First-passage Problem Of Random Vibration Systems With Multi-Degrees-of-Freedom

For a physical or structural system excited by external or parametric stochasticexcitations, the response will spread randomly in the phase space. First-passage failurehappens when the state of the system leaves a certain domain (safe or admissible domain)for the first time. First-passage problem has a long history, and it emerges from a widerange of stochastic phenomena, such as neuron firing, chemical reaction rates, thetriggering of stock options and stochastic structural dynamics, etc.. In engineering, largescale bridges and high buildings are usually excited by wind and atmospheric turbulence,which will cause random vibration. When the amplitude of the response exceeds a criticalvalue, the structure will fail. First-passage is a major failure mode of structural ormechanical system. For such systems, reliability is defined as the probability of the systemresponse being in a safe or admissible domain. First-passage time is the lifetime ofmechanical or structural systems. First-passage problem is related with the reliability of astructural or mechanical system. So it is theoretically and practically significant toevaluate the reliability and lifetime of stochastically excited dynamical systems.However, the first-passage problem is among the most difficult problems in the theoryof stochastic dynamics. At present, a mathematical exact solution is possible only if therandom phenomenon in question can be treated as a diffusion Markovian process. Forhomogeneous diffusion processes, the conditional reliability function, namely, theprobability that the response remains within the safety domain with a given initial state init, is governed by the backward Kolmogorov (BK) equation and the moments of thefirst-passage time is governed by generalized Pontryagin equations. Known exactlyanalytical solutions are limited to one-dimensional case. In practice, random vibrationssystems are often strongly nonlinear systems with multi-degrees-of-freedom (MDOF). Toobtain the conditional reliability function and mean first-passage time, one has to solve thecorresponding BK equation and Pontryagin equation, which are both high-dimensionalpartial differential equations (PDEs). So it is difficult to study the first-passage problem ofMDOF random vibration systems.Based on averaging method and the theory of diffusion Markovian processes,first-passage problems of several classes of strong nonlinear random vibration systemswith MDOF are studied in this paper. In chapter1, some background introduction is given. The main work of this paper isalso listed.In chapter2, the stochastic averaging method based on generalized harmonic functionis introduced. The stochastic excitations are often modeled as Gaussian white noise andwide-band noise. The stochastic averaging methods for MDOF strongly nonlinear systemssubject to Gaussian white noise excitation, wide-band noise excitations, combinedharmonic and wide-band noise excitations, are introduced. The averaged It stochasticdifferential equations (SDEs) about amplitude and the combinations of phase angles areobtained, which facilitates the establishment of BK equation and Pontryatin equation.In chapter3, the first-passage problem of MDOF strongly nonlinear systems subject toGaussian white noise excitations is studied. The BK equation and Pontryagin equation areestablished based on the averaged It SDEs about amplitude in the case of non-internalresonance. Numerical example is given to show the effectiveness of the theoreticalmethod.In chapter4, the first-passage problem of MDOF strongly nonlinear systems subject towide-band noise excitations is studied. Wide-band noise is a more appropriate model inengineering. To use averaging method in this case, both stochastic averaging anddeterministic averaging must be performed, which is different from the case of Gaussianwhite noise excitations. Similar as the procedures in chapter3, BK equation andPontryagin equation are established for obtaining the conditional reliability function andmean first-passage time. A numerical example is given to illustrate the theoretical method.In chapter5, the first-passage problem of MDOF strongly nonlinear systems subject tocombined harmonic and wide-band noise excitations is studied. Due to the presence ofharmonic excitation and the coupling effects of MDOF system, various resonant casesmust be distinguished during analysis. In the case of resonance (external resonance, bothexternal and internal resonance), the combinations of phase angles are introduced. Theaveraging procedures are completed to obtain the It SDEs about the amplitudes and thecombinations of phase angles. Then the BK equation and Pontryagin equation areestablished to obtain the reliability function and mean first-passage time. Two numericalexamples are also given.In chapter6, the summary of this paper is given. Some research directions about thefirst-passage problem in near future are also pointed out.