Probability Density Function Solution Research Articles

Non-stationary probabilistic analysis of non-linear ship roll motion due to modulated periodic and random excitations

This paper is about the study on the non-stationary probability density function (PDF) solution of the non-linear ship rolling system under modulated periodic and random excitations. The quadratic and cubic damping terms are introduced to describe the non-linear damping moment and the power terms in state variables are adopted to describe the righting moment in the ship rolling system, respectively. The relevant Fokker–Planck–Kolmogorov (FPK) equation of the ship rolling system is constructed and the non-stationary probabilistic solutions are obtained by the presented solution procedure. In the solution process, the time variable t is introduced to the traditional exponential-polynomial-closure (EPC) method to obtain the non-stationary PDF solutions of the ship rolling system. The obtained non-stationary PDF solutions of the non-linear ship rolling systems demonstrate that the presented solution procedure can give high solution accuracy even in the PDF tails when they are compared with Monte Carlo simulation (MCS). Moreover, owing to the strong system nonlinearity and the interaction of the periodic excitation and random excitation, the achieved non-stationary PDF solutions of the system responses can deviate significantly from the Gaussian PDF. It is also found that the PDFs of responses are not even symmetrical about the means of responses due to the interaction of the periodic excitation and the random excitation even if the system only contains odd nonlinear terms. In addition, the computational efficiency is increased by over five hundred times by the presented solution procedure compared to MCS.

Identification of nonlinear aerodynamic damping of cross-wind excited slender structures in atmospheric boundary layer flow

Due to the low natural frequency and low damping of slender structures, the crosswind aero-elastic effect become dominant around the vortex-resonance wind speed and can lead to undesired crosswind responses. For understanding and describing the aero-elastic effect of slender structures, aerodynamic damping is usually adopted. This study suggested an identification procedure for crosswind aerodynamic damping of slender structures by curve-fitting the probability density function (PDF) of the crosswind response. Firstly, the aerodynamic damping is described by a generalized Van der Pol (GVDP) model, and the analytical solution of response PDF, which is a function of the parameters of GVDP model, is derived by the equivalent nonlinear equation method. Next, the proposed identification method was validated by stochastic simulation of a crosswind-excited tall building. Finally, free-vibration wind tunnel test of a slender structure with slenderness ratio equal to 12 and 20 at open flat terrain was conducted and aerodynamic damping was then quantified by the present method. It is found that the present method provides a robust and accurate identification of crosswind aerodynamic damping of slender structures in atmospheric boundary flow.

Transient solution of stochastic oscillators with both odd and even nonlinearity under modulated Gaussian white noise.

This study presents an improved solution procedure for determining the transient probability density function (PDF) solutions of the nonlinear oscillators with both odd and even nonlinearity under modulated random stimulation, which is an extension of the exponential-polynomial-closure approach. An evolutionary exponential-polynomial function with time-varying undetermined variables is considered as the transient probabilistic solution. By selecting a set of independent evolutionary base functions spanning a R^{n} space as weight functions, a set of ordinary differential equationscan be formulated by integrating the weighted residual error. The undetermined variables can be determined numerically by solving those ordinary differential equations. Three numerical examples illustrate that the improved solution procedure can acquire the transient probabilistic responses of the stochastic dynamic systems effectively and efficiently even in the PDF solution tails when compared with Monte Carlo simulation. Moreover, the results indicate that the PDF solutions of the oscillators are asymmetrical at their nonzero means due to the influence of even nonlinearity. The nonstationary behaviors of the system responses are also investigated along with the behavior of modulated Gaussian white noise.

Transient response of energy harvesting systems with multi-well potential under Poisson white noise excitations

This paper studies the transient response of vibration-based energy harvesting (VEH) systems characterized by multi-well potential functions exposed to Poisson white noise excitations. With the recently developed radial basis function neural networks (RBFNN) approach, the generalized Fokker–Planck–Kolmogorov (GFPK) equation in three-dimensional state space for the transient probability density function (PDF) is solved. The trial solution is composed of a series of radial basis Gaussian functions, each multiplied by a time-varying weight coefficient. The method of Lagrange multiplier is introduced for the constraint of weight coefficients to normalized the trial solution of the PDF solution. As examples to validate the proposed solution approach, the mono-stable and tri-stable energy harvester systems are studied. Monte Carlo simulations (MCS) have been applied to check the analytical solution. Some theoretical trends of the harvested mean square voltage (MSV) as a function of system parameters are examined. The non-Gaussian excitation and time-varying nature of the system can significantly affect the MSV output. It is noted that this work provides researchers and engineers a tool to optimize energy harvesting systems operated in random environment.

Transient PDF Solution of Nonlinear Stochastic Oscillator Subjected to Modulated Gaussian White Noise

In this paper, an extended exponential-polynomial-closure (EPC) procedure is presented to investigate the transient probability density function (PDF) of stochastic oscillator under the stimulation being modulated white noise based on the conventional EPC approach. By the presented EPC procedure, an evolutionary exponential–polynomial function is defined as the PDF solution and a set of evolutionary independent base functions are introduced to span a [Formula: see text] space. By integrating the residual error with those evolutionary functions spanning the [Formula: see text] space, a group of ordinary differential equations can then be formulated. Three nonlinear oscillators are analyzed numerically and the results comparisons of various techniques validate the presented EPC procedure in solution efficiency and accuracy, particularly in the PDF solution tails which play a crucial role in reliability analysis. The transient PDF solutions acquired by the presented EPC procedure match well with Monte Carlo simulation even in the PDF solution tails, whereas the solutions obtained by the equivalent linearization are deemed to be unacceptable for the strongly nonlinear oscillators subjected to modulated stimulation. Notably, the presented EPC procedure can highly increase the computational efficiency by about 1[Formula: see text]000 times in all the presented numerical examples in comparison to Monte Carlo simulation.

The generalized EPC method for the non-stationary probabilistic response of nonlinear dynamical system

It is well known that the non-stationary response probability density function (PDF) plays an important role in the reliability and failure analysis. With current approximate or numerical methods, the non-stationary approximate PDF is generally obtained in terms of the ones at the discrete time instants. Repeated computation must be conducted for the ones at other time instants since the response PDF at only one time instant can be obtained after performing the solution procedure. Thus, the computational efficiency suffers a major setback. In this paper, the exponential polynomial closure (EPC) method is further improved and enhanced to obtain the completely non-stationary PDF solution, which is distributed continuously in the time domain. It takes the temporal base function into the PDF approximation. The unknown coefficients in the EPC solution are generalized to be explicit functions of the time parameter. With the least squares method, the explicit time functions can be determined based on the few simulated response PDF values. This implementation makes the continues PDF distribution in the time domain available, which greatly improves the computational efficiency without the repeating computation. Two typical nonlinear systems under stationary and non-stationary random excitations are taken as examples to illustrate the efficiency of the proposed method. Numerical results show that the results obtained by the proposed method agree well with the simulated results. In addition, the relationship between the explicit time function and the modulate function is discussed.

Random vibration analysis of nonlinear structure with pounding tuned mass damper

To mitigate the undesired vibrations of the structure, a cost-effective approach is to set up the nonlinear dynamic vibration absorber (NLDVA) on the structure. As an alternative NLDVA, the Pounding Tuned Mass Damper (PTMD) gathered extensively attention in recent years. However, the random vibration analysis of structure coupled with PTMD still poses a challenge. Besides, there remains an urgent need to construct a more comprehensive recovery coefficient (RC) for the two-stage damping mechanism of PTMD. To fill in this gap, this work involves the nonlinear random vibration of a type of two degree-of-freedom system composing of a main structure (MS) coupled to a PTMD device, in which the collision process of the system is portrayed by a segmented recovery coefficient (RC) model. Specifically, an equivalent system of structure-PTMD is firstly constructed by coordinate transformation. Then, the approximate probability density function (PDF) of equivalent system is performed by combining non-smooth conversion and stochastic averaging method (SAM). Further, the approximate PDF solution of the structure-PTMD can be determined by resorting to Jacobi transformation and verified numerically. Remarkably, the approximate PDF solutions of the system can also be applied to estimate the power spectral density of the system response. Being analytical, the analysis in this work yields results in the closed-form expressions that enable one to acquire new insight into the dynamic features of the structure-PTMD system. Moreover, the proposed scheme will promote the design and application of PTMD in practical projects from a theoretical point of view.

RBFNN for Calculating the Stationary Response of SDOF Nonlinear Systems Excited by Poisson White Noise

Random perturbations in nature described by non-Gaussian excitation models are far more widely applied and development prospects than that of Gaussian excitation models in practice. However, the stochastic dynamics research of non-Gaussian excitation is still not very mature. In this work, radial-basis-function-neural-network (RBFNN) is applied for calculating the stationary response of single-degree-of-freedom (SDOF) nonlinear system excited by Poisson white noise. Specifically, the trial probability-density-function (PDF) solution of reduced generalized-Fokker–Plank–Kolmogorov (GFPK) equation is constructed by a suitable number of Gaussian basis functions (GBFs) with a fixed set of means and standard deviations. Subsequently, an approximate squared error of the GFPK equation in a finite domain is considered. Together with the normalization condition, the approximate squared error can be minimized by establishing a Lagrangian function, and then the optimal weight coefficients associated with the approximate PDF solution are solved from a system of linear algebraic equations. For demonstrating the effectiveness of the proposed procedure, two specific examples are presented. The corresponding reduced GFPK equation is truncated with higher order for the strong non-Gaussian case. The precision of the analytical solution is verified against the Monte Carlo simulation (MCS) data. In addition, all the results indicate that RBFNN shows fairly high efficiency under the premise of ensuring high precision in the whole computational procedure.

Dilution of Reactive Plumes: Evolution of Concentration Statistics Under Diffusion and Nonlinear Reaction

Concentration fields of solutes in porous media often exhibit large fluctuations, driven by physical and chemical heterogeneity from the pore to the Darcy scale. For many applications, ranging from reactive transport modeling to toxicology, the knowledge of mean concentrations is not sufficient, and quantifying concentration variability is necessary. The probability density function (PDF) of concentration quantifies the frequency of occurrence of concentration values throughout a spatial domain. While evolution equations and analytical solutions for the concentration PDF exist for conservative solutes, less is known about its evolution under the joint action of transport and reaction. In this work, we investigate how dilution of a reactive plume by diffusion affects the statistics of concentrations. While mixing has no effect on first-order reactions, its coupling with nonlinear reactions leads to non-trivial effective kinetics relevant for a broad range of reactive transport problems. We study the evolution of the concentration PDF under diffusion and nonlinear reaction in one spatial dimension, which represents a critical step toward further coupling with heterogeneous advection. We show that the dependence of the scalar dissipation rate on concentration encodes the impact of diffusive transport on the concentration PDF and derive a dynamical equation for its time evolution. Using a weak-coupling approximation for the reaction and diffusion dynamics, we derive analytical predictions for the concentration PDF and its moments. Our results provide new insights into how diffusion and reaction control concentration variability and open new opportunities for coupling mixing models with chemical reactions.

Transient probabilistic solutions of stochastic oscillator with even nonlinearities by exponential polynomial closure method

The current work is devoted to analyze the transient probability density function solutions of stochastic oscillator with even nonlinearities under external excitation of Gaussian white noise by applying the extended exponential polynomial closure method. Specifically, the Fokker–Planck–Kolmogorov equation which governs the probability density function solutions of the nonlinear system is presented first. The residual error of the Fokker–Planck–Kolmogorov equation is then derived by assuming the probability density function solution as the type of exponential polynomial with time-dependent variables. Finally, by making the projection of the residual error vanish, a set of nonlinear ordinary differential equations is established and solved numerically. Numerical analysis show that the extended exponential polynomial closure method with polynomial order being six is both effective and efficient for solving the transient analysis of the stochastic oscillator with even nonlinearities by comparing the numerical results obtained by the proposed method with those obtained by Monte Carlo simulation method. Numerical results also show that the transient probability density function solutions of the system responses are not symmetric about their nonzero means due to the existence of even nonlinearities.

Probabilistic analysis on parametric random vibration of a marine riser excited by correlated Gaussian white noises

A path integration method is used to discuss the stochastic response of a marine riser excited parametrically and externally by correlated Gaussian white noises. On the basis of the nonlinear vibration equation of a marine riser, the first-order vibration mode is studied and its governing equation is formulated. The response probability density function (PDF) is further developed by a path integration method. During the solution procedure, the Gaussian closure method and the short-time Gaussian approximation method are used to develop the transition probability density function. The results of different time intervals are compared with the results of Monte-Carlo simulation (MCS) to determine the adequate time interval. After that, the path integration method with a Gauss–Legendre scheme is used to obtain the stationary PDF solution of displacement and velocity. The path integration solutions (PIS) are compared with those of the equivalent linearization method (EQL) and MCS. Five demonstrative cases are analyzed to evaluate the effectiveness of the path integration method. The influences of nonlinearity and excitations are considered. It shows that the results of PIS coincide better with the results of MCS than those of EQL. The analysis shows that parametric excitation causes the softening behavior of the PDF distribution compared with those of EQL. The increase of excitation correlation results in the nonsymmetrical PDF distribution of response. The increase of nonlinearity in displacement can lead the PDF of displacement to showing a hardening behavior.

Probabilistic solution of nonlinear ship rolling in random beam seas

In this paper, the probability density function (PDF) and the mean up-crossing rate of nonlinear ship rolling in random beam seas are investigated. The excitation of stationary random sea waves is approximated as a second-order linear filtered white noise. The Fokker–Planck–Kolmogorov (FPK) equation governing the probability density function of ship rolling is a four-dimensional linear partial differential equation with varying coefficients, and obtaining its exact solution is much more sophisticated. The exponential-polynomial closure (EPC) method is applied to solve the corresponding FPK equation of the system. In numerical examples, linear-plus-cubic damping model and linear-plus-quadratic damping model with three different sea states are further examined. Comparison with the equivalent linearisation (EQL) method and Monte Carlo simulated results show that the proposed procedure is effective to obtain a satisfactory probability density function solution, especially in the tail region.

Stochastic Response of SDOF Self-Centering System

As a new type of seismic resisting device, the self-centering system is attractive due to its excellent re-centering capability, but research on such a system under random seismic loadings is quite limited. In this paper, the stochastic response of a single-degree-of-freedom (SDOF) self-centering system driven by a white noise process is investigated. For this purpose, the original self-centering system is first approximated by an auxiliary nonlinear system, in which the equivalent damping and stiffness coefficients related to the amplitude envelope of the response are determined by a harmonic balance procedure. Subsequently, by the method of stochastic averaging, the amplitude envelope of the response of the equivalent nonlinear stochastic system is approximated by a Markovian process. The associated Fokker–Plank–Kolmogorov (FPK) equation is used to derive the stationary probability density function (PDF) of the amplitude envelope in a closed form. The effects of energy dissipation coefficient and yield displacement on the response of system are examined using the stationary PDF solution. Moreover, Monte Carlo simulations (MCS) are used for ascertaining the accuracy of the analytical solutions.

Description of instabilities in Uhlenbeck–Ornstein process

The method of Riemannian geometry has been successful in the context of equilibrium thermodynamics. In this work, we extend this approach to non-equilibrium processes. As a geometric-differential frame of non-equilibrium systems, we consider in our study the geometric properties of a manifold associated with simple but typical non-equilibrium models. We consider a Uhlenbeck–Ornstein process and the formal structure of the probability density function solution of the Fokker–Planck equation. We propose a general geometric strategy for the construction of macroscopic potentials in non-equilibrium problems. This macroscopic potential is a function of the transport coefficient and is associated with system instabilities.

Electric field induced coil-stretch transition of DNA molecule

The dynamics of single DNA molecules in a homogeneous planar elongational electric field is investigated in the framework of the Hookean Dumbbell model. For a Gaussian, rapidly changing advecting flow the Fokker-Planck equation for time-dependent probability density function of DNA elongation is obtained. The analytical solution for probability density function of time independent problem is derived. The coil-stretch transition is studied when the elongation of the DNA molecule is very small than the maximum length. The characteristic relaxation time is calculated as a function of the Weissenberg number. It is observed that the coil-stretch transition and characteristic relaxation time of DNA molecule strongly depend on the angle of the principal axis of molecule with respect to axis of elongation.

Entropy-based China income distributions and inequality measures

ABSTRACTWe use information theoretic information recovery methods, on a 2005 sample of household income data from the Chinese InterCensus, to estimate the income distribution for China and each of its 31 provinces and to obtain corresponding measures of income inequality. Using entropy divergence methods, we seek a probability density function solution that is as close to a uniform probability distribution of income (with the least inequality), as the data will permit. These entropy measures of income inequality reflect how the allocation and distribution systems are performing, and we show the advantages of investigating province variation in income inequality using entropy measures rather than Gini coefficients. Finally, we use a sample of data from the China Family Panel Study to recover an estimate of the 2010 and the 2016 to investigate possible directions of inequality changes using these different additional data sources, given that the 2015 Inter-Census is not yet available.

Probabilistic response of nonsmooth nonlinear systems under Gaussian white noise excitations

This paper proposes an approach to obtain the steady state probability density function (PDF) of nonsmooth nonlinear dynamical systems driven by the Gaussian white noise. The steady state PDF solution is assumed to contain two probability flows obtained by the detailed balance method. The weighted mean squared residuals of the error of the FPK equation are minimized with respect to the unknown coefficients in the assumed solution. The integrability of the weighted mean squared residuals is found to provide a means to deal with the singularities in the steady state PDF that arise from the nonsmooth dynamics of the nonlinear stochastic system. The accuracy of the proposed approach is estimated by the Monte Carlo simulations via two examples.

Transient responses of stochastic systems under stationary excitations

Stochastic systems to stationary random excitations may fail long before stationarity is achieved. Transient state has to be taken into account. It has been a challenge to obtain exact transient response properties. A novel approximate technique for determining non-stationary probability density function (PDF) of randomly excited nonlinear oscillators is developed. Specifically, it expresses the PDF approximation in terms of polynomial functions with time-dependent coefficients. Based on the results from statistical linearization, residual error of the FPK equation associated with proposed approximation solution can be treated by weighted residual method. As a result, nonlinear ordinary differential equations are produced. Numerical method is adopted to solve these equations and approximate PDF solutions are then obtained. In order to verify the efficiency of the proposed procedure, four examples of stochastic vibrating systems with additional excitations or/and parametric excitations are considered. It is shown that the results obtained by the proposed procedure agree well with those from Monte Carlo simulation.

Entropy Based China Income Distributions and Inequality Measures

We use entropy based methods on a sample of household income data from the 2005 Chinese Inter-Census as a basis for summarizing the income data in the form of a probability density function(PDF), and to obtain measures of income inequality. We estimate the income probability density functions using information theoretic entropy based divergence measures and recover the corresponding inequality measures for each of China’s provinces. In the entropy criterion-measure we seek a probability density function solution that is as close to a uniform PDF-distribution of income (an equal distribution with the least inequality), as the micro sample data will permit. These entropy measures reflect cross sectional measures of income inequality and reflect how the province level economic behavioral system is functioning and how the allocation and distribution systems is performing. Finally we use a sample of data from the China Family Panel Study to recover an estimate of the income PDF for China in 2016.

Remaining Useful Life Prediction Based on a General Expression of Stochastic Process Models

In remaining useful life (RUL) prediction, stochastic process models are widely used to describe the degradation processes of systems. For age-dependent stochastic process models, the RUL probability density function (PDF) can be calculated using a closed-form solution. For state-dependent models, however, it is difficult to calculate such a closed-form solution. Therefore, the RUL is always approximately estimated using a sequential Monte Carlo-based method, but this method has some limitations. First, it only provides a numerical approximation result whose accuracy highly relies on the quality and quantity of the simulated degradation trajectories. Second, the time interval is unable to be adjusted during the state transition process, resulting in too few discrete probability densities in the result near the end-of-life. This paper describes the degradation processes using a general expression of age- and state-dependent models. The analytical solution of the RUL PDF is derived from the general expression. After that, a new RUL prediction method is proposed. In this method, a series of degradation trajectories are generated through degradation process simulation. The RUL PDF is estimated by inputting the state values of the degradation trajectories into the analytical solution. The validity of the proposed method is verified using fatigue-crack-growth data.