On the Characteristics of Bifurcation and Nonlinear Dynamic Response

Spectral analysis has been widely applied to the detection of bifurcation and the determination of the extent to which dynamic instability and chaotic responses develop. However, because spectral analysis employs stationary sinusoids in representing time-varying signals of inherent nonlinearity, the use of Fourier domain methodologies would inexorably risk misinterpreting the true characteristics and obscuring the underlying physics of the nonlinear system being investigated. The fact that the amplitude and frequency of all the individual spectral component of a nonlinear, nonstationary dynamic response are modulated and coupled in time necessarily implies that, if the inception and transition of a bifurcated state of unstable motion is to be fully characterized, amplitude modulation and frequency modulation need to be temporally decoupled. The fundamental notion of instantaneous frequency defines frequency as the temporal gradient of phase and thus provides a powerful mechanism through which amplitude modulation and frequency modulation can be disassociated. Results of applying instantaneous frequency to the characterization of bifurcation and evolution of instability for a cracked rotor also indicate that instantaneous frequency interprets nonlinear rotary responses with sound physical bases.