Stochastic harmonic function and spectral representations

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