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.

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.

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.

The Stochastic Response of a Class of Impact Systems Calculated by a New Strategy Based on Generalized Cell Mapping Method

2018 ◽  
Vol 85 (5) ◽  
Author(s):  
Liang Wang ◽  
Shichao Ma ◽  
Chunyan Sun ◽  
Wantao Jia ◽  
Wei Xu
Keyword(s):  
Transition Probability ◽  
Transition Probability Matrix ◽  
Mapping Method ◽  
Cell Mapping ◽  
Stochastic Response ◽  
Generalized Cell Mapping ◽  
Significant Difference ◽  
New Strategy ◽  
Almost All ◽  
The Impact

In this paper, a new strategy based on generalized cell mapping (GCM) method will be introduced to investigate the stochastic response of a class of impact systems. Significant difference of the proposed procedure lies in the choice of a novel impact-to-impact mapping, which is built to calculate the one-step transition probability matrix, and then, the probability density functions (PDFs) of the stochastic response can be obtained. The present strategy retains the characteristics of the impact systems, and is applicable to almost all types of impact systems indiscriminately. Further discussion proves that our strategy is reliable for different white noise excitations. Numerical simulations verify the efficiency and accuracy of the suggested strategy.

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.

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.

Ideal flow of Markov Chain

2018 ◽  
Vol 10 (06) ◽  
pp. 1850073
Author(s):  
Kardi Teknomo
Keyword(s):  
Transition Probability ◽  
Transition Probability Matrix ◽  
Scaling Factor ◽  
Matrix Perturbation ◽  
Flow Network ◽  
Ideal Flow ◽  
The Matrix ◽  
Strongly Connected ◽  
Flow Equilibrium ◽  
Matrix Perturbation Analysis

Ideal flow network is a strongly connected network with flow, where the flows are in steady state and conserved. The matrix of ideal flow is premagic, where vector, the sum of rows, is equal to the transposed vector containing the sum of columns. The premagic property guarantees the flow conservation in all nodes. The scaling factor as the sum of node probabilities of all nodes is equal to the total flow of an ideal flow network. The same scaling factor can also be applied to create the identical ideal flow network, which has from the same transition probability matrix. Perturbation analysis of the elements of the stationary node probability vector shows an insight that the limiting distribution or the stationary distribution is also the flow-equilibrium distribution. The process is reversible that the Markov probability matrix can be obtained from the invariant state distribution through linear algebra of ideal flow matrix. Finally, we show that recursive transformation [Formula: see text] to represent [Formula: see text]-vertices path-tracing also preserved the properties of ideal flow, which is irreducible and premagic.

scholarly journals A note on the matrix renewal function

1972 ◽  
Vol 13 (4) ◽  
pp. 417-422 ◽  
Author(s):  
A. M. Kshirsagar ◽  
Y. P. Gupta
Keyword(s):  
Generalized Inverse ◽  
Transition Probability ◽  
Renewal Theory ◽  
Transition Probability Matrix ◽  
Markov Renewal Process ◽  
Renewal Function ◽  
Stieltjes Transform ◽  
Imbedded Markov Chain ◽  
The Matrix ◽  
Passage Times

AbstractThe Laplace-Stieltjes Transform m(s) of the matrix renewal function M(t) of a Markov Renewal process is expanded in powers of the argument s, in this paper, by using a generalized inverse of the matrix I–P0, where P0 is the transition probability matrix of the imbedded Markov chain. This helps in obtaining the values of moments of any order of the number of renewals and also of the moments of the first passage times, for large values of t, the time. All the results of renewal theory are hidden under the Laplacian curtain and this expansion helps to lift this curtain at least for large values of t and is thus useful in predicting the number of renewals.

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.

scholarly journals Stochastic LU factorizations, Darboux transformations and urn models

2018 ◽  
Vol 55 (3) ◽  
pp. 862-886 ◽  
Author(s):  
F. Alberto Grünbaum ◽  
Manuel D. de la Iglesia
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Free Parameter ◽  
Transition Probabilities ◽  
Transition Probability ◽  
Transition Probability Matrix ◽  
Urn Models ◽  
Triangular Matrices ◽  
New Family ◽  
Step Transition

Abstract We consider upper‒lower (UL) (and lower‒upper (LU)) factorizations of the one-step transition probability matrix of a random walk with the state space of nonnegative integers, with the condition that both upper and lower triangular matrices in the factorization are also stochastic matrices. We provide conditions on the free parameter of the UL factorization in terms of certain continued fractions such that this stochastic factorization is possible. By inverting the order of the factors (also known as a Darboux transformation) we obtain a new family of random walks where it is possible to state the spectral measures in terms of a Geronimus transformation. We repeat this for the LU factorization but without a free parameter. Finally, we apply our results in two examples; the random walk with constant transition probabilities, and the random walk generated by the Jacobi orthogonal polynomials. In both situations we obtain urn models associated with all the random walks in question.

Some explicit results for correlated random walks

1989 ◽  
Vol 26 (4) ◽  
pp. 757-766 ◽  
Author(s):  
Ram Lal ◽  
U. Narayan Bhat
Keyword(s):  
Markov Chain ◽  
State Space ◽  
Equilibrium Distribution ◽  
Transition Probabilities ◽  
Transition Probability ◽  
Transition Probability Matrix ◽  
Correlated Random Walks ◽  
Finite State ◽  
Direction Of Movement ◽  
Step Transition

In a correlated random walk (CRW) the probabilities of movement in the positive and negative direction are given by the transition probabilities of a Markov chain. The walk can be represented as a Markov chain if we use a bivariate state space, with the location of the particle and the direction of movement as the two variables. In this paper we derive explicit results for the following characteristics of the walk directly from its transition probability matrix: (i) n -step transition probabilities for the unrestricted CRW, (ii) equilibrium distribution for the CRW restricted on one side, and (iii) equilibrium distribution and first-passage characteristics for the CRW restricted on both sides (i.e., with finite state space).

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