Introduction

In the field of radio physics and signal processing, the Hilbert Transform (HT) is taken as a standard procedure and has long been abandoned. The HT and also its properties as applied to analysis of linear and non-linear vibrations are discussed theoretically at great length in [83]. The application of the HT to the initial signal provides some additional information about amplitude, instantaneous phase and frequency of vibrations. This information was valid when applied to analysis of vibration motions [Hammond, 87]. What's more, it was understood that the HT also should be used for solving an inverse problem - the problem of vibration system identification.

The first attempts to use the HT for vibration system identification were made in the frequency domain [Tomlinson, 84]. The HT of the Frequency Response Function (FRF) of a linear structure reproduces the original FRF, and any departure from this, i.e. distortion, can be attributed to non-linear effects. It is possible to distinguish the common types of non-linearity in mechanical structures from the distortions in the FRF.

Other approaches, published in English [Feldman, 85], were devoted to the application of the HT in the time domain, where the simplest natural vibration system, having mass and linear stiffness element, gives rise to pure harmonic motion. Real vibration always gradually decreases in amplitude because of energy losses from the system. If the system has non-linear elastic forces, the natural frequency will depend decisively on the amplitude of vibration. Energy dissipation lowers the instantaneous amplitude according to a non-linear dissipative function. As non-linear dissipative and elastic forces have totally different effects on free vibrations, the HT identification methodology [Feldman, 85] enables us to determine some aspects of the behavior of these forces. For this identification in the time domain, it was proposed that relationships be constructed between the damping coefficient (or decrement) as a function of amplitude plus relationships between the instantaneous frequency and the amplitude [Feldman, 85]. Lately, determination of the linear damping coefficient by extracting the slope of the vibration envelope was suggested in [Agneni, 89] .

Further development of the HT-based methods was made for generalized identification of non-linear free and forced vibrations [Feldman, 91]. Some studies [Feldman, 94] provide the reader with a comprehensive concept for dealing with free and forced response data involving HT identification of SDOF non-linear systems under free or forced vibration conditions. These methods, among them non-parametric ones, were recommended for instantaneous modal parameter identification, including system backbone and damping curve determination. Recent efforts have been directed towards refining the present technique with respect to analysis of a two-component vibration signal [Feldman & Braun, 95]. If a vibration signal consists of two non-stationary components, the time domain decomposition could be calculated based on using the HT twice. This technique is able to decouple two-degrees-of-freedom systems and to estimate each non-stationary (or non-linear) component separately. One study [Feldman, 97] describes how the results obtained from the HT identification method can be put to use in non-symmetric vibration systems identification. The strength of the considered identification method lies in its potential to yield a physical model for non-symmetric elements of the system.

An improvement of the HT identification approach was made by [Feldman, 97] for free vibration non-stationary signals with overlapping spectrums. The improved method now enables identification of non-linear and non-stationary SDOF systems without any assumption of their weak non-linearity. The paper proves that a smooth skeleton curve with regard to a single principal component of the solution has the same qualitative relation with amplitude, but shifted with bias, as nonlinear system static force characteristics. That is, the estimated average natural frequency includes all information about the initial non-linear elastic characteristics and can be used for identification of non-linear systems.

An original technique, known as the Empirical Mode Decomposition (EMD), first introduced by Huang et al. [Huang 98], adaptively decomposes a signal into the simplest intrinsic oscillatory modes (components). The EMD method is based on a spline algorithm, which constructs upper and lower envelopes that are fitted to the local maxima of the initial wideband signal. Signal decomposition is a powerful approach, and has become extremely popular in various areas, including non-linear and non-stationary mechanics and acoustics.

A different technique, called the Hilbert Vibration Decomposition (HVD) method, dedicated to the same problem of decomposition of non-stationary wideband vibration, was developed by [Feldman 06]. The proposed HVD method is based on the HT presentation of the instantaneous frequency and does not involve the spline algorithms. The first component separated from the initial vibration contains the varying highest amplitude. The residual signal contains information about other lower amplitude components.

A new general method of nonlinear SDOF system identification, based on the modern non-stationary signal decompositions, takes into account other high components of motion [Feldman 06]. In addition to the principal component, the generalized HT identification considers the next high harmonics of the nonlinear solution. Such consideration is based on a new notion known as the Envelope of Envelope function. The accuracy of the generalized HT method depends on the total number of high harmonics (intrinsic signal components) considered, and theoretically defines the exact solution for the identified nonlinear static force characteristics. As a result the HT identification restores the adequate nonlinear differential equation of vibration motion.

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