Cepstral operational modal analysis for multiple-input systems based on the real cyclic cepstrum

Operational modal analysis aims at identifying modal parameters of a system from response measurements only. Cepstrum-based operational modal analysis (OMA) has an advantage over conventional OMA methods, since it relaxes the assumption of white noise excitation and is able to retain the relative scaling between modes, which is important in many applications such as force identification. In previous research, the cepstrum-based OMA has been applied to reconstruct the forcing functions generated by a gear pair from response measurements, in order to obtain more direct diagnostic information. However, the cepstrum-based OMA can only cope with systems with a single excitation. To overcome this challenge, this paper proposes a novel technique, which can be applied to minimum-phase multiple input systems that have a cyclostationary excitation with a unique cyclic frequency. The technique first extracts the single-input system from the multiple-input case using the cyclic spectral density (CSD) of the measurement, equal to the spectral correlation density (SCD) evaluated at a particular cyclic frequency. The method then identifies the frequency response function (FRF) by fitting the real cyclic cepstrum of the measurement, this being the inverse Fourier transform of the log magnitude of the CSD. This method elaborates on a technique developed in previous research, which extracted the signal of interest from the complex cyclic cepstrum instead, this being the inverse Fourier transform of the logarithm of the CSD with unwrapped phase. In this paper, a limitation of the previously developed technique is identified and discussed for the first time, viz. problems with the phase unwrapping of the CSD. The technique in this paper overcomes the drawback in the previous research and exhibits better performance in noisy conditions. Both methods have been validated on synthetic signals as well as vibration measurements from a beam test rig. An attempt has also been made to apply them to vibration signals from a single-stage spur gear test rig. The method based on the real cyclic cepstrum is considered to have great potential to reconstruct the forcing functions from vibration signals measured from multi-stage gearboxes.