• Research paper •

Stochastic harmonic function and spectral representations

Chen Jianbin,Li Jie

1. State Key Laboratory of Disaster Reduction in Civil Engineering & School of Civil Engineering, Tongji University, Shanghai 200092, China
• Received:2010-06-02 Revised:2010-11-25 Online:2011-05-25 Published:2011-05-16
• Contact: Chen Jianbin

Abstract: Stochastic harmonic function representations and their properties are studied. In the paper, it is firstly proved that as the distributions of the random frequencies are consistent with the target power spectral density function, the power spectral density of the stochastic harmonic process is identical to the target power spectral density. Further, it is proved that the stochastic harmonic process is asymptotically normally distributed. The rate of approaching normal distribution is discussed by adopting Pearson distribution to describe the one-dimensional distribution of the stochastic harmonic process. Compared to existing representations of stochastic process, very few stochastic harmonic components can capture the exact target power spectral density. This greatly reduces the number of the random variables and thus eases the difficulty of stochastic dynamics. Finally, linear and nonlinear responses of a multi-degree-of-freedom system subjected to random ground motions are carried out to exemplify the effectiveness and advantages of the stochastic harmonic representations. Keywords: Stochastic harmonic function, power spectral density function, covariance function, stationary process, nonlinearity

Key words: nonlinearity

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