The analytical trispectrum for multiple degree-of-freedom systems possessing cubic nonlinearity

Higher-order spectra have become a useful tool in spectral analysis, particularly for identifying the presence and sometimes type of nonlinearity in a system. Two such spectra that have figured prominently in signal processing are the bispectrum and trispectrum. The bispectrum is well-suited to capturing the presence of quadratic nonlinearities in system response data while the trispectrum has proved useful in detecting cubic nonlinearities. In a previous work, the authors developed an analytical solution for the auto-bispectrum for multi-degree-of-freedom systems. Here this analysis is extended to the trispectrum. Specifically, an expression is developed for the trispectral density of a multi-degree-of-freedom system subject to Gaussian excitation applied at an arbitrary location. The analytical expression is compared to those obtained via estimation using the direct method.