狭帯域不規則励振を受けるDuffing系のジャンプ現象を伴う応答の確率密度の解析(機械力学,計測,自動制御)

In our life there are various infectious diseases which lead to negative effects on the economy. Therefore, it is highly important to develop mathematical models to describe the spread of the pandemic. In this study, we first develop a deterministic SEI1I2R epidemic model incorporating double epidemic hypothesis. We first verify that there exists a unique global positive solution of the system with any positive initial value, then we investigate the local and global asymptotical stability of the equilibria. Following that, we incorporate the inherent stochasticity of the virus spread by reinterpreting the previous SEIR-type epidemic model of infectious diseases as stochastic systems modelled via the log-normal standard Ornstein–Uhlenbeck process. After that we study the stochastic dynamics of the model in detail. More precisely, we first analyze the existence and uniqueness of the global solution of the system. Then by using the Markov semigroup theory, we establish sufficient criteria for the existence and uniqueness of an invariant probability measure of the system, which implies the persistence of the disease in a community. Especially, under the same conditions as the local asymptotical stability of the endemic equilibrium, we get the specific form of the local probability density function around the quasi-endemic equilibrium of the stochastic system. Meanwhile, we also verify that the invariant global probability density function can be approximated by the local probability density function as the noise intensity tends to zero. In addition, we give some sufficient conditions for elimination of the disease. Finally, numerical simulations are carried out to demonstrate the validity of the derived results and the availability of the proposed models by using MATLAB. This work can improve our insight into epidemic dynamics and help us implement interventions to regulate the disease dynamics.

An intensity-modulated Gaussian process {x(t)} is defined as the product of a non-negative stationary random process {σx(t)} and a stationary Gaussian process {z(t)} that is statistically independent of {σx(t)}. When such an intensity-modulated process is the excitation to a linear time-invariant system, the response process {y(t)}, conditioned on the excitation-modulating function σx(t), is a generally nonstationary Gaussian process. This property is used to develop an approximation to the probability density function (PDF) of the system response by assuming that the time-varying conditional response variance σ2y(t), conditioned on the excitation-modulating function σx(t), is governed by the gamma PDF. The resulting response process PDF is characterized by two parameters: the mean and variance of the conditional response variance σ2y(t). An alternative approximation to the response PDF, characterized by these same two parameters, is developed by using a Taylor’s series representation in the variable σ2y of the response PDF conditioned on σ2y. This latter PDF approximation is shown to be identical to the Gram–Charlier series of type A through terms involving the Hermite polynomial of degree 6. The two response PDF approximations are shown to give essentially identical results for values of the coefficient of variation of σ2y of one-half or less. Simple, easily evaluated expressions for the mean and variance of σ2y are derived in terms of measurable quantities by utilizing the assumption that fluctuations in σx(t) occur slowly in comparison with those of z(t).

The joint probability density function (PDF) of different structural responses is a very important topic in the stochastic response analysis of nonlinear structures. In this paper, the probability density evolution method, which is successfully developed to capture the instantaneous PDF of an arbitrary single response of interest, is extended to evaluate the joint PDF of any two responses. A two-dimensional partial differential equation in terms of the joint PDF is established. The strategy of selecting representative points via the number theoretical method and sieved by a hyper-ellipsoid is outlined. A two-dimensional difference scheme is developed. The free vibration of an SDOF system is examined to verify the proposed method, and a frame structure exhibiting hysteresis subjected to stochastic ground motion is investigated. It is pointed out that the correlation of different responses results from the fact that randomness of different responses comes from the same set of basic random parameters involved. In other words, the essence of the probabilistic correlation is a physical correlation.

Velocity analysis with local event slopes related probability density function

The influence of the tail features of the local magnetic field probability density function (PDF) on the ferromagnetic Ising model is studied in the limit of infinite range interactions. Specifically, we assign a quenched random field whose value is in accordance with a generic distribution that bears platykurtic and leptokurtic distributions depending on a single parameter tau<3 to each site. For tau<5/3, such distributions, which are basically Student-t and r distribution extended for all plausible real degrees of freedom, present a finite standard deviation, if not the distribution has got the same asymptotic power-law behavior as a alpha-stable Lévy distribution with alpha=(3-tau)/(tau-1). For every value of tau, at specific temperature and width of the distribution, the system undergoes a continuous phase transition. Strikingly, we impart the emergence of an inflexion point in the temperature-PDF width phase diagrams for distributions broader than the Cauchy-Lorentz (tau=2) which is accompanied with a divergent free energy per spin (at zero temperature).

RESPONSE STATISTICS OF OCEAN STRUCTURES TO NON-LINEAR HYDRODYNAMIC LOADING PART II: NON-GAUSSIAN OCEAN WAVES

Constructing transient response probability density of non-linear system through complex fractional moments

Direct probability integral method for stochastic response analysis of static and dynamic structural systems

New statistical theory was developed by the authors in the previous paper, which gives probability density function (p. d. f.) and extreme values of total second order responses of moored vessels in random seas. It assumes the response represented in the form of two term Volterra functional series and is based on the generalized Laguerre polynomial expansion of p. d. f. of which the first term is a gamma p. d. f. consisting of three parameters.This paper shows non-normality of surge responses of moored tanker and semi-submersible measured in a wave basin, and determines that the theory estimates it very well. Furthermore effect of time varying wave drift damping and other slow motion hydrodynamic force coefficients on p. d. f. and extreme values are discussed by time domain simulations. This effect cannot be represented in the form of two term Volterra functional series.

The paper describes the development and validation of an efficient and cost effective method for the prediction of the NOx emissions of turbulent gas turbine burners in the early burner design phases which are usually focused on the optimization of the swirler aerodynamics and the fuel air mixing. As the method solely relies on non-reacting tests of burner models in the water channel it can be applied before any test equipment for combustion experiments exists. In order to achieve optimum similarity of fuel air mixing in the water channel tests with engine operation the model is operated at the engine momentum ratio. During the LIF measurements the water flow representing the fuel is doped with fluorescent dye, a plane perpendicular to the length axis near the burner exit plane is illuminated with a 5W Ar-Ion Laser and the fluorescence is recorded with a video camera from downstream. From the video sequences the local probability density functions (PDF) of the dye concentration fluctuations are calculated from the data. Furthermore, the time mean velocity fields are measured with PIV. From the LIF data the PDFs of the local equivalence ratio are derived. Assuming flamelets, the NOx generation in the entire equivalence ratio range observed in the water channel tests is computed using the unstrained freely propagating one-dimensional flame model in Cantera and the GRI3.0 reaction scheme. Although neither flame stretch nor post flame NOx generation were considered the NOx values computed were in excellent agreement with the experimental data from perfectly premixed combustion experiments. The local time averaged NOx mole fraction is obtained by integrating the flamelet NOx over the mixture PDF. Finally the global NOx emission of the burner at the considered operating point is obtained by spatial integration considering the measured velocity field. The method was validated using a conical swirl burner with two fuel injection stages, allowing the degree of premixedness to be adjusted over a wide range depending on the specific fuel injection scenario. For the case with fuel injection along the air inlet slots NOx values slightly above the minimum NOx limit for perfectly premixed combustion were computed. This is consistent with the emission measurements and indicates finite mixing quality of this injection method. In the partially premixed regime the configurations with potential for low NOx emissions were reliably identified with the LIF and PIV based water channel method. The method also shows the steep increase of the NOx emissions with decreasing degree of premixing observed in the experiments but quantitative predictions would have required a postprocessing of the data from the LIF mixing study with a higher spacial resolution than available.

Due to the growing concern on the issues of global warming, climate change, and energy shortage, more rigorous requirements are imposed on the energy conversion process of combustion. Due to the relative safety and convenience in transportation and storage, liquid fuels are commonly used in many practical combustion systems such as diesel engines, gas turbines and industrial furnaces. In these combustion devices, turbulent spray flows are involved. The character of spray dispersion, evaporation, mixing and combustion of fuel droplets strongly determines the performance of these systems with respect to the combustion efficiency, stability, and emissions. Therefore, an improved understanding of turbulent spray combustion and the development of predictive models are needed for a better design of more efficient combustion systems. The present work focuses on the development of a computational methodology based on the transported joint probability density function (PDF) method for the modeling and simulation of two-phase turbulent spray flows without and with chemical reactions. In the non-reacting situation, the dependent variables of the joint PDF include the gas velocity and the mixture fraction. For the simulation of reacting spray flows, a three-variate joint PDF transport equation is derived and modeled in order to account for the pre-vaporization effects and partially premixed regime in turbulent spray flames. The dependent variables include the gas phase mixture fraction, the reaction progress variable and gas enthalpy. Detailed combustion chemistry is considered through an extended spray flamelet model by including a reaction progress variable in addition to the classical formulation. The dilute spray is simulated using a Lagrangian discrete parcel method for the description of droplet motion, heating and evaporation. The infinite conductivity model with consideration of non-equilibrium effects based on the Langmuir-Knudsen law is considered. The spray evolution and flame structures in the frame of the polydisperse reacting spray flows are investigated. Numerical results are compared with experimental data provided by Prof. Masri at the University of Sydney, Australia, and the test cases include three different turbulent non-reacting acetone spray flows in air and turbulent spray flames with the liquid fuel ethanol. For the acetone spray flows, computational results generally show good agreement with experimental data in terms of the droplet size and mean velocity distribution, as well as the liquid volume flux. The results show that the inflow liquid mass loading hardly affects the droplet diameter distribution, whereas the inlet turbulence level has a pronounced effect. The tendency of droplet accumulation near the jet centerline is found with a somewhat overprediction of liquid volume flux at downstream locations. A more sophisticated turbulence model is expected to eliminate this discrepancy. Moreover, the local joint PDF of the gas velocity and the mixture fraction is analyzed. A linear correlation of the gas velocity and the mixture fraction exists close to the nozzle exit outside the main spray jet, and no regions are found where statistical independence prevails. For the simulation of spray flames, computations with the newly developed spray flamelet/progress variable approach and with the previous standard spray flamelet formulation are carried out and compared with the experiments by Prof. Masri at the University of Sydney. A good agreement between the computations with the new formulation and the experiments for gas temperature and droplet size and velocity is achieved. The major spray and combustion properties are correctly captured using this new formulation, which is compared with an unphysically attached flame near the nozzle exit predicted by the previous model. A partially premixed combustion prevails in this piloted turbulent spray flame. Due to the prevaporization of the ethanol droplets near the nozzle exit, a lean premixed gas mixture is found at the inner side of spray jet. Moving downstream, the lean-sided diffusion flame is promoted towards the inner fuel-rich side by heating up the inner premixed core that is controlled by the droplet evaporation. Additionally, it is observed that in the far-field region, the diffusion flame becomes the dominant combustion mode. In summary, an efficient computational model based on the transported joint PDF method is developed to two-phase turbulent spray flows. The combined transported joint PDF and a newly proposed spray flamelet/progress variable approach shows an improved performance in the prediction of complex turbulent spray flames, and new insights on the local flame structure influenced by evaporating sprays are obtained.

The approximate analytical method using the maximum entropy method (MEM) is proposed to estimate the stationary probability density function of nonlinear systems subjected to non-white random excitation. The MEM is often used to estimate the response distribution of nonlinear systems subjected to Gaussian white noise. In order to obtain the response distribution, the moment equations are used as the constraint conditions on the MEM. However, in the case of nonlinear systems under non-white random excitation, the term of the cross-correlation between the excitation and the response in the moment equations is generally not able to be calculated. In the proposed method, the equivalent linearization technique is applied to calculate the cross-correlation approximately. Using the method, we estimate the stationary response probability density functions of a Duffing oscillator and an asymmetric nonlinear system subjected to non-white excitation with the exponentially decaying correlation function. In the analysis, a wide range of the excitation bandwidth is considered. To demonstrate the effectiveness of the method, the results are compared with those of Monte Carlo simulation.

Dynamic reliability analysis of Aerial Building Machine under extreme wind loads using improved QBDC-based active learning

Advanced Probabilistic Neural Network for the Prediction of Concrete Strength Doo Kie Kim1, Seong Kyu Chang1 and Sang Kil Chang1 Summary Accurate and realistic strength estimation before the placement of concrete is highly desirable. In this study, the advanced probabilistic neural network (APNN) was proposed to reflect the global probability density function by summing the heterogeneous local probability density function automatically determined in the individual standard deviation of variables. Currently, the estimation of the compressive strength of concrete is performed by a probabilistic neural network (PNN) on the basis of concrete mix proportions, and the PNN is improved by the iteration method. However, an empirical method has been incorporated to specify the smoothing parameter in the PNN technique, causing significant uncertainty in the estimation results. In addition, the probability density function (PDF) is the sum of homogeneous multivariate Gaussian distribution because only one global smoothing parameter is used. The APNN was applied to predict the compressive strength of concrete using actual test data from two concrete companies, and the estimated results of APNN were compared with those of PNN. APNN showed better results than PNN in predicting the compressive strength of concrete and provided the promising probabilistic viewpoints by using the individual standard deviation in a variable.

Open and closed-loop stochastic dynamics of a class of nonlinear chemical processes with multiplicative noise