Accurate frequency domain measurement of the best linear time-invariant approximation of linear time-periodic systems including the quantification of the time-periodic distortions

• Published in: Mechanical systems and signal processing 2014-10, Vol.48 (1-2), p.274-299
• Main Authors: Louarroudi, E., Pintelon, R., Lataire, J.
• Format: Article
• Language: English
• Subjects: Anisotropy; Approximation; Best linear time-invariant (BLTI) approximation; Deviation; Distortion; Dynamical systems; Dynamics; Floquet resonances; Frequency response function (FRF); Linear time-periodic (LTP); Mathematical analysis; Mechanical systems; Nonparametric identification; Time-periodic (TP) distortions
• ISSN: 0888-3270; 1096-1216
• DOI: 10.1016/j.ymssp.2014.03.002

Time-periodic (TP) phenomena occurring, for instance, in wind turbines, helicopters, anisotropic shaft-bearing systems, and cardiovascular/respiratory systems, are often not addressed when classical frequency response function (FRF) measurements are performed. As the traditional FRF concept is based on the linear time-invariant (LTI) system theory, it is only approximately valid for systems with varying dynamics. Accordingly, the quantification of any deviation from this ideal LTI framework is more than welcome. The “measure of deviation” allows us to define the notion of the best LTI (BLTI) approximation, which yields the best – in mean square sense – LTI description of a linear time-periodic LTP system. By taking into consideration the TP effects, it is shown in this paper that the variability of the BLTI measurement can be reduced significantly compared with that of classical FRF estimators. From a single experiment, the proposed identification methods can handle (non-)linear time-periodic [(N)LTP] systems in open-loop with a quantification of (i) the noise and/or the NL distortions, (ii) the TP distortions and (iii) the transient (leakage) errors. Besides, a geometrical interpretation of the BLTI approximation is provided, leading to a framework called vector FRF analysis. The theory presented is supported by numerical simulations as well as real measurements mimicking the well-known mechanical Mathieu oscillator. •A close connection exists between the BLTI approximation of an LTP system and classical FRF estimators.•Traditional FRF measurements underestimate the higher order Floquet resonances in LTP systems.•The dispersion of the cyclic variations is determined by the harmonic FRFs.•The geometrical interpretation of the BLTI concept leads to the technique of vector FRF analysis.•Successful demonstration of the theory on the emulated Mathieu oscillator.