Probabilistic Response Analysis Research Articles

Probabilistic Response Analysis of Nonlinear Tristable Energy Harvester Under Gaussian Colored Noise

Probabilistic Response Analysis of Nonlinear Tristable Energy Harvester Under Gaussian Colored Noise

Probabilistic response analysis of nonlinear vibro-impact systems with two correlated Gaussian white noises

The response distributions of a dynamic system under the effect of correlation noise not only demonstrated a Gaussian symmetric distribution, but also exhibited a non-Gaussian skewed distribution. Thus, correlation noise can induce more complex dynamic phenomena in the response of a nonlinear system. The purpose of this study is to explore the stochastic response and bifurcation of nonlinear vibro-impact systems driven by correlated additive and multiplicative Gaussian white noises using the path integral (PI) method. First, we derived an equivalent correlated stochastic system without collision conditions in the new ancillary phase space using the non-smooth Ivanov transform method. We then used Novikov’s theorem to derive an effective Fokker–Planck–Kolmogorov equation for the equivalent correlated stochastic system. Second, we used PI technology, which combines the three-point Gauss–Legendre quadrature rule to solve the effective Fokker–Planck–Kolmogorov equation and obtained the probability density functions (PDFs) of the equivalent correlated stochastic system. Subsequently, we obtained the PDFs of the original vibro-impact stochastic system using the relationship between the initial state space and the new auxiliary phase space. Finally, the proposed method was used to investigate the transient and stationary responses of a Duffing-Van der Pol Vibro-impact system perturbed by additive and multiplicative random noises. In addition, this technique can also assess the effects of restitution coefficients and barrier position on stochastic P-bifurcation. The accuracy of this method was verified by comparing it with a Monte Carlo simulation.

A simplified weak simulation method for the probabilistic response analysis of nonlinear random vibration problems

Numerical simulation methods are among the most used techniques to evaluate the stochastic response of engineering systems subjected to random excitations. In many practical situations the system should be integrated over long time intervals and a large number of sample trajectories need to be generated to assess the quantities of interest. In this circumstance conventional methods usually show exploding behavior or their implementation is computationally very demanding. In this paper we propose a new numerical simulation method for the probabilistic response analysis of nonlinear random vibration problems. We devise a simplified weak method having the simplicity of basically needing only three-point distributed random variable per time step for its implementation. Also, we prove that it reproduces the same statistical properties that characterize the exact solution to the linearized equation corresponding to the underlying system. This makes the method an efficient tool for stable long term simulations. Numerical experiments are carry out to illustrate the practical performance of the method.

Simulation of fully nonstationary random processes using generalized harmonic wavelets

Simulation of nonstationary random processes plays an important role to probabilistic response analysis of structures and systems under stochastic excitations via Monte Carlo methods. Based on the evolutionary spectral representation theory, this article proposes a novel approach to simulate one-dimensional nonstationary random processes by using generalized harmonic wavelet (GHW). According to the explicit relationship between the evolutionary power spectral density (EPSD) and GHW coefficients, the samples of a random process can be generated by superposing GHWs with stochastic phase angles. The attractive features of the proposed approach are (1) the sample time histories obtained by the proposed method can accurately match the target EPSD and reflect the evolutionary probabilistic characteristics of the random process, (2) the number of GHWs used in the simulation is fixed due to the constraint relationship between the frequency and time resolutions of GHWs, and (3) the proposed method has better computational efficiency than the classical spectrum representation method (SRM). To demonstrate the accuracy and efficiency of the proposed GHW-based approach, three case studies involving the generation of time history samples from a given EPSD model, the simulation of an actual earthquake record, and a comparison between the proposed method and the SRM are presented respectively.

Human-Induced Response and Reliability Analysis of Stochastic Dynamic System

This paper proposed a method analyzing crowd-induced stochastic vibration response by the dynamic reliability method. The efficiency of the probability density evolution method (PDEM) is improved by embedding the Kullback-Leibler (K-L) relative sensitivity analysis in response analysis of stochastic dynamic system. Then the complicated point selection technique of high dimension uncertain variables is avoided. The proposed method is illustrated by the response analysis of random crowd-structure interaction system where the load parameter randomness is considered. The acceleration induced by the crossing of crowd is predicted with the proposed method. Results obtained highlight the computation improvement in PDEM probabilistic response analysis in high dimension dynamic system. And the method could be used as an efficient way to predict and evaluate vibration serviceability of structures with human-induced vibration.

Response and reliability analysis of a high-dimensional stochastic system

The efficiency of the probability density evolution method (PDEM) is improved in this paper by embedding the Kullback–Leibler (K–L) relative sensitivity in the response analysis of a stochastic dynamic system. The response reliability obtained and the probability density function of the response peaks are used for ranking to get a reduced set of random variables for the PDEM analysis. The need of complicated point selection technique with the high-dimensional uncertain variables is therefore alleviated. The proposed method is illustrated with the response analysis of a random crowd-structure system where the load randomness is considered. The acceleration response induced by the presence of the crowd is evaluated with the proposed method. Results obtained highlight the significant improvements in the computation efficiency of the probabilistic response analysis of a high-dimensional dynamic system.

Probabilistic response analysis for a class of nonlinear vibro-impact oscillator with bilateral constraints under colored noise excitation

Abstract In this paper, a new stochastic method combining the stochastic averaging of quasi-conservative and energy loss is developed to analyze the probabilistic response of a class of nonlinear vibro-impact oscillator with bilateral barriers driven by colored noise. According to the relationships between the total energy and the potential energy of impact positions of the nonlinear system, the energy loss of every movement period of the unperturbed nonlinear vibro-impact system with given total energy is analyzed in detail. Based on this analysis, an averaged It o ^ stochastic differential equation is derived through the stochastic method which combining a nonlinear transformation and the stochastic averaging of quasi-conservative, in which the averaged It o ^ equation has the typical characteristics of piecewise smooth. Next, the stationary probability density function (SPDF) is obtained by solving the averaged drift and diffusion terms of the corresponding piecewise smooth Fokker-Planck equation for the averaged It o ^ equation with the Fourier series expansion. Finally, an example is given to validate the effectiveness of the proposed method. The effects of four physical quantities of the stochastic vibro-impact system: the positions of impact, restitution coefficient, noise intensity and correlation time of the colored noise, on the probabilistic response are discussed by the SPDFs in detail, and the results are verified through the Monte Carlo simulation.

Probabilistic response of dynamical systems based on the global attractor with the compatible cell mapping method

A generalized compatible cell mapping (CCM) method is proposed in this paper to take advantages of the simple cell mapping (SCM) method, the generalized cell mapping (GCM) method together with a subdivision procedure. A coarse cell partition is first used to obtain a covering set of the global attractor. Then, a finer global attractor is obtained by the subdivision process. The probabilistic response of stochastic dynamic systems is obtained by the sparse matrix analysis algorithm applied to the covering set of the global attractor. Because the computational domain is the covering set of the global attractor rather than the whole state space, the numerical efficiency of the proposed method can be greatly improved as compared to the GCM. A three-dimensional and a four-dimensional dynamical system under Poisson white noise excitation are studied to demonstrate the effectiveness of the proposed method for the probabilistic response analysis. Monte Carlo simulations show a good agreement with the proposed method.

Probabilistic response and analysis for a vibro-impact system driven by real noise

In this paper, a new stochastic averaging method is proposed to derive the stationary probabilistic response of vibro-impact system subject to real noise excitation, and the effects of real noise and the restitution coefficient on the response are discussed in detail. According to the size relationship between the system energy and the potential energy of the impact position $$\varDelta $$ , the motions of the unperturbed vibro-impact system are divided into two types: the free motion with no energy loss and the impact motion with energy loss. Based on those relationships, a new method that combines a transformation and the Fourier expansion scheme by introducing, the mean drift and diffusion coefficients of the averaged Ito stochastic differential equation for the two types of motions are obtained, which are depended on the bandwidth and the noise density of real noise. The probability density function of stationary response of vibro-impact system is derived through solving the associated Fokker–Planck–Kolmogorov equation. Finally, the effects of the restitution coefficient and the noise intensity of real noise in different bandwidths are detailedly analyzed through an example. The main results on probabilistic response analysis of vibro-impact Duffing system are obtained to illustrate the proposed stochastic averaging method, and Monte Carlo simulation method is also conducted to show the effectiveness of the proposed method.

Probabilistic response analysis of nonlinear vibration energy harvesting system driven by Gaussian colored noise

A new quasi-conservative stochastic averaging method is proposed to analyze the Probabilistic response of nonlinear vibration energy harvesting (VEH) system driven by exponentially correlated Gaussian colored noise. By introducing a method combining a transformation and the residual phase, the nonlinear vibration electromechanical coupling system is equivalent to a single degree of freedom system, which contains the energy-dependent frequency functions. Then the corresponding drift and diffusion coefficients of the averaged Ito^ stochastic differential equation for the equivalent nonlinear system are derived, which are dependent on the correlation time of Gaussian colored noise. The probability density function (PDF) of stationary responses is derived through solving the associated Fokker–Plank–Kolmogorov (FPK) equation. Finally, the mean-square electric voltage and mean output power are analytically obtained through the relation between the electric voltage and the vibration displacement, and the output power has a linear square relationship with the electric voltage, respectively. The main results on probabilistic response of VEH system are obtained to illustrate the proposed stochastic averaging method, and Monte Carlo (MC) simulation method is also conducted to show that the proposed method is quite effective.

Probabilistic seismic assessment of seismically isolated electrical transformers considering vertical isolation and vertical ground motion

This study presents a probabilistic response analysis of seismically isolated electrical transformers with emphasis on comparing the performance of equipment that are non-isolated to equipment that are isolated only in the horizontal direction or are isolated by a three-dimensional isolation system. The performance is assessed by calculating the probability of failure as function of the seismic intensity with due consideration of: (a) horizontal and vertical ground seismic motions, (b) displacement capacity of the seismic isolation system, (c) limit states of electrical bushings, (d) details of construction of the isolation system, (e) weight of the isolated transformer, and (f) bushing geometry and configuration. Calculations of the probability of failure within the lifetime of isolated and non-isolated transformers at selected locations are also performed. The results of this study demonstrate that seismic isolation systems can improve the seismic performance for a wide range of parameters and that systems which isolate in both the horizontal and vertical directions can be further effective. The seismic assessment methodology presented can be used for: (a) deciding on the need to use seismic isolation and selecting the properties of the isolation system for transformers depending on the design limits, location, and configuration of transformer and (b) calculating the mean annual frequency of functional failure and the corresponding probability of failure over the lifetime of the equipment. The results may also be used to assess the seismic performance of electric transmission networks under scenarios of component failures.

Probabilistic Nonlinear Response Analysis of Steel-Concrete Composite Beams

This paper employs a methodology for probabilistic response analysis based on the first-order second moment (FOSM) method in conjunction with response sensitivity computation through the direct differentiation method (DDM), to study the variability of the structural response of steel-concrete composite (SCC) beams. This methodology is applied to compute the first-order and second-order statistical moments of the response of two actual structural systems for which experimental data are available. The results of the DDM-based FOSM method are compared with the experimental measurements and with the results of the computationally more expensive Monte Carlo-Simulation (MCS) method. Different modeling hypotheses for the material parameter uncertainty are considered. The DDM-based FOSM method agrees very well with the MCS results for low-to-moderate levels of response nonlinearity under low-to-moderate material parameter uncertainty and up to high level of response nonlinearity under low material parameter uncertainty. The DDM-based FOSM method is shown to correctly describe the effects of random spatial variability of material parameters.

Ultimate Limit State Model Basis for Assessment of Offshore Wind Energy Converters

This paper establishes the model basis regarding the ultimate limit state consisting of structural, loading, and probabilistic models of the support structure of offshore wind energy converters together with a sensitivity study. The model basis is part of a risk based assessment and monitoring framework and will be applied for establishing the “as designed and constructed” reliability as prior information for the assessment and as a basis for designing a monitoring system. The model basis is derived considering the constitutive physical equations and the methodology of solving these which then in combination with the ultimate limit state requirements leads to the specific constitutive relations. As a result finite element models based on shell elements incorporating a structural and a loading model are introduced and described in detail. Applying these models the ultimate capacity of the support structure and the tripod structure are determined with a geometrically and materially nonlinear finite element analysis. The observed failure mechanisms are the basis for the definition of the ultimate limit state responses. A probabilistic model accounting for the uncertainties involved is derived on the basis of literature review and measurement data from a prototype Multibrid M5000 support structure. In combination with the developed structural and loading models, sensitivity analyses in regard to the responses are performed to enhance the understanding and to refine the developed models. To this end, as the developed models necessitate substantial numerical efforts for the probabilistic response analysis predetermined designs of numerical experiments are applied for the calculation of the sensitivities using the Spearman rank correlation coefficient. With this quantification of the sensitivity of the random variables on the responses including nonlinearity the refinement of the model is performed on a quantitative basis.

Application of artificial neural networks to the response prediction of geometrically nonlinear truss structures

This paper examines the application of artificial neural networks (ANN) to the response prediction of geometrically nonlinear truss structures. Two types of analysis (deterministic and probabilistic analyses) are considered. A three-layer feed-forward backpropagation network with three input nodes, five hidden layer nodes and two output nodes is firstly developed for the deterministic response analysis. Then a back propagation training algorithm with Bayesian regularization is used to train the network. The trained network is then successfully combined with a direct Monte Carlo Simulation (MCS) to perform a probabilistic response analysis of geometrically nonlinear truss structures. Finally, the proposed ANN is applied to predict the response of a geometrically nonlinear truss structure. It is found that the proposed ANN is very efficient and reasonable in predicting the response of geometrically nonlinear truss structures.

Probabilistic Response Analysis of Cracked Prestressed Concrete Beams

In the response analysis of cracked prestressed concrete beams deterministic procedure is insufficient to provide complete information. Probabilistic analysis is a holistic approach to predicting structural responses considering uncertainties in structural and load parameters. This paper proposes an efficient and accurate algorithm to predict responses of cracked prestressed concrete beams with parameter uncertainties. The proposed algorithm integrates the advantages of the response surface method, section curvature method and Monte Carlo simulation. Uncertainties in the structure and load parameters can be taken into account in this algorithm. The algorithm is verified using the Monte Carlo simulation. The proposed algorithm is then used to predict the response of a simply supported prestressed beam with parameter uncertainties. The results show that neglecting the effect of uncertainty in the prestressing force results in a significant underestimation of response statistics of cracked prestressed concrete beams. Finally, the most influential random variables on the response statistics of cracked prestressed concrete beams are identified by using a sensitivity analysis.

Probabilistic Response Analyses to Earthquakes

Probabilistic Response Analyses to Earthquakes

Closure to “Probabilistic Response Analyses to Earthquakes”

Closure to “Probabilistic Response Analyses to Earthquakes”

Discussion of “Probabilistic Response Analysis to Earthquakes”

Discussion of “Probabilistic Response Analysis to Earthquakes”

Probabilistic Response Analyses to Earthquakes

A single degree-of-freedom system is analyzed probabilistically for earthquake-like excitations. A Gaussian nonstationary random process model developed previously has been used for the ground acceleration. After finding the variance of the response, the distribution of the highest peak is determined approximately. The average of this is closely related to the conventional average response spectra. Numerical results are presented for four earthquake-like excitations which differ widely in their properties such as duration, maximum acceleration, rate of zero crossings, etc. The four excitations are supposed to simulate the Port Hueneme (NS, March 18, 1957); Vernon, CA (S82E, October 2, 1933); Taft, CA (S69E, July 21, 1952); and Tower Latino Americana, Mexico (N09E, May 11, 1962) earthquakes. The present approximate analysis indicates that long duration earthquakes may have more than one peak in their undamped average velocity spectra.

Closure to "Probabilistic Response Analyses to Earthquakes"

Closure to "Probabilistic Response Analyses to Earthquakes"