4.1. Input-Output Spectral Relationship

Figure 3 outlines the broad steps in the cumulant spectral analysis of a LTI system. The relationship of the nth-order ( n ? 2) cumulant spectra of the stationary input Q and output y is given as45 ^, 46:

(46)

where H _Qy( ?) is the linear transfer function:

(47)

for the governing equation:

(48)

In modal analysis of a fixed platform, m, c and k are structural modal mass, damping and stiffness, respectively. Here, the first vibration mode is considered for its largest contribution to the platform response, with the natural frequency . Obviously S _y( ? ₁, ? ₂, , ? _{n -1}) depends on S _Q( ? ₁, ? ₂, , ? _{n -1}), where the latter is available through the ( n - 1)-dimensional Fourier transform of the nth-order cumulant functions of Q44:

(49)

Similarly, if S _y( ? ₁, , ? _{n -1}) is known, the nth-order cumulant functions of displacement can be estimated through the inverse Fourier transform:

(50)

By choosing n = 2, 3 and 4, S( ? ₁, , ? _{n -1}) represents the power-spectrum, bi-spectrum and tri-spectrum respectively and R( ? ₁, , ? _{n -1}) represents the autocorrelation function, third- and fourth-order cumulant functions. Setting ?