1. 西安电子科技大学 First Passage of Fractional-derivative Stochastic Systems with Power-form Restoring Force 1) Wei. Li ( 李伟 ), 2) Natasa. Trisovic 1) School of Mathematics and Statistics, Xidian University, China 2) Faculty of Mechanical Engineering, University of Belgrade, Serbia 11/14/2015

2. 西安电子科技大学 Part 1 Background and Introduction Part 2 Mathematical model and formulations Part 3 BK equation and GP equation associated with first-passage Part 4 Numerical results and Monte-Carlo simulation OUTLINE

3. 西安电子科技大学 Background of Stochastic Dynamical System

4. 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?

5. SD S Damping Restoring force Random excitation Fractional derivative Integer order order Power-form polynomial Gaussian white Background R. L. Bagley, P. J. Torvik. (1980) Ivana Kovacic, Zvonko Rakaric (2012, 2013) whitecolor

6. 西安电子科技大学 Background Fractional derivative has been successfully used in Environmental Engineering; Viscoelasticity Material; Biomedical Science Vibrated dynamical systems.

7. 西安电子科技大学 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

8. 西安电子科技大学 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

9. 西安电子科技大学 Our Work Fractional derivative Power-form restoring force Gaussian excitation First passage of Stochastic dynamical system

10. 西安电子科技大学 Mathematical model

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16. 西安电子科技大学 Reliability function Energy process H(t) of the system

17. 西安电子科技大学 Reliability function satisfies a Backward Kolmogorov (BK) equation

18. 西安电子科技大学 Moment of first-passage time Satisfies the Generalized Pontryagin (GP) equation

19. 西安电子科技大学 Example

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21. 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.

22. 西安电子科技大学 Thank you for your attention ！