Weak-signal Detection Based On The Chaotic And Stochastic-resonance Characteristics Of Duffing Oscillator

The fault diagnostics of machinery is an important sub-discipline of mechanical engineering. The fault diagnostics includes not only the fault mechanism of machinery, but also the diagnosis methods. The vibration diagnosis method is one of the most important methods among all the diagnosis methods. The key challenge for this method is the detection ability of weak signal. Weak signal exists extensively in practical engineering, so the weak-signal detection methods determine the accuracy of vibration signal analysis and even fault diagnosis to some extent. This problem can be extracted as the weak-signal detection in fault diagnosis of machinery.The weak-signal detection methods can be divided into the traditional linear methods and the developing nonlinear methods. Taking a nonlinear Duffing oscillator as study object, a weak-signal detection approach based on this oscillator is studied; the detection theory, models and results are analyzed in-depth; the problems in practical application are solved; the weak-signal detection and fault diagnosis of machinery are realized. This is the significance and purpose of this dissertation.The Duffing oscillator has both chaotic and stochastic-resonance characteristics. These two nonlinear characteristics can generate corresponding weak-signal detection methods. Thus this dissertation can be divided into two parts.The chaos of Duffing oscillator is studied first. Because of the sensibility to parameter and immunity to noise in critical chaotic state, the oscillator can detect the signal of very low signal-to-noise ratio(SNR). By using the basic theory of nonlinear dynamics and Mel’nikov method, the dynamic characteristics of this oscillator are studied qualitatively and quantitatively, and on this basis a weak-signal detection approach based on chaotic Duffing oscillator is obtained. Although this method can detect low-SNR signal, it has some shortages: it can only detect the small-frequency signal; the parameters of detection model is hard to choose; the noise and the initial phase may affect the detection result. A detection model combining scale-transfored method based on a group of arrayed parameter-fixed chaotic Duffing oscillator is proposed to realize the detection of weak signal for any frequency and initial phase.The stochastic resonance(SR) of Duffing oscillator is then investigated. This oscillator is a potential weak-signal detection model because it can amplify the low-frequency weak signal when SR occurs. In order to reveal its SR mechanism, the phenomenon is qualitatively analyzed through the derivation and evolution of the equation; the occurrence conditions are quantitatively analyzed from the perspective of Kramers rate; the small-parameter limits of SR are deduced according to the adiabatic approximation theory; and the concept of stationary-solution curve of Duffing oscillator is introduced, which is helpful to understand the response curve and SR mechanism of Duffing oscillator vividly and intuitively.In order to overcome the limits of Duffing oscillator that cannot produce SR under large-parameter conditions(large amplitude, frequency and noise-intensity), the amplitude-transformed method, scale-transformed/parameter-adjusted method and damping-ratio-adjusted method are creatively introduced to realize the SR of this oscillator under large-parameter conditions. The equivalence of scale-transformed and parameter-adjusted methods of Duffing oscillator under large-frequency parameters is proved as well. A method to realize the SR of Duffing oscillator under any large-parameter conditions is finally proposed.A cascade Duffing oscillator which can also produce SR and realize large-parameter SR is introduced. The parameter conditions under which this model can produce better response result than single Duffing oscillator are investigated. The parameters include the amplitude, frequency, noise-intensity of signal and the damping ratio, oscillator numbers of the system. Research results show that the cascade Duffing oscillator can further improve the output SNR under appropriate parameter conditions, and it is a potential detection model for square signal for its excellent plastic and filtering features.A generalized parameter-adjusted SR model of Duffing oscillator, which roundly contains the system parameters, signal parameters and adjusted parameters, is proposed. A discrimination function for the SR of Duffing oscillator is put forward based on the Kramers rate, and is used to deeply study the effect of the parameters of the model on the occurrence of SR. When the system, signal and noise are not matching, the model can realize the SR of Duffing oscillator through generalized parameter-adjusted method. The parameter-adjusted rules are summarized at last.Based on these results, a weak-signal detection method based on the SR of Duffing oscillator is proposed. The detailed detection model and procedure are given out. Three fault-diagnosis examples are employed to demonstrate the feasibility and reliability of the proposed approach in practical engineering application.This dissertation not only perfects the weak-signal detection method based on the chaotic Duffing oscillator, but also establishes the basic theoretical system of the weak-signal detection approach based on the SR of Duffing oscillator. The research results provide new ideas for weak-signal detection, and establish a basis of theory and experiment for the next research of the proposed methods.