Abstract Summary
In the area of passive vibration mitigation, nonlinear energy sinks (NESs) have been extensively studied. They are an alternative to traditional tuned mass-dampers as they can mitigate vibrations over larger frequency ranges while being efficient in both transient and stationary regimes presenting period and/or non period responses. NESs are characterized by nonlinear stiffness properties (e.g., cubic stiffness), leading, in some cases, to irreversible transfers of energy between the main system and a NES. The literature on nonlinear absorbers contains several variations on the type of nonlinearity used. However, in all cases, nonlinear properties are mainly constant over time. In this work, a nonlinear absorber with a time-dependent nonlinear stiffness is considered. The nonlinearity, which can be adjusted in an acoustical application, follows a specific functional form, which is enforced through a control system. The objective of this work is to optimize the characteristics of the nonlinear absorber, including the time varying nonlinear stiffness. However, the optimal design of nonlinear absorbers is known to present several challenges. Specifically, their behavior can be acutely sensitive to uncertainties. In fact, it is well known that a NES exhibits a discontinuity in its response due to the presence of an activation threshold. This discontinuity is typically associated with significantly different dynamic behaviors that can be reached for small perturbations of the loading conditions or the design. This work presents a dedicated stochastic design optimization algorithm for the proposed absorber with time-varying nonlinear stiffness. It is tailored to identify and tackle the discontinuous behavior during the optimization process. The approach is based on support vector machine (SVM), clustering, Gaussian processes, and adaptive sampling. One of the key elements of the approach is the approximation of the absorber activation threshold through an SVM classifier. The stochastic design optimization enables the computation of probabilistic constraints, expected value and variance of the response. Both uncertainties in ``design” variables and loading conditions are included in the optimization. In addition, revealing fast and slow system dynamics which leads to detection of Slow Invar-iant Manifolds (SIMs), equilibrium and singular points are analytically/numerically derived through the Manevitch complexification approach. SIMs and characteristic points, corresponding to various designs, are used to physically interpret the optimal solutions and their robustness to perturbations in loading conditions
