Stochastic Harmonic Functions of Second Kind for Spectral Representations

Abstract:

The stochastic harmonic function of the second kind is proposed for representations of stochastic processes.It is firstly proved that,as long as the random phase angles and random circular frequencies are independent and uniformly distributed whereas the amplitudes are related to the target power spectral density function in a specified way,the power spectral density of the process represented by the stochastic harmonic function of the second kind is identical to the target power spectral density.Then,it is demonstrated that the process represented by the stochastic harmonic function of the second kind is asymptotically Gaussian.The rate of approaching Gaussian distribution is further studied by obtaining the onedimensional distribution via Pearson distribution.The study reveals the similarities between the stochastic harmonic functions of the first kind and the second kind.However,the application of the stochastic harmonic function of the second kind is more convenient than that of the first kind because the random circular frequencies are uniformly distributed.Finally,linear and nonlinear responses of a multidegreeoffreedom system subjected to random ground motions are analyzed to exemplify the effectiveness and superiorities of the proposed approach.