dimensional displacement of the main structure

\(\theta\) :

Angular position of the RNES

\(\omega _{n,y}=\sqrt{\frac{\bar{k}}{M+m}}\) :

Reference frequency

m, r and \(c_\theta\) :

Mass, radius and damping of the RNES, respectively

\(y=Y/D\) :

Dimensionless displacement of the main structure

\(\hat{m}=\frac{m}{M}\) :

Dimensionless suppressor mass

\(\hat{r}=\frac{r}{D}\) :

Dimensionless suppressor radius

\(\zeta _y=\frac{c}{2(M+m)\omega _{n,y}}\) :

Damping ratio associated with the main structure

\(\zeta _\theta =\frac{c_\theta }{2mr^2 \omega _{n,y}}\) :

RNES damping ratio

\(\delta =\frac{\Delta k}{\bar{k}}\) :

Dimensionless parametric excitation amplitude

\(n=\frac{\Omega }{\omega _{n,y}}\) :

Dimensionless parametric excitation frequency

\(y_{max}\) :

Maximum displacement of the main structure obtained in the periodic orbit

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Acknowledgements

The first author acknowledges the São Paulo Research Foundation (FAPESP) for the financial support given during his period as a visiting research at Polytechnic University of Marche, grant 2019/15046-2 and the Brazilian Research Council (CNPq) for the grant 305945/2020-3. FAPESP is also acknowledged for sponsoring a Thematic Project that focuses on the study of nonlinear dynamics applied to engineering systems (grant 2022/00770-0).

Funding

This paper results from funding from the São Paulo Research Foundation (FAPESP), Grants 2019/15046-2 and 2022/00770-0. The Brazilian Research Council (CNPq) also supported the research (Grant 305945/2020-3).

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Authors and Affiliations

  1. Offshore Mechanics Laboratory (LMO), Escola Politécnica University of São Paulo, Av. Prof. Lucio Martins Rodrigues, São Paulo, 05508-020, Brazil

    Guilherme Rosa Franzini

  2. Department of Construction, Civil Engineering and Architecture, Polytechnic University of Marche, Via Brecce Bianche, 60131, Ancona, Italy

    Pierpaolo Belardinelli & Stefano Lenci

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  1. Guilherme Rosa Franzini
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Correspondence to Guilherme Rosa Franzini.

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Appendix: Additional three-dimensional basins of attraction

Appendix: Additional three-dimensional basins of attraction

Figure 13 shows 3D sections of the 4D basins of attraction on the space \(y(0)\times \dot{y}(0) \times \theta (0)\) for different values of initial angular velocities, namely \(\dot{\theta }(0)\) = \(-0.07\), 0.07, 0.35 and 0.5.

Fig. 13
figure 13

3D sections of the calculated 4D basins. In the caption the section position in the space \(y(0)\times \dot{y}(0) \times \theta (0) \times \dot{\theta }(0)\)

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Franzini, G.R., Belardinelli, P. & Lenci, S. Dynamics analysis of a nonlinear energy sink for passive suppression of a parametrically excited system. Meccanica 59, 703–715 (2024). https://doi.org/10.1007/s11012-024-01812-3

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