The Generalized Cell Mapping Method in Nonlinear Random Vibration Based Upon Short-Time Gaussian Approximation

1990 ◽  
Vol 57 (4) ◽  
pp. 1018-1025 ◽  
Author(s):  
J. Q. Sun ◽  
C. S. Hsu
Keyword(s):  
Steady State ◽  
Gaussian Approximation ◽  
Transition Probability ◽  
Transition Probability Matrix ◽  
Generalized Cell Mapping ◽  
Mean Square Response ◽  
One Step ◽  
Step Transition ◽  
The One ◽  
Short Time

A short-time Gaussian approximation scheme is proposed in the paper. This scheme provides a very efficient and accurate way of computing the one-step transition probability matrix of the previously developed generalized cell mapping (GCM) method in nonlinear random vibration. The GCM method based upon this scheme is applied to some very challenging nonlinear systems under external and parametric Gaussian white noise excitations in order to show its power and efficiency. Certain transient and steady-state solutions such as the first-passage time probability, steady-state mean square response, and the steady-state probability density function have been obtained. Some of the solutions are compared with either the simulation results or the available exact solutions, and are found to be very accurate. The computed steady-state mean square response values are found to be of error less than 1 percent when compared with the available exact solutions. The efficiency of the GCM method based upon the short-time Gaussian approximation is also examined. The short-time Gaussian approximation renders the overhead of computing the one-step transition probability matrix to be very small. It is found that in a comprehensive study of nonlinear stochastic systems, in which various transient and steady-state solutions are obtained in one computer program execution, the GCM method can have very large computational advantages over Monte Carlo simulation.

The Reliability Analysis of Mechanical Leg (2UPS+UP) Based on a Discrete Time Repairable Markov Model

2015 ◽  
Vol 713-715 ◽  
pp. 760-763
Author(s):  
Jia Lei Zhang ◽  
Zhen Lin Jin ◽  
Dong Mei Zhao
Keyword(s):  
Steady State ◽  
Discrete Time ◽  
Transition Probability ◽  
Continuous Model ◽  
Transition Probability Matrix ◽  
State Equations ◽  
One Step ◽  
The Difference ◽  
Step Transition ◽  
The One

We have analyzed some reliability problems of the 2UPS+UP mechanism using continuous Markov repairable model in our previous work. According to the check and repair of the robot is periodic, the discrete time Markov repairable model should be more appropriate. Firstly we built up the discrete time repairable model and got the one step transition probability matrix. Secondly solved the steady state equations and got the steady state availability of the mechanical leg, by the solution of the difference equations the reliability and the mean time to first failure were obtained. In the end we compared the reliability indexes with the continuous model.

A Statistical Study of Generalized Cell Mapping

1988 ◽  
Vol 55 (3) ◽  
pp. 694-701 ◽  
Author(s):  
Jian-Qiao Sun ◽  
C. S. Hsu
Keyword(s):  
Stochastic Systems ◽  
Transition Probability ◽  
Transition Probability Matrix ◽  
Mapping Method ◽  
Cell Mapping ◽  
Generalized Cell Mapping ◽  
One Step ◽  
The Mean ◽  
Step Transition ◽  
The One

In this paper a statistical error analysis of the generalized cell mapping method for both deterministic and stochastic dynamical systems is examined, based upon the statistical analogy of the generalized cell mapping method to the density estimation. The convergence of the mean square error of the one step transition probability matrix of generalized cell mapping for deterministic and stochastic systems is studied. For stochastic systems, a well-known trade-off feature of the density estimation exists in the mean square error of the one step transition probability matrix, which leads to an optimal design of generalized cell mapping for stochastic systems. The conclusions of the study are illustrated with some examples.

A Nonuniform Cell Partition for the Analysis of Nonlinear Stochastic Systems

1998 ◽  
Vol 65 (4) ◽  
pp. 867-869 ◽  
Author(s):  
J. Q. Sun
Keyword(s):  
Ad Hoc ◽  
Stochastic Systems ◽  
Transition Probability ◽  
Nonlinear Stochastic Systems ◽  
Transition Probability Density ◽  
Generalized Cell Mapping ◽  
One Step ◽  
Step Transition ◽  
The One ◽  
Cell Partition

This paper presents a study of nonuniform cell partition for analyzing the response of nonlinear stochastic systems by using the generalized cell mapping (GCM) method. The necessity of nonuniform cell partition for nonlinear systems is discussed first. An ad hoc scheme is then presented for determining optimal cell sizes based on the statistical analysis of the GCM method. The proposed nonuniform cell partition provides a roughly uniform accuracy for the estimate of the one-step transition probability density function over a large region in the state space where the system varies significantly from being linear to being strongly nonlinear. The nonuniform cell partition is shown to lead to more accurate steady-state solutions and enhance the computational efficiency of the GCM method.

On the Data-Driven Generalized Cell Mapping Method

2019 ◽  
Vol 29 (14) ◽  
pp. 1950204 ◽  
Author(s):  
Zigang Li ◽  
Jun Jiang ◽  
Ling Hong ◽  
Jian-Qiao Sun
Keyword(s):  
Global Analysis ◽  
Transition Probability ◽  
Transition Probability Matrix ◽  
Mapping Method ◽  
Data Driven ◽  
Cell Mapping ◽  
Time Step ◽  
Generalized Cell Mapping ◽  
Step Transition ◽  
The One

Global analysis is often necessary for exploiting various applications or understanding the mechanisms of many dynamical phenomena in engineering practice where the underlying system model is too complex to analyze or even unavailable. Without a mathematical model, however, it is very difficult to apply cell mapping for global analysis. This paper for the first time proposes a data-driven generalized cell mapping to investigate the global properties of nonlinear systems from a sequence of measurement data, without prior knowledge of the underlying system. The proposed method includes the estimation of the state dimension of the system and time step for creating a mapping from the data. With the knowledge of the estimated state dimension and proper mapping time step, the one-step transition probability matrix can be computed from a statistical approach. The global properties of the underlying system can be uncovered with the one-step transition probability matrix. Three examples from applications are presented to illustrate a quality global analysis with the proposed data-driven generalized cell mapping method.

Stabilité de la récurrence nulle pour certaines chaines de Markov perturbées

1983 ◽  
Vol 20 (3) ◽  
pp. 482-504 ◽  
Author(s):  
C. Cocozza-Thivent ◽  
C. Kipnis ◽  
M. Roussignol
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Transition Probability ◽  
The Other ◽  
Barrier Functions ◽  
One Step ◽  
Null Recurrence ◽  
Step Transition ◽  
The One ◽  
Taylor's Formula

We investigate how the property of null-recurrence is preserved for Markov chains under a perturbation of the transition probability. After recalling some useful criteria in terms of the one-step transition nucleus we present two methods to determine barrier functions, one in terms of taboo potentials for the unperturbed Markov chain, and the other based on Taylor's formula.

Stochastic Response of a Vibro-Impact System via a New Impact-to-Impact Mapping

2021 ◽  
Vol 31 (08) ◽  
pp. 2150139
Author(s):  
Liang Wang ◽  
Bochen Wang ◽  
Jiahui Peng ◽  
Xiaole Yue ◽  
Wei Xu
Keyword(s):  
Transition Probability ◽  
Transition Probability Matrix ◽  
Significant Feature ◽  
Stochastic Response ◽  
Impact Surface ◽  
Impact System ◽  
The Matrix ◽  
Step Transition ◽  
The One ◽  
The Impact

In this paper, a new impact-to-impact mapping is constructed to investigate the stochastic response of a nonautonomous vibro-impact system. The significant feature lies in the choice of Poincaré section, which consists of impact surface and codimensional time. Firstly, we construct a new impact-to-impact mapping to calculate the one-step transition probability matrix from a given impact to the next. Then, according to the matrix, we can investigate the stochastic responses of a nonautonomous vibro-impact system at the impact instants. The new impact-to-impact mapping is smooth and it effectively overcomes the nondifferentiability caused by the impact. A linear and a nonlinear nonautonomous vibro-impact systems are analyzed to verify the effectiveness of the strategy. The stochastic P-bifurcations induced by the noise intensity and system parameters are studied at the impact instants. Compared with Monte Carlo simulations, the new impact-to-impact strategy is accurate for nonautonomous vibro-impact systems with arbitrary restitution coefficients.

Stabilité de la récurrence nulle pour certaines chaines de Markov perturbées

1983 ◽  
Vol 20 (03) ◽  
pp. 482-504
Author(s):  
C. Cocozza-Thivent ◽  
C. Kipnis ◽  
M. Roussignol
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Transition Probability ◽  
The Other ◽  
Barrier Functions ◽  
One Step ◽  
Null Recurrence ◽  
Step Transition ◽  
The One ◽  
Taylor's Formula

We investigate how the property of null-recurrence is preserved for Markov chains under a perturbation of the transition probability. After recalling some useful criteria in terms of the one-step transition nucleus we present two methods to determine barrier functions, one in terms of taboo potentials for the unperturbed Markov chain, and the other based on Taylor's formula.

Perturbation theory for unbounded Markov reward processes with applications to queueing

1988 ◽  
Vol 20 (01) ◽  
pp. 99-111 ◽  
Author(s):  
Nico M. Van Dijk
Keyword(s):  
Transition Probabilities ◽  
Arrival Rate ◽  
Average Reward ◽  
Reward Function ◽  
One Step ◽  
Reward Process ◽  
Markov Reward ◽  
Step Transition ◽  
The One ◽  
Queueing Processes

Consider a perturbation in the one-step transition probabilities and rewards of a discrete-time Markov reward process with an unbounded one-step reward function. A perturbation estimate is derived for the finite horizon and average reward function. Results from [3] are hereby extended to the unbounded case. The analysis is illustrated for one- and two-dimensional queueing processes by an M/M/1-queue and an overflow queueing model with an error bound in the arrival rate.

scholarly journals Transient analysis of a queue with queue-length dependent MAP and its application to SS7 network

1999 ◽  
Vol 12 (4) ◽  
pp. 371-392
Author(s):  
Bong Dae Choi ◽  
Sung Ho Choi ◽  
Dan Keun Sung ◽  
Tae-Hee Lee ◽  
Kyu-Seog Song
Keyword(s):  
Congestion Control ◽  
Queue Length ◽  
Transient Analysis ◽  
Transition Probabilities ◽  
Transient Behavior ◽  
Complex Model ◽  
Arrival Process ◽  
One Step ◽  
Step Transition ◽  
The One

We analyze the transient behavior of a Markovian arrival queue with congestion control based on a double of thresholds, where the arrival process is a queue-length dependent Markovian arrival process. We consider Markov chain embedded at arrival epochs and derive the one-step transition probabilities. From these results, we obtain the mean delay and the loss probability of the nth arrival packet. Before we study this complex model, first we give a transient analysis of an MAP/M/1 queueing system without congestion control at arrival epochs. We apply our result to a signaling system No. 7 network with a congestion control based on thresholds.

scholarly journals On the Convergence Rate of Bonus-Malus Systems

1992 ◽  
Vol 22 (2) ◽  
pp. 217-223 ◽  
Author(s):  
Heikki Bonsdorff
Keyword(s):  
Markov Chain ◽  
Probability Distribution ◽  
Convergence Rate ◽  
Transition Probabilities ◽  
Transition Probability ◽  
Transition Probability Matrix ◽  
Limit Probability ◽  
Step Transition

AbstractUnder certain conditions, a Bonus-Malus system can be interpreted as a Markov chain whose n-step transition probabilities converge to a limit probability distribution. In this paper, the rate of the convergence is studied by means of the eigenvalues of the transition probability matrix of the Markov chain.

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