Probabilistic optimization of vibrational systems under stochastic excitation containing extreme forcing events
Author(s)Joo, Han Kyul
Massachusetts Institute of Technology. Department of Mechanical Engineering.
Themistoklis P. Sapsis.
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For the past few decades there has been an increased interest for efficient quantification schemes of the response statistics of vibrational systems operating in stochastic settings with the aim of providing optimal parameters for design and/or operation. Examples include energy harvesting configurations from ambient vibrations and stochastic load mitigation in vibrational systems. Although significant efforts have been made to provide computationally efficient algorithms for the response statistics, most of these efforts are restricted to systems with very specific characteristics (e.g. linear or weakly nonlinear systems) or to excitations with very idealized form (e.g. white noise or deterministic periodic). However, modern engineering applications require the analysis of strongly nonlinear systems excited by realistic loads that have radically different characteristics from white noise or periodic signals. These systems are characterized by essentially non-Gaussian statistics (such as bimodality of the probability distributions, heavy tails, and non-trivial temporal correlations) caused by the nonlinear characteristics of the dynamics, the correlated (non-white noise) structure of the excitation, and the possibility of non-stationary forcing characteristics (intermittency) related to extreme events. In this thesis, we first address the problem of deriving semi-analytical approximations for the response statistics of strongly nonlinear systems subjected to stationary, correlated (colored) excitation. The developed method combines two-times moment equations with new non-Gaussian closures that reflect the underlying nonlinear dynamics of the system. We demonstrate how the proposed approach overcomes the limitations of traditional statistical linearization schemes and can approximate the statistical steady state solution. The new method is applied for the analysis of bistable energy harvesters with mechanical and electromagnetic damping subjected to correlated excitations. It allows for the computation of semi-analytical expressions for the non-Gaussian probability distributions of the response and the temporal correlation functions, with minimal computational effort involving the solution of a low-dimensional optimization problem. The method is also assessed in higher-dimensional problems involving linear elastic rods coupled to a nonlinear energy harvester. In the second part of this thesis, we consider the problem of mechanical systems excited by stochastic loads with non-stationary characteristics, modeling extreme events. Such excitations are common in many environmental settings and they lead to heavy-tailed probability distribution functions. For both design and operation purposes it is important to efficiently quantify these high-order statistical characteristics. To this end, we apply a recently developed approach, the probabilistic decomposition-synthesis (PDS) method. Under suitable but sufficiently generic assumptions, the PDS method allows for the probabilistic and dynamic decoupling of the regime associated with extreme events from the "background" fluctuations. Using this approach we derive fully analytical formulas for the heavy tailed probabilistic distribution of linear structural modes subjected to stochastic excitations containing extreme events. The derived formulas can be evaluated with very small computational cost and are shown to accurately capture the complicated heavy-tailed and asymmetrical features in the probability distribution many standard deviations away from the mean. We finally extend the scheme to quantify the response statistics of nonlinear multi-degree-of-freedom systems under extreme forcing events, emphasizing again accurate heavy-tail statistics. The developed scheme is applied for the design and optimization of small mechanical attachments that can mitigate and suppress extreme forcing events delivered to a primary system. Specifically, we consider the suppression of extreme impacts due to slamming in high speed craft motion via optimally designed nonlinear springs/attachments. The very low computational cost for the quantification of the heavy tail structure of the response allows for direct optimization on the nonlinear characteristics of the attachment. Based on the results of this optimization we propose a new asymmetric nonlinear spring that far outperforms optimal cubic springs and tuned mass dampers, which have been used in the past. Accuracy of the developed method is illustrated through direct comparisons with Monte-Carlo simulations.
Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mechanical Engineering, 2017.Cataloged from PDF version of thesis.Includes bibliographical references (pages 221-232).
DepartmentMassachusetts Institute of Technology. Department of Mechanical Engineering.
Massachusetts Institute of Technology