An approximate method is presented for estimating the power spectral density of the response of Duffing's oscillator when it is driven with Gaussian white noise.
When T R ≥ 0.2 the power spectral density of the response shows power only at integer multiples of the pulsing frequency, indicating that the response is fully entrained by the forcing.
The right panel, which is a heat map showing the power spectral density of the response as a function of pulsing frequency, shows that similar entrainment occurs over a wide range of frequencies: the bright lines which correspond to peaks in the power spectral density have integer slopes, indicating that the power in the response is concentrated at harmonics of the forcing frequency.
For a given subject, the likelihood function of the observed data and the missing data indicator is L(, ψ, r, y obs, z obs ) = ∫ pψ (r| y obs, y mis, z obs, z mis ) p (y obs, y mis, z obs, z mis ) d (y mis, z mis ), where the density p (y obs, y miss, z obs, z mis ) denotes the joint density of the responses and the covariates.
The apparent mass and the head vibration transmission responses were derived using two different frequency response function estimators based upon the cross- and auto-spectral densities of the response and excitation signals, denoted as H1 and Hv estimators, respectively.
The coupling effects of dual-axis vibration were investigated using two different frequency response function estimators based upon the cross- and auto-spectral densities of the response and excitation signals, denoted as H1 and Hv estimators, respectively.
Koenker (2005, Chapter 5, Subsection 5.3) suggested that when the conditional densities of the response are heterogeneous, it is natural to consider whether weighted quantile regression might lead to efficiency improvements.
The following are the four reasons for the proposed weight in (7): (1) Koenker (2005, Chapter 5, Subsection 5.3) suggested that when the conditional densities of the response are heterogeneous, it is natural to consider whether weighted quantile regression might lead to efficiency improvements.
The idea is to approximate the cross power spectral density matrix of the response by a low rank matrix.
For White excitations, we solve the Fokker Plank Kolmogorov equation for the exact joint probability density function of the response.
The RSE allows to determine the mean-value and power spectral density function of the response as approximate explicit functions of the design parameters.
