Reliability Of Internally Resonant Random Vibration System Based On First-passage Model

Stochastic excitation is ubiquitous in engineering,such as ground vibration caused by earthquake,atmospheric turbulence,the sea wave,road surface roughness and other incentives.They are all stochastic excitation.Because the engineering structures are under a variety of stochastic vibration while working,the application of stochastic vibration is very broad,such as aviation,automobile,civil engineering,etc.Also the influence of stochastic vibration on the engineering structure reliability is obviously important.In the classical vibration theory,the situation of engineering structures under the action of deterministic excitations such as harmonic force has been studied for a long time and the theory has been well developed.But it is obviously not enough now.Neglecting the influence of stochastic excitation in some cases may be wrong.As a result stochastic vibration research has received widespread attention recently.Studying the reliability of engineering structure under the condition of stochastic vibration has practical significance.Multi-degree-of-freedom(MDOF)nonlinear system is often studied in the vibration analysis.There may be internal resonance in the MDOF system.The internal resonance has a great influence on the reliability of the system.This dissertation is funded by the National Natural Science Foundation of China(Grant Nos.11272201,11372271,11132007)and investigates the reliability of MDOF system under stochastic excitation in the case of resonance.The main research and achievements are as follows:(1)The first-passage problem of MDOF system under Gaussian white noise in the case of internal resonance is studied.The average It? stochastic differential equations in the case of internal resonance are obtained by using averaging method based on the generalized harmonic function.Based on the average It? stochastic differential equation,the backward Kolmogorov equation which determines the conditional reliability function and the Pontryagin equation which determines the averaged life of system are established.The initial and boundary conditions of the two high dimensional partial differential equations have also been defined.The first passage reliability problem in the case of 1:1 internal resonance is studied by an example of a two-degree-of-freedom damped Duffing-van der Pol system.The conditional reliability functions and the averaged first-passage time are obtained.The results of 1:1 internal resonance are compared with those of non-internal resonance to discuss the influence of internal resonance on the reliability of the system.Then the numerical results are compared with the results of Monte Carlo simulation in order to verify the correctness of the theoretical analysis.(2)The first-passage reliability of MDOF quasi-integrable Hamiltonian systems under wide-band noise in the case of internal resonance is studied.The average It? stochastic differential equations are obtained based on Stratonovich-Khasminskii limit theorem.The backward Kolmogorov equation which determines the conditional reliability function and the Pontryagin equation which determines the averaged life of system in the case of internal resonance have been established.The first passage reliability problem in the case of 1:1 internal resonance is studied by an example of a two degree of freedom damped and coupled Duffing-van der Pol system.Then the numerical results are compared with the Monte Carlo simulation results in order to verify the correctness of the theoretical analysis.(3)The first-passage reliability of MDOF nonlinear system under combined harmonic force and Gaussian white noise excitations in the case of internal/external resonance is studied.The average It? stochastic differential equations are obtained by using generalized function average method.The conditional reliability functions in the case of internal/external are defined.The backward Kolmogorov equation which determines the conditional reliability function and the Pontryagin equation which determines the averaged life of system are established on the basis of the average It? stochastic differential equation.The initial and boundary conditions of the two high dimensional partial differential equations have also been defined.The first passage reliability problem in the case of 1:1 internal resonance and(or)1:1 external resonance is studied by an example of a two degree of freedom damped coupled Duffing-van der Pol system.The influence of internal resonance and external resonance on the reliability of the system is discussed.Then the numerical results are compared with the Monte Carlo simulation results in order to verify the correctness of the theoretical analysis.