A comparison between non-gaussian closure and statistical linearization techniques for random vibration of a nonlinear oscillator

Noori, Mohammad; Saffar, Ali; Davoodi, H.

doi:10.1016/0045-7949(87)90109-x

Cited by 17 publications

(4 citation statements)

References 18 publications

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“…Linear random vibration theory is thus not applicable. In this section, the stochastic linearization method (SLM) [60] is applied to solve Equation (4) using an equivalent linear-viscous damping eq c and a linear stiffness coefficient eq k to represent the bilinear model. Equation (4) can thus be rewritten in terms of SLM as follows:…”

Section: Stochastic Dynamic Analysis and Stochastic Linearization Methodsmentioning

confidence: 99%

Displacement-Dependent Damping Inerter System for Seismic Response Control

et al. 2019

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Various inerter systems utilizing velocity-dependent damping for vibration control have been developed. However, a velocity-dependent damping element may exhibit relatively poor performance compared to a displacement-dependent damping element (DDE) of equivalent damping ratio, when the structural deformation is small in the early stage of the seismic response. To address this issue, the advantage of DDE in generating a larger control force in the early stage of excitation is promoted here and enhanced by a supplemental inerter-spring-system, thus realizing a proposed novel displacement-dependent damping inerter system (DDIS). First, the influence of various DDIS-parameters is carried out by resorting to the stochastic linearization method to handle non-linear terms. Then, seismic responses of the DDIS-controlled system are evaluated in the time domain taking the non-linearity into account, thus validating the accuracy of the stochastic dynamic analysis. Several design cases are considered, all of which demonstrated damping enhancement and timely control achieved by the DDIS. The results show that the energy dissipation as well as reduction of structural displacement and acceleration accomplished by the proposed system are significant. DDIS suppresses structural responses in a timely manner, as soon as the peak excitation occurs. In addition, it is demonstrated that interactions among the inerter, spring, and DDE, which constitute the damping-enhancement mechanism, lead to a higher energy-dissipation efficiency compared to the DDE alone. enhancement mechanism. Arakaki et al. [27,28] utilized the rotation mechanism to amplify the effective damping force of a viscous damper, which is a type of velocity-dependent damping element (VDE). However, these devices did not explicitly use the mass enhancement effect until Ikago et al. [1] proposed the tuned viscous mass damper, which belongs to a type of velocity-dependent damping inerter system (VDIS). The performance of the tuned viscous mass damper control system was subsequently investigated via shaking table tests conducted on single-story systems equipped with scaled-down versions of the damper [29]. Garrido et al. [30] proposed a rotational inertia double-tuned mass damper by replacing the viscous damping of the tuned mass damper with a tuned viscous mass damper, which achieved significantly greater control than the tuned mass damper with similar additional mass ratio. Through the incorporation of an electromagnetic damper, which is a type of VDE, Nakamura et al. [31] developed an electromagnetic inerter mass damper with variable damping force. Asai et al. [32,33] achieved enhanced energy-harvesting performance using a tuned inerter. Zhang et al. [10,34] proposed an isolation inerter system that used an inerter and a VDE to mitigate the vibration of a storage tank. The effect of the mechanical layout of the system was also investigated and considered in the development of a demand-based optimal design method for the system. Ikago et al. [1] presented the closed-form ...

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Section: Stochastic Dynamic Analysis and Stochastic Linearization Methodsmentioning

confidence: 99%

Displacement-Dependent Damping Inerter System for Seismic Response Control

et al. 2019

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“…These methods were built to calculate the statistical properties of the responses of nonlinear system based on the assumption that the distribution of response is Gaussian. On the other hand, a majority of methodologies were proposed based on the non-Gaussian approximation of oscillator responses, such as multi-Gaussian closure method (Er, 1998b), series of Gram–Charlier expansion (Noori et al, 1987; Liu and Davies, 1990), and stochastic averaging method (Stratonovich, 1967; Jin and Huang, 2010). As is well known, the Fokker–Planck–Kolmogorov (FPK) equation rules the probabilistic solutions of the nonlinear stochastic oscillators under the white noise excitation (Soong and Grigoriu, 1993).…”

Section: Introductionmentioning

confidence: 99%

Investigation on the EPC method in analyzing the nonlinear oscillators under both harmonic and Gaussian white noise excitations

Zhu

Gong

et al. 2022

Journal of Vibration and Control

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The objective of the current paper is to study and extend the exponential-polynomial-closure (EPC) method in analyzing the non-stationary response probabilistic solutions of nonlinear stochastic oscillators excited by combined deterministic harmonic and Gaussian white noise excitations. The probabilistic solution of the non-stationary responses of nonlinear stochastic oscillator is expressed as an exponential function of polynomial with time-variant coefficients and then the Fokker–Planck–Kolmogorov equation is solved approximately. Two validation examples are presented. The results obtained by Monte Carlo simulation (MCS) and EPC method are presented to show their good agreement, which confirms the effectiveness of the EPC method for the probabilistic solutions to nonlinear stochastic oscillators. The advantage of the EPC method for analyzing the oscillators with strong nonlinearities is investigated by comparison with Equivalent Linearization method. Furthermore, compared with MCS, the computational efficiency by using the EPC method is increased by about 56 times in the first example and 44 times in the second example.

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“…[29][30][31][32][33][34][35][36][37]). In most cases, these nonlinear approaches may offer some improvement compared with the stochastic linearization approach applied to nonlinear systems but the associated computational cost is considerably larger [38]. For strongly nonlinear systems, such as bistable systems, these improvements can be very small.…”

Section: Introductionmentioning

confidence: 99%

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

Joo

Sapsis

2016

Probabilistic Engineering Mechanics

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We develop a moment-equation-copula-closure method for the inexpensive approximation of the steady state statistical structure of strongly nonlinear systems 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, based on copula functions 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 the 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-offreedom energy harvesters with mechanical and electromagnetic damping, and we show that the results compare favorably with direct Monte-Carlo simulations.

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A comparison between non-gaussian closure and statistical linearization techniques for random vibration of a nonlinear oscillator

Cited by 17 publications

References 18 publications

Displacement-Dependent Damping Inerter System for Seismic Response Control

Displacement-Dependent Damping Inerter System for Seismic Response Control

Investigation on the EPC method in analyzing the nonlinear oscillators under both harmonic and Gaussian white noise excitations

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

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