Efficient and Accurate Calculation of Discrete Frequency Response Functions and Impulse Response Functions

Modal properties of a structure can be identified by experimental modal analysis (EMA). Discrete frequency response functions (FRFs) and impulse response functions (IRFs) between responses and excitation are bases for EMA. In calculation of a discrete FRF, discrete Fourier transform (DFT) is applied to both response and excitation data series, and a transformed data series in DFT is virtually extended to have an infinite length and be periodic with a period equal to the length of the series; the resulting periodicity can be physically incorrect in some cases, which depends on an excitation technique used. There are various excitation techniques in EMA, and periodic extension in DFT for EMA using periodic random and burst random excitation is physically correct. However, EMA using periodic random excitation needs a relatively long excitation time to have responses to be steady-state and periodic, and EMA using burst random excitation needs a long sampling period for responses to decay to zero, which can result in relatively long response and excitation data series and necessitate a large number of spectral lines for associated DFTs, especially for a high sampling frequency. An efficient and accurate methodology for calculating discrete FRFs and IRFs is proposed here, by which fewer spectral lines are needed and accuracies of resulting FRFs and IRFs can be maintained. The relationship between an IRF from the proposed methodology and that from the least-squares method is shown. A new coherence function that can evaluate qualities of FRFs and IRFs from the proposed methodology in the frequency domain is used, from which meaningful coherence function values can be obtained even with response and excitation series of one sampling period. Based on the new coherence function, a fitting index is used to evaluate overall qualities of the FRFs and IRFs. The proposed methodology was numerically and experimentally applied to a two-degree-of-freedom mass-spring-damper system and an aluminum plate to estimate their FRFs, respectively. In the numerical example, FRFs from the proposed methodology agree well with the theoretical one; in the experimental example, a FRF from the proposed methodology with a random impact series agreed well with the benchmark one from a single impact test.