Performance measures for single-degree-of-freedom energy harvesters under stochastic excitation

We develop performance criteria for the objective comparison of different classes of single-degree-of-freedom oscillators under stochastic excitation. For each family of oscillators, these objective criteria take into account the maximum possible energy harvested for a given response level, which is a quantity that is directly connected to the size of the harvesting configuration. We prove that the derived criteria are invariant with respect to magnitude or temporal rescaling of the input spectrum and they depend only on the relative distribution of energy across different harmonics of the excitation. We then compare three different classes of linear and nonlinear oscillators and using stochastic analysis tools we illustrate that in all cases of excitation spectra (monochromatic, broadband, white-noise) the optimal performance of all designs cannot exceed the performance of the linear design. Subsequently, we study the robustness of this optimal performance to small perturbations of the input spectrum and illustrate the advantages of nonlinear designs relative to linear ones.Comment: 24 pages, 12 figure

A moment-equation-copula-closure method for nonlinear vibrational systems subjected to correlated noise

We develop a moment equation closure minimization method for the inexpensive approximation of the steady state statistical structure of nonlinear systems whose potential functions have bimodal shapes and which are subjected to correlated excitations. Our approach relies on the derivation of moment equations that describe the dynamics governing the two-time statistics. These are combined with a non-Gaussian pdf representation for the joint response-excitation statistics that has i) single time statistical structure consistent with the analytical solutions of the Fokker-Planck equation, and ii) two-time statistical structure with Gaussian characteristics. Through the adopted pdf representation, we derive a closure scheme which we formulate in terms of a consistency condition involving the second order statistics of the response, the closure constraint. A similar condition, the dynamics constraint, is also derived directly through the moment equations. These two constraints are formulated as a low-dimensional minimization problem with respect to unknown parameters of the representation, the minimization of which imposes an interplay between the dynamics and the adopted closure. The new method allows for the semi-analytical representation of the two-time, non-Gaussian structure of the solution as well as the joint statistical structure of the response-excitation over different time instants. We demonstrate its effectiveness through the application on bistable nonlinear single-degree-of-freedom energy harvesters with mechanical and electromagnetic damping, and we show that the results compare favorably with direct Monte-Carlo Simulations

Closure Schemes for Nonlinear Bistable Systems Subjected to Correlated Noise: Applications to Energy Harvesting from Water Waves

The moment equation closure minimization (MECM) method has been developed for the inexpensive approximation of the steady-state statistical structure of bistable systems, which have bimodal potential shapes and which are subjected to correlated excitation. Our approach relies on the derivation of moment equations that describe the dynamics governing the two-time statistics. These are then combined with a closure scheme that arises from a non-Gaussian probability density function (PDF) representation for the joint response-excitation statistics. We demonstrate its effectiveness through the application on a bistable nonlinear single-degree-of-freedom (SDOF) ocean wave energy harvester with linear damping, and the results compare favorably with direct Monte Carlo simulations.Samsung Scholarship Program (grant “Nonlinear Energy Harvesting From Broad-Band Vibrational Sources By Mimicking Turbulent Energy Transfer Mechanisms)MIT Energy Initiative (grant “Nonlinear Energy Harvesting From Broad-Band Vibrational Sources By Mimicking Turbulent Energy Transfer Mechanisms)American Bureau of Shipping (Career Development Chair

Probabilistic optimization of vibrational systems under stochastic excitation containing extreme forcing events

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mechanical Engineering, 2017.Cataloged from PDF version of thesis.Includes bibliographical references (pages 221-232).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.by Han Kyul Joo.Ph. D

Single-degree-of-freedom energy harvesters by stochastic excitation

Thesis: S.M., Massachusetts Institute of Technology, Department of Mechanical Engineering, 2014.Cataloged from PDF version of thesis.Includes bibliographical references (pages 97-100).In this thesis, the performance criteria for the objective comparison of different classes of single-degree-of-freedom oscillators under stochastic excitation are developed. For each family of oscillators, these objective criteria take into account the maximum possible energy harvested for a given response level, which is a quantity that is directly connected to the size of the harvesting configuration. We prove that the derived criteria are invariant with respect to magnitude or temporal rescaling of the input spectrum and they depend only on the relative distribution of energy across different harmonics of the excitation. We then compare three different classes of linear and nonlinear oscillators and using stochastic analysis tools we illustrate that in all cases of excitation spectra (monochromatic, broadband, white-noise) the optimal performance of all designs cannot exceed the performance of the linear design. Subsequently, we study the robustness of this optimal performance to small perturbations of the input spectrum and illustrate the advantages of nonlinear designs relative to linear ones.by Han Kyul Joo.S.M

Performance measures for single-degree-of-freedom energy harvesters under stochastic excitation

We develop performance criteria for the objective comparison of different classes of single-degree-of-freedom oscillators under stochastic excitation. For each family of oscillators, these objective criteria take into account the maximum possible energy harvested for a given response level, which is a quantity that is directly connected to the size of the harvesting configuration. We prove that the derived criteria are invariant with respect to magnitude or temporal rescaling of the input spectrum and they depend only on the relative distribution of energy across different harmonics of the excitation. We then compare three different classes of linear and nonlinear oscillators and using stochastic analysis methods we illustrate that in all cases of excitation spectra (monochromatic, broadband, white-noise) the optimal performance of all designs cannot exceed the performance of the linear design. Subsequently, we study the robustness of this optimal performance to small perturbations of the input spectrum and illustrate the advantages of nonlinear designs relative to linear ones

Performance Barriers for Single-Degree-of-Freedom Energy Harvesters Under Generic Stochastic Excitation

We develop performance criteria for the objective comparison of different classes of single-degree-of-freedom oscillators under stochastic excitation. For each family of oscillators, these objective criteria take into account the maximum possible energy harvested for a given response level, which is a quantity that is directly connected to the size of the harvesting configuration. We prove that the derived criteria are invariant with respect to magnitude or temporal rescaling of the input spectrum and they depend only on the relative distribution of energy across different harmonics of the excitation. We then compare three different classes of linear and nonlinear oscillators and using stochastic analysis tools we illustrate that in all cases of excitation spectra (monochromatic, broadband, white-noise) the optimal performance of all designs cannot exceed the performance of the linear design.Kwanjeong Educational Foundation (Korea)Massachusetts Institute of Technology (Startup Grant

Extreme events and their optimal mitigation in nonlinear structural systems excited by stochastic loads: Application to ocean engineering systems

We develop an efficient numerical method for the probabilistic quantification of the response statistics of nonlinear multi-degree-of-freedom structural systems under extreme forcing events, emphasizing accurate heavy-tail statistics. The response is decomposed to a statistically stationary part and an intermittent component. The stationary part is quantified using a statistical linearization method while the intermittent part, associated with extreme transient responses, is quantified through i) either a few carefully selected simulations or ii) through the use of effective measures (effective stiffness and damping). The developed approach is able to accurately capture the extreme response statistics orders of magnitude faster compared with direct methods. The scheme is applied to the design and optimization of small attachments that can mitigate and suppress extreme forcing events delivered to a primary structural system. Specifically, we consider the problem of suppression of extreme responses in two prototype ocean engineering systems. First, we consider linear and cubic springs and perform parametric optimization by minimizing the forth-order moments of the response. We then consider a more generic, possibly asymmetric, piecewise linear spring and optimize its nonlinear characteristics. The resulting asymmetric spring design far outperforms the optimal cubic energy sink and the linear tuned mass dampers.Office of Naval Research (Grants N00014-14-1-0520 and N00014-15-1-2381)Air Force Office of Scientific Research (Grant FA9550-16-1-0231

Heavy-Tailed Response of Structural Systems Subjected to Stochastic Excitation Containing Extreme Forcing Events

We characterize the complex, heavy-tailed probability density functions (pdfs) describing the response and its local extrema for structural systems subject to random forcing that includes extreme events. Our approach is based on recent probabilistic decompositionsynthesis (PDS) technique (Mohamad, M. A., Cousins, W., and Sapsis, T. P., 2016, "A Probabilistic Decomposition-Synthesis Method for the Quantification of Rare Events Due to Internal Instabilities," J. Comput. Phys., 322, pp. 288-308), where we decouple rare event regimes from background fluctuations. The result of the analysis has the form of a semi-analytical approximation formula for the pdf of the response (displacement, velocity, and acceleration) and the pdf of the local extrema. For special limiting cases (lightly damped or heavily damped systems), our analysis provides fully analytical approximations. We also demonstrate how the method can be applied to high dimensional structural systems through a two-degrees-of-freedom (TDOF) example system undergoing extreme events due to intermittent forcing. 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, through comparisons with expensive Monte Carlo simulations.United States. Office of Naval Research (Grant FA9550-16-1-0231)MIT Energy InitiativeUnited States. Office of Naval Research (Grant N00014-14-1-0520)United States. Office of Naval Research (Grant N00014-15-1-2381