Non-linear Free Vibration Identification via the Hilbert Transform

The Hilbert Transform (HT) and its properties, applied to the analysis of linear and non-linear vibrations, are discussed at great length in [1]. The application of the HT to a signal provides some additional information about amplitude, instantaneous phase and vibration frequency. This information is valid when applied to the analysis of non-linear vibration motions [2]. Moreover, it has been noted that the HT should also be used for solving an inverse problem—the problem of vibration system identification. Previous results of applying the HT to time domains for non-linear vibration system identification are presented in reference [3]. The simplest natural vibration system, having mass and a linear stiffness element, in a time domain gives rise to pure harmonic motion. In the presence of non-linear elastic forces, the natural frequency will depend decisively on the amplitude of vibration. Real free vibration always gradually decreases in amplitude due to system energy losses. Energy dissipation lowers the instantaneous amplitude according to a non-linear dissipative function. As non-linear dissipative and elastic forces have distinct effects on free vibrations, the HT identification methodology [4] enables the determination of 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 amplitude (system backbone) [5]. Note, however, that previous work incorporated a theoretical assumption about a given vibration signal, which should have a slowly varied envelope (weak non-linearity condition). This paper describes a generalized HT identification approach for non-linear free vibration of one or two degrees of freedom system without the assumption of weak non-linearity.