西安电子科技大学 first passage of fractional-derivative stochastic systems with power-form...

西安电子科技大学西安电子科技大学

First Passage of Fractional-derivative Stochastic Systems with Power-form

Restoring Force1)Wei. Li (李伟 ), 2)Natasa. Trisovic

1)School of Mathematics and Statistics, Xidian University, China2)Faculty of Mechanical Engineering, University of Belgrade, Serbia

04/20/23


Part 1 Background and Introduction

Part 2 Mathematical model and formulations

Part 3 BK equation and GP equation associated with first-passagePart 4 Numerical results and Monte-Carlo simulation

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Background of Stochastic Dynamical System

Background

• Whether the structures can keep safe and

stable?

•Is it possible to break down during the

vibration?

•How much probability the structures can

survive from stochastic vibration?

SDSDSS

DampiDampingng

RestorinRestoring forceg force

Random Random excitatioexcitatio

nn

Fractional Fractional derivative derivative

IntegerInteger

orderorder

Power-Power-formform

polynomipolynomialal

Gaussian Gaussian whitewhite

Background

•R. L. Bagley, P. J. Torvik. (1980)•Ivana Kovacic, Zvonko Rakaric (2012, 2013)

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Background

Fractional derivative has been successfully used in

• Environmental Engineering;

• Viscoelasticity Material;

• Biomedical Science

• Vibrated dynamical systems.


L. C. Chen, W. Q. Zhu. ——fractional derivative damping——stochastic averaging method ——Stochastic jump and bifurcation

•Z. L. Huang, X. L. Jin——fractional derivative damping——stochastic averaging method——stationary response and stability•P. D. Spanos, G. I. Evangelatos——fractional derivative restoring force——time domain simulation and statistical linearization——response

Applications in SDS


Aims to determine the probability that the response of a randomly excited dynamical system reaches the boundary of a bounded domain of state space within its lifetime.

First Passage


Our Work

Fractional derivative

Power-formrestoring force

Gaussianexcitation

First passage of Stochastic dynamical

system


Mathematical model


0( ) ( ) ( )

tt a d t


Reliability function

Energy process H(t) of the system


Reliability function satisfies a Backward Kolmogorov (BK) equation


Moment of first-passage time

Satisfies the Generalized Pontryagin (GP)

equation


Example

0 0.05, 3.5, 11 0.05,D 1,c 5,cH

First passage of SDS with fractional derivative and power-form restoring fore

Conclusions

Bigger boundary value of safe domain and strong nonlinearity of restoring force are advantageous to improve system reliability

Monte-Carlo simulation proved that the proposed methods and procedures are correct and efficient.


Thank you for your attention！

西安电子科技大学 first passage of fractional-derivative stochastic systems with power-form...

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