Application of the Duffing Chaotic Oscillator Model for Early Fault Diagnosis- Ⅰ.Basic Theory

摘要

In this paper, the well-known Duf fing equation and the nonlinear equation describing vibration of the human eardrum are introduced from elastic nonlinear system theory. According to the fact that the human ear can distinguish weak sound with smalldifference, the idea that the Duffing oscillator can be used to detect a weak signal and diagnose early fault of machinery is proposed. In order to obtain a model for weak signaldetection via the Duffing oscillator, the first step is to seek all forms of solutions of theDu f fing equation. The second step is to study global bifurcations of the Du f fing equationusing qualitative analysis theory of a dynamic system. That is to say, a series of bifurcations thresholds of the Duffing equation can be analyzed by the Melnikov function and asubharmonics Melnikov function. Then the three types of bifurcations thresholds varyingwith damping and external exciting amplitude are discussed. The analysis concludes thatthe bifurcation threshold corresponding to the maximum orbit of solutions outside the homoclinic orbit of the Duffing equation can be used to detect a weak signal. Finally, the implementing model of the Du ffing oscillator for weak signal detection is given.