Stochastic optimization of nonlinear energy sinks

Abstract

Nonlinear energy sinks (NES) are a promising technique to achieve vibration mitigation. Through nonlinear stiffness properties, NES are able to passively and irreversibly absorb energy. Unlike the traditional Tuned Mass Damper (TMD), NES absorb energy from a wide range of frequencies. Many studies have focused on NES behavior and dynamics, but few have addressed the optimal design of NES. Design considerations of NES are of prime importance as it has been shown that NES dynamics exhibit an acute sensitivity to uncertainties. In fact, the sensitivity is so marked that NES efficiency is near-discontinuous and can switch from a high to a low value for a small perturbation in design parameters or loading conditions. This article presents an approach for the probabilistic design of NES which accounts for random design and aleatory variables as well as response discontinuities. In order to maximize the mean efficiency, the algorithm is based on the identification of regions of the design and aleatory space corresponding to markedly different NES efficiencies. This is done through a sequence of approximated sub-problems constructed from clustering, Kriging approximations, a support vector machine, and Monte-Carlo simulations. The refinement of the surrogates is performed locally using a generalized max-min sampling scheme which accounts for the distributions of random variables. The sampling scheme also makes use of the predicted variance of the Kriging surrogates for the selection of aleatory variables values. The proposed algorithm is applied to three example problems of varying dimensionality, all including an aleatory excitation applied to the main system. The stochastic optima are compared to NES optimized deterministically.

Appendix

1.1 Clustering

It is well known that clustering can be a tedious process since finding “optimal” clusters is dependent on the criterion used. Although clustering would seem to be straightforward in the case of two clusters due to discontinuities, the NES case presents difficulties when clustering is left fully unsupervised. For example, in some cases, the K-means algorithm (Hartigan and Wong 1979) can generate non-meaningful clusters. Figure 16 provides such an example of poor clustering for $E_{NES_{inf}}$ as a function of the cubic stiffness α. The figure clearly shows that a plateau of low efficiency points are classified as efficient by the clustering algorithm.

Fig. 16

Clustering result from K-means algorithm with 2 clusters applied to $E_{NES_{inf}}$ values

However, this difficulty can be overcome by injecting some a priori knowledge from the physics of NES. Indeed, some NES configurations can be automatically classified as inefficient, thus leading to a much clearer discontinuity. The discarded configurations are those with an $E_{NES_{inf}}$ lower than that obtained when the cubic stiffness is minimal. K-means is then applied to the remaining subset of samples. The lower efficiency points as designated by K-means are combined with the previously discarded points to form one cluster, while the remaining points with higher $E_{NES_{inf}}$ values make up the other cluster. This procedure is illustrated in Figs. 16 and 17. Note that clustering is performed based only on the energy dissipated, so the rightmost green triangular points are classified as inefficient. In higher dimensions, the $E_{NES_{inf}}$ for the minimum α is still used while the other variables are set to their means.

Fig. 17

Clustering result from clustering scheme applied to $E_{NES_{inf}}$ values. First points below threshold (red) are removed, then K-means algorithm with 2 clusters is applied to remaining points