# Optimizing Bifurcations and Isolated Response Curves to Enhance the Performances of Nonlinear Energy Sinks

Adrien Mélot

► To cite this version:

Adrien Mélot. Optimizing Bifurcations and Isolated Response Curves to Enhance the Performances of Nonlinear Energy Sinks. NODYCON 2025 - 4th International Nonlinear Dynamics Conference, Jun 2025, Hoboken, NJ, United States. pp.1-1. hal-05243651

HAL Id: hal-05243651

https://inria.hal.science/hal-05243651v1

Submitted on 7 Sep 2025

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# Optimizing Bifurcations and Isolated Response Curves to Enhance the Performances of Nonlinear Energy Sinks

Adrien Mélot*

*Univ. Gustave Eiffel, Inria, COSYS-SII, I4S team, Rennes, France

**Abstract.** This work introduces a computational framework to enhance the performances of nonlinear energy sinks (NES) by tailoring their bifurcation diagram through direct optimization of bifurcation points and isolated response curves. To this end, an original multi-objective optimization problem is developed. The proposed approach is illustrated on an academic example of nonlinear energy sink.

## Introduction

Nonlinear energy sinks (NES) have proven to be effective for mitigating vibrations in numerous engineering applications. Their inherent nonlinearity provides an efficient mechanism for dissipating the energy of the structure to which they are attached, across a wide range of frequencies. However, this same nonlinearity can also induce adverse dynamics, such as isolated response curves—closed solution branches in the parameter space—that are challenging to predict. These dynamics can result in unanticipated high-amplitude vibrations, which may compromise the system's performance, reliability, or safety. Classical methods for designing NES often rely on analytical rules or numerical optimization of the maximum amplitude of the primary system or time for energy dissipation [1],[2],[3]. These approaches, however, lack a global perspective on the system's dynamics. To address these limitations, this work introduces a computational framework to enhance the performance of nonlinear energy sinks by tailoring both the existence and location of bifurcation points and possible isolated response curves in the bifurcation diagram.

## Results and Discussion

The core of the framework involves multi-parametric optimization to simultaneously control bifurcation points and isolated response curves. The multi-objective problem is regularized by formulating an objective functional as a weighted sum of test functions stemming from bifurcation and singularity theories [4],[5]. We employ the harmonic balance method to solve the dynamics, although the optimization problem is formulated with a high degree of generality and can be easily adapted to other computational methods. The approach is illustrated on an academic NES system, shown in Fig. 1, which consists of a linear primary system of mass $m_1$, linear stiffness $k$ and linear damping $c$ to which a NES of mass $m_2$ is attached. The NES is connected to the primary system through a nonlinear stiffness $k_{nl}$ and a damping term $c_{nl}$, which can be either linear or nonlinear, depending on the specific application, without loss of generality. Particular emphasis is laid on the case of harmonic forcing in a 1:1 resonance scenario. This approach allows for controlling the system's bifurcation structure, making it a powerful tool for the design of NES that can be tailored to meet the specific needs of different vibration mitigation scenarios. This computational approach represents a significant step forward in the optimization and design of nonlinear energy sinks.

<figure><img src="image_2.png" /><figcaption>Figure 1: Illustration of the considered system.</figcaption></figure>

## References

[1] Mattei P.O., Côte R. (2023) Optimization of a dynamic absorber with nonlinear stiffness and damping for the vibration control of a floating offshore wind turbine toy model. J. Theor. Comput. Appl. Mech.

[2] Gourc E., Michon G., Seguy S., Berlioz A. (2014) Experimental Investigation and Design Optimization of Targeted Energy Transfer Under Periodic Forcing. J. Vib. Acoust., **136**(2):021021.

[3] Dekemele K. (2023) Tailored nonlinear stiffness and geometric damping: Applied to a bistable vibration absorber. Int. J. Non-Linear Mech., **157**:104548.

[4] Mélot A., Denimal E., Renson L. (2024) Multi-parametric optimization for controlling bifurcation structures. Proc. R. Soc. A: Math. Phys. Eng. Sci., **480**:20230505.

[5] Mélot A., Denimal E. Goy, Renson L. (2024) Control of isolated response curves through optimization of codimension-1 singularities. Comput. Struct., **299**:107394.