The trispectrum for Gaussian driven, multiple degree-of-freedom, non-linear structures

Abstract Higher-order spectra have become a useful tool in spectral analysis, particularly for identifying the presence and type of system non-linearity. Two such spectra that have figured prominently in signal processing are the bispectrum and trispectrum. In a previous work, the authors developed an analytical solution for the bispectrum for multi-degree-of-freedom systems. Here this analysis is extended to the trispectrum. Specifically, an expression is developed for the trispectrum 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.

[1]  Loren D. Lutes,et al.  Trispectrum for the response of a non-linear oscillator , 1991 .

[2]  J. A. Brandon,et al.  Diagnostics of a System with an Interface Nonlinearity using Higher Order Spectral Estimators , 2001 .

[3]  L. Isserlis.  ON A FORMULA FOR THE PRODUCT-MOMENT COEFFICIENT OF ANY ORDER OF A NORMAL FREQUENCY DISTRIBUTION IN ANY NUMBER OF VARIABLES , 1918 .

[4]  M. Hinich.  Higher order cumulants and cumulant spectra , 1990, Fifth ASSP Workshop on Spectrum Estimation and Modeling.

[5]  Jonathan E. Cooper,et al.  Higher-order spectra for identification of nonlinear modal coupling , 2009 .

[6]  Lennart Ljung,et al.  System Identification: Theory for the User , 1987 .

[7]  Keith Worden,et al.  Nonlinearity in Structural Dynamics , 2019 .

[8]  M. Schetzen,et al.  Nonlinear system modeling based on the Wiener theory , 1981, Proceedings of the IEEE.

[9]  P. R. White,et al.  Two-Stage Adaptive Line Enhancer and Sliced Wigner Trispectrum for the Characterization of Faults from Gear Box Vibration Data , 1999 .

[10]  A. C. McCormick,et al.  Bispectral and trispectral features for machine condition diagnosis , 1999 .

[11]  Paul R. White,et al.  HIGHER-ORDER TIME-FREQUENCY ANALYSIS AND ITS APPLICATION TO FAULT DETECTION IN ROTATING MACHINERY , 1997 .

[12]  Jonathan M. Nichols,et al.  Second-order spectra for quadratic nonlinear systems by Volterra functional series: Analytical description and numerical simulation , 2008 .

[13]  P. White,et al.  HIGHER-ORDER SPECTRA: THE BISPECTRUM AND TRISPECTRUM , 1998 .

[14]  M. Schetzen.  The Volterra and Wiener Theories of Nonlinear Systems , 1980 .

[15]  John W. Dalle Molle,et al.  Trispectral analysis of stationary random time series , 1995 .

[16]  Jonathan M. Nichols,et al.  On the use of the auto-bispectral density for detecting quadratic nonlinearity in structural systems , 2008 .