Characterization Of Ambient Vibration Data By Response Shape Vectors
Authors: Burke B.G. , Sundararajan C.G. , Safaie F.M. 
Published: October 2008

ABSTRACT
Ambient vibration monitoring methods are based on the calculation and interpretation of auto- and cross-spectral densities of accelerations recorded simultaneously at several selected locations on a structure. In principle, these data are used to identify natural frequencies and mode shapes that, presumably, will remain stable with time unless significant structural damage occurs. In practice, although the identification of peaks in the spectral density functions is straightforward, interpretation of natural frequencies and mode shapes is a tedious and difficult task because of the quantity and complex nature of the data.
This paper describes a least-squares method for calculating a response shape vector at each discrete frequency of the matrix of auto- and cross-spectral density functions. The resulting shape vectors provide (1) a single function of frequency, the squared norm, for displaying peaks in the spectral density function, (2) a quantitative measure of the "mode shape" associated with the peaks, and (3) a measure of goodness of fit of the vector to the data. These results provide a quantitative and greatly simplified basis for interpretation.
The paper includes (1) a derivation of' the method, (2) a description of its implementation including a listing of a Fortran computer code, (3) an example using measured data, and (4) suggestions for further development. The examples illustrate the ability of the method to quantify mode shapes, difficulties associated with distinguishing between closely spaced modes, and the degree to which resulting response shapes are stable in time.

INTRODUCTION
The detection of structural damage by ambient vibration monitoring relies on the identification of significant changes in natural frequencies and mode shapes of the structure by interpretation of auto- and cross-spectral density functions of acceleration measurements. The concept, identification of significant changes, implies that:a positive identification is made of a number of natural frequencies and mode shapes in the measured data;a quantified' measure of stability is established for modal frequency and shape for the undamaged structure, anda basis is available for determining that a particular damage level will cause changes in the detectable modes that are measurably greater than established stable ranges.
ignificant improvements to the ambient vibration monitoring are achieved when:the number of identifiable modes that exhibit significant changes with member damage is increased, andthe measure of stability for frequency or shape is improved.
This paper describes results of an attempt to develop a method for computing response shapes from measured cross-spectra; the computed response shape vector is envisioned as a means to provide a quantifiable measure of mode shape and an additional basis for identifying natural modes of the structure. The method employs a least-squares fit to the auto- and cross-spectral densities of measured data to determine a "best" shape vector at each frequency. A theoretical basis for the method is presented, and the equations for the method are derived. Results from measurements on an oil producing platform in the Gulf of Mexico are presented to illustrate the method and to provide a basis for its evaluation.