Time-Frequency Analysis of Systems with Changing Dynamic Properties

Abstract

Time-frequency analysis methods transform a time series into a two-dimensional representation of frequency content with respect to time. The Fourier Transform identifies the frequency content of a signal (as a sum of weighted sinusoidal functions) but does not give useful information regarding changes in the character of the signal, as all temporal information is encoded in the phase of the transform. A time-frequency representation, by expressing frequency content at different sections of a record, allows for analysis of evolving signals. The time-frequency transformation most commonly encountered in seismology and civil engineering is a windowed Fourier Transform, or spectrogram; by comparing the frequency content of the first portion of a record with the last portion of the record, it is straightforward to identify the changes between the two segments. Extending this concept to a sliding window gives the spectrogram, where the Fourier transforms of successive portions of the record are assembled into a time-frequency representation of the signal. The spectrogram is subject to an inherent resolution limitation, in accordance with the uncertainty principle, that precludes a perfect representation of instantaneous frequency content. The wavelet transform was introduced to overcome some of the shortcomings of Fourier analysis, though wavelet methods are themselves unsuitable for many commonly encountered signals. The Wigner-Ville Distribution, and related refinements, represent a class of advanced time-frequency analysis tools that are distinguished from Fourier and wavelet methods by an increase in resolution in the time-frequency plane. I introduce several time-frequency representations and apply them to various synthetic signals as well as signals from instrumented buildings. vi For systems of interest to engineers, investigating the changing properties of a system is typically performed by analyzing vibration data from the system, rather than direct inspection of each component. Nonlinear elastic behavior in the forcedisplacement relationship can decrease the apparent natural frequencies of the system - these changes typically occur over fractions of a second in moderate to strong excitation and the system gradually recovers to pre-event levels. Structures can also suffer permanent damage (e.g., plastic deformation or fracture), permanently decreasing the observed natural frequencies as the system loses stiffness. Advanced time-frequency representations provide a set of exploratory tools for analyzing changing frequency content in a signal, which can then be correlated with damage patterns in a structure. Modern building instrumentation allows for an unprecedented investigation into the changing dynamic properties of structures: a framework for using time-frequency analysis methods for instantaneous system identification is discussed.

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