A coupled model for train-track-bridge stochastic analysis with consideration of spatial variation and temporal evolution

Due to random characteristics of system parameters and excitations, the dynamic assessment and prediction for the train-track-bridge interaction systems become rather complex issues needing to be addressed, especially considering the longitudinal inhomogeneity and uncertainty of dynamic properties in physics and correspondingly their temporal evolutions. In this paper, a temporal-spatial coupled model is developed to fully deal with the deterministically/non-deterministically computational and analytical matters in the train-track-bridge interactions with a novelty, where a train-track-bridge interaction model is newly developed by effectively coupling the three-dimensional nonlinear wheel-rail contact model and the finite element theory, moreover, the Monte-Carlo method (MCM) and Karhunen–Loève expansion (KLE) are effectively united to model the random field of track-bridge systems, and a spectral evolution method accompanied by a track irregularity probabilistic model are introduced to select the most representative track irregularity sets and to characterize their random evolutions in temporal dimension. In terms of random vibration analysis, the high-efficiency and effectiveness of this developed model is validated by comparing to a robust method, i.e., MCM. Apart from validations, multi-applications of the temporal-spatial coupled model from aspects of deterministic computation, random vibration, resonant analysis and long-term dynamic prediction, etc., have been fully presented to illustrate the universality of the proposed model.     

Predicting stochastic characteristics of generalized eigenvalues via a novel sensitivity-based probability density evolution method

This paper proposes a novel numerical method for predicting the probability density function of generalized eigenvalues in the mechanical vibration system with consideration of uncertainties in structural parameters. The eigenproblem of structural vibration is presented by first and the sensitivity of generalized eigenvalues with respect to structural parameters can be derived. The probability density evolution method is then developed to capture the probability density function of generalized eigenvalues considering uncertain material properties. Within the proposed method, the probability density evolution equation for the generalized eigenvalue problem is established accounting for the sensitivity of generalized eigenvalues with respect to structural parameters. A new variable which connects generalized eigenvalues to structural parameters is then introduced to simplify the original probability density evolution equation. Next, the simplified probability density evolution equation is solved by using the finite difference method with total variation diminishing schemes. Finally, the probability density function as well as the second-order statistical quantities of generalized eigenvalues can be predicted. Numerical examples demonstrate that the proposed method yields results consistent with Monte-Carlo simulation method within significantly less computation time and the coefficients of variation of uncertain parameters as well as the total number of them have remarkable effects on stochastic characteristics of generalized eigenvalues.     

Towards a general condition based maintenance model for a stochastic dynamic system

This paper considers a stochastic dynamic system subject to random deterioration, with regular condition monitoring and preventive maintenance. A model is presented to advise at a monitoring check what maintenance action to take based upon the condition monitoring and preventive maintenance information obtained to date. A general assumption adopted in the paper is that the performance of the system concerned can not be described directly by the monitored information, but is correlated with it stochastically. The model is relevant to a large class of condition monitoring techniques currently employed in industry including vibration and oil analysis. The model is constructed under fairly general conditions and includes two novel developments. Firstly, the concept of the conditional residual time is used to measure the condition of the monitored system at the time of a monitoring check, and secondly, contrary to previous practice, the monitored observation is now assumed to be a function of the system condition. Relationships between the observed history of condition monitoring, preventive maintenance actions, and the condition of the system are established. Methods for estimating model parameters are discussed. Since the model presented is generally beyond the scope for an analytical solution, a numerical approximation method is also proposed. Finally, a case example is presented to illustrate the modelling concepts in the case of non-maintained plant.     

Parameter analysis based on stochastic model for differential evolution algorithm

In this paper, a stochastic model is used to describe and analyze the evolution process of differential evolution (DE) for numerical optimization. With the model, it illustrates how the probability distribution of the whole population is changed by mutation, selection and crossover operations. Based on the theoretical analysis, some guidelines about the parameter setting for DE are provided. In addition, numerical simulations are carried out to verify the conclusions drawn from model analysis.     

A probability model for computer system overhead

Two models for computer system overhead are developed by considering the number of jobs in the system as an immigration-death process. The models developed relate the death rates to the state of the process. The first model uses the total numver of jobs in the system as the state of the processe. The second model classifies the jobs in the system according to the priority classes unsed in the computer system.In the model based on the total number of jobs in the system, the death rate when the system is in the state i, μ_i, is μ_i = μ min(i, x₀) where μ and x₀ are parameters to be estimated by maximum likelihood. In the second model, the death rate for jobs in the k^th priority class when the state of the system is i = (i₁…,i_p), μ_i^(k) is given by $\text{μ}{}_{\mathrm{i}}{}^{\mathrm{(k)}}\text{=μ}{}^{\mathrm{(k)}}\text{i}{}_{\mathrm{k}}\text{min}\text{(1,}\text{x}{}_0\text{i·1}\text{)}$. Computational difficulties in finding the maximum likelihood estimates of the parameters for this model are noted.     

A new forecasting method for time continuous model of dynamic system

Usually a linear differential equation is used to represent continuous dynamic systems, but a linear difference equation is used to represent discrete dynamic systems. AGO is one of the most important characteristics of grey theory, and its main purpose is to reduce the randomness of data. A linear differential equation, instead of a linear difference equation, is used to replace the grey differential equation to analyze discrete systems in this paper. Approximating a k-order derivative by operating after spline curve fitting of 1-AGO data, a model is directly established by means of the least square method. ARIMA models are used to analyze the leading indicator in advance, and the Fourier series with suitably chosen values of parameters is used in the fitting of leading indicator. This model is called the GDM(2, 2, 1) model.     

A dynamic stochastic programming model for international portfolio management

We develop a multi-stage stochastic programming model for international portfolio management in a dynamic setting. We model uncertainty in asset prices and exchange rates in terms of scenario trees that reflect the empirical distributions implied by market data. The model takes a holistic view of the problem. It considers portfolio rebalancing decisions over multiple periods in accordance with the contingencies of the scenario tree. The solution jointly determines capital allocations to international markets, the selection of assets within each market, and appropriate currency hedging levels. We investigate the performance of alternative hedging strategies through extensive numerical tests with real market data. We show that appropriate selection of currency forward contracts materially reduces risk in international portfolios. We further find that multi-stage models consistently outperform single-stage models. Our results demonstrate that the stochastic programming framework provides a flexible and effective decision support tool for international portfolio management.     

Nonlinear dynamic analysis of a complex dual rotor-bearing system based on a novel model reduction method

In this paper, a dynamic model of a complex dual rotor-bearing system of an aero-engine is established based on the finite element method with three types of beam elements (rigid disc, cylindrical beam element and conical beam element), as well as taking into account the nonlinearities of all of the supporting rolling element bearings. To rapidly and accurately analyze dynamic behaviors of the complex dual rotor-bearing system, a two-level model order reduction (MOR) method is proposed by combining component mode synthesis (CMS) method and proper orthogonal decomposition (POD) technique. The first-level reduced-order model (ROM) of the dual rotors is obtained by CMS method with a high precision for the original system. Then, the POD method is applied to second-level model order reduction to further decrease the degrees of freedom (DOFs) of first-level ROM. Second-level ROM with mode expansion and direct second-level ROM are obtained, and the nonlinear displacement responses of the two ROMs are compared with the first-level ROM. The numerical results demonstrate that the proposed method has a higher computational efficiency and accuracy in terms of mode expansion than the direct model reduction by using POD method. In addition, the nonlinear vibration responses of the dual rotor-bearing system are studied by this second-level ROM in the case of different clearances of the inter-shaft bearing. The results indicate that the dynamic characteristics of the dual rotor-bearing system are very complicated for a large clearance.     

Analysis of stochastic dual dynamic programming method

In this paper we discuss statistical properties and convergence of the Stochastic Dual Dynamic Programming (SDDP) method applied to multistage linear stochastic programming problems. We assume that the underline data process is stagewise independent and consider the framework where at first a random sample from the original (true) distribution is generated and consequently the SDDP algorithm is applied to the constructed Sample Average Approximation (SAA) problem. Then we proceed to analysis of the SDDP solutions of the SAA problem and their relations to solutions of the “true” problem. Finally we discuss an extension of the SDDP method to a risk averse formulation of multistage stochastic programs. We argue that the computational complexity of the corresponding SDDP algorithm is almost the same as in the risk neutral case.     

A stochastic production planning model with a dynamic chance constraint

This paper deals with the optimal production planning for a single product over a finite horizon. The holding and production costs are assumed quadratic as in Holt, Modigliani, Muth and Simon (HMMS) [7] model. The cumulative demand is compound Poisson and a chance constraint is included to guarantee that the inventory level is positive with a probability of at least α at each time point. The resulting stochastic optimization problem is transformed into a deterministic optimal control problem with control variable and of the optimal solution is presented. The form of state variable inequality constraints. A discussion the optimal control (production rate) is obtained as follows: if there exists a time t₁ such that t₁?[O, T]where T is the end of the planning period, then (i) produce nothing until t₁ and (ii) produce at a rate equal to the expected demand plus a ‘correction factor’ between t₁ and T. If t₁ is found to be greater than T, then the optimal decision is to produce nothing and always meet the demand from the inventory.     

Effective dynamics of a coupled microscopic-macroscopic stochastic system

A conceptual model for microscopic-macroscopic slow-fast stochastic systems is considered. A dynamical reduction procedure is presented in order to extract effective dynamics for this kind of systems. Under appropriate assumptions, the effective system is shown to approximate the original system, in the sense of a probabilistic convergence.     

A dynamic analysis of a coupled beam/slider system

The paper deals with transverse vibration of a beam with moving boundary conditions. In order to examine the transfer of energy between a moving support and a vibrating beam under assumption of zero slope of the beam elastic line at the moving support, dynamic analysis of a coupled beam/slider system is carried out. The moving support is modelled as a slider attached to a spring which realizes definite boundary conditions. Equations of motion are derived using Hamilton’s principle. Because length of the beam varies appropriate transformations of time and position variables are made to convert the solution space into a rectangle and, subsequently, to solve the partial differential equation of transverse vibration of the beam using the FDM. The phenomenon of the energy flow between the slider and the beam is a subject of the detailed analysis. The beam vibration with a fixed formula of length is examined, too. The dynamic characteristics of the system is brought forward from spectral analysis of numerical solutions.     

Synchronized stationary distribution of stochastic coupled systems based on graph theory

This paper deals with the synchronized stationary distribution of stochastic coupled systems. The response system is constructed to help achieve a synchronized stationary distribution. Firstly, an error system obtained by the drive system and the response system is given and an appropriate Lyapunov function for the error system is constructed. On the basis of the graph theory and the Lyapunov method, some sufficient conditions are proposed to guarantee the existence of a stationary distribution for the error system, which reflects the coupling structure has a close relationship with synchronized stationary distribution. Then, an application to stochastic coupled oscillators is presented and sufficient conditions are obtained to illustrate the feasibility of the theoretical results. Finally, a numerical example is provided to demonstrate the effectiveness of theoretical results.     

Generalized dynamic model of a system moving in an external field with stochastic components

Models are proposed to describe the dynamics of a system affected by external perturbations. Multicomponent fields with stochastic components are assumed to play a part in the perturbations.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 107, No. 3, pp. 433–438, June, 1996.     

A probability wave description of stochastic variables whose means satisfy a system of difference equations

Probability wave theory is used to study the behavior of stochastic vectors whose means satisfy ordinary first-order difference equations. Difference-differential equations are given for the probability waves corresponding to the difference model for the means. Analogues of the Liouville and Ehrenfest theorems are proved. A first-order difference equation for the evolution of the component dispersion of the random vector is obtained. An algorithm for solving the wave equations is proposed. The results from analyzing some solutions to the probability wave equations are presented. The relationship of the finite-difference method to the manifestation of the particle-wave dualism is discussed. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 115. No. 1, pp. 56–76. April. 1998.     

Quasilinearization method for coupled dynamic problems of thermoviscoelasticity

Use of the quasilinearization method is proposed for the solution of coupled dynamic (particularly, quasistatic) problems of thermoviscoelasticity under cyclic loading. The coupled problems under consideration here include vibrations of a one-dimensional body (beam, plate, shell) and shear vibrations of a hollow cylinder made of a material with temperature-dependent properties, with the principle of temperature-time analogy applicable in the latter case. The quasilinearization method is shown to be a fast converging one when applied to the solution of these problems.Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Mekhanika Polimerov, No. 2, pp. 310–316, March–April, 1976.     

A stochastic and dynamic model for the single-vehicle pick-up and delivery problem

In this paper a stochastic and dynamic model for the Pick-up and Delivery Problem is developed and analyzed. Demands for service arrive according to a Poisson process in time. The pick-up locations of the demands are independent and uniformly distributed over a service region. A single vehicle must transport the demands from the pick-up to the delivery location. Once a demand has been picked up it can only be dropped off at its desired delivery location. The delivery locations are independent and uniformly distributed over the region, and they are independent of the pick-up locations. The objective is to minimize the expected time in the system for the demands. Unit-capacity vehicle and multiple-capacity vehicle variations are considered. For each variation, bounds on the performance of the routing policies are derived for light and heavy traffic. The policies are analyzed using both analytical methods and simulation.     

Prediction of output response probability of sound environment system using simplified model with stochastic regression and fuzzy inference

The traditional standard stochastic system models, such as the autoregressive (AR), moving average (MA) and autoregressive moving average (ARMA) models, usually assume the Gaussian property for the fluctuation distribution, and the well-known least squares method is applied on the basis of only the linear correlation data. In the actual sound environment system, the stochastic process exhibits various non-Gaussian distributions, and there exist potentially various nonlinear correlations in addition to the linear correlation between input and output time series. Consequently, the system input and output relationship in the actual phenomenon cannot be represented by a simple model. In this study, a prediction method of output response probability for sound environment systems is derived by introducing a correction method based on the stochastic regression and fuzzy inference for simplified standard system models. The proposed method is applied to the actual data in a sound environment system, and the practical usefulness is verified.     

A stochastic dynamic programming model for valuing exclusivity from encroachment in franchising

Valuing territorial exclusivity in franchising is difficult because of the uncertainty associated with variables such as future franchise sales and brand strength. We present a stochastic dynamic programming model to value the exclusivity option from the perspective of both the franchisor and the franchisee. When there is positive value to the franchisor of including the exclusivity option in the contract, and to the franchisee of purchasing this option, the likelihood of franchisor-franchisee encroachment-related conflict is reduced. We also discuss structural results and explain our results using a numerical example.