Normalization of Experimental Modal Vectors to Remove Modal Vector Contamination

Abstract

When modal vectors are estimated from measured frequency response function (FRF) data, some amount of contamination in terms of random and bias errors is always present. The sources of these errors may be the experimental data acquisition process (calibration inconsistencies, averaging limitations, leakage errors, etc.) or due to limitations of the modal parameter estimation methods (mismatch between measured FRF data and the model form). These random and bias errors include uncertainty in complex magnitude about the central axis of the modal vector as well as rotation of the central axis. In a number of practical applications, particularly those involving close modal frequencies, the contamination of a modal vector will often have a significant influence from the modal vector that is near in frequency. In these situations, the numerical procedure for estimating the final, scaled modal vector, in terms of residue, generally involves a linear estimation method that, with MIMO FRF data, utilizes a weighted least squares solution procedure. This numerical solution process is reviewed and a real normalization of the weighting vectors used for estimating each modal vector in the MIMO FRF case is shown to reduce the contamination from nearby modal vectors. Theoretical evaluations for both proportional and non-proportional analytical cases are evaluated, as well as, results for a real application with pseudo-repeated modal frequencies and associated modal vectors that has historically demonstrated the problem.