Variations in the Dynamic Properties of Structures: the Wigner-ville Distribution

The Wigner-Ville Distribution (WVD) is a promising method for analyzing frequency variations in seismic signals, including those of interest for structural monitoring. Nonlinearities in the force displacement relationship will temporarily decrease the apparent natural frequencies of structures during strong to moderate excitation, and earthquake damage can permanently change building stiffnesses. A Fourier Transform of a building record contains information regarding frequency content, but it can not resolve the exact onset of changes in natural frequency – all temporal resolution is contained in the phase of the transform. The spectrogram is better able to resolve temporal evolution of frequency content, but has a trade-off in time resolution versus frequency resolution in accordance with the uncertainty principle. To overcome this restriction, several distributions have been proposed, including wavelet methods. Wavelets have very good temporal resolution, but generate a wavelet “scale” instead of frequency, which complicates analysis of evolving frequency content. Time-frequency transformations such as the WVD allow for instantaneous frequency estimation at each data point, for a typical temporal resolution of fractions of a second. We develop a mathematical foundation for analyzing the evolution of frequency content in a signal, and apply these techniques to synthetic records from linear and nonlinear FEM analysis (including plastic rotation and weld fractures). Our analysis techniques are then applied to earthquake records from damaged buildings.