Nonlinear System Identification By Using Parameterized Time-frequency Analysis

In practice, there are a variety of nonlinear phenomena and nonlinearsystems in engineering mechanical equipments and structures, which canbe taken advantage of or should be avoided. In terms of these widespreadnonlinear problems, modeling the system to characterize its inherentnonlinear behaviors and properties is the key point in analyzing andresolving these problems. Thus, the nonlinear system identification hasdeveloped to be a popular research interest in the field of systemidentification. Traditional linear methods are no longer applicable for thenonlinear identification because of non-neglected error introduced. In spiteof some developments of nonlinear system identification theories havebeen gained in recent two decades, the identification of nonlinear systemis always a challenging task in reverse engineering problems.Nowadays, kinds of popular methods have been proposed to tacklethe identification of nonlinear structural dynamic systems. However,compared with their own deficiencies, the time-frequency domain basedidentification methods are a class of newly developed methods in recentdecades and draw more and more attention of researchers. These time- frequency domain based methods are mainly based on time-frequencyanalysis methods in the signal processing. With analyzing the input-outputsignals to identify the nonlinearities and estimate system parameters, theclass of methods takes advantage of other common methods in somedegrees.As we known, traditional time-frequency methods have somedrawbacks in analyzing the multi-component non-stationary signals. As aconsequence, recently the theory of parameterized time-frequency analysis(PTFA) has been founded and gradually improved and a series of relativetime-frequency analysis methods have been proposed. Thus, the mainresearch point in this paper is to use a kind of typical PTFA, i.e.,polynomial chirplet transform (PCT), to realize the identification ofnonlinear systems.Theoretically, the PCT is a newly developed time-frequency methodsbased on broadening and extending of the traditional chirplet transform.Based on the polynomials as kernel functions and the parameterizedoperators as frequency transformation means, it appropriatelyapproximates and characterize the instantaneous frequency of the signal.With ideal kernel parameters applied, the PCT can extract theinstantaneous frequency curve accurately.Typically, for a nonlinear vibration system, the time-varyinginstantaneous amplitude and instantaneous frequency are the two key characteristics to characterize the system motion response. Based on theanalysis of vibration properties of the nonlinear vibration system, the paperuses the PCT to analysis the system response signal, and then extracts thetwo key instantaneous parameters. With the extracted instantaneouscharacteristics, one can get the estimation of instantaneous modalparameters of the system, i.e., the instantaneous natural frequency and theinstantaneous damping coefficient. And the backbone and the dampingcurve, which characterize the inherent properties of the system, can bederived from the relations between the instantaneous modal parameters andthe instantaneous characteristics of the system response. The backbone andthe damping curve act as indicators of the type and intensity of thenonlinearities. In addition, the paper proposed the average nonlinearcharacteristic forces, with which one can estimate specific physicalparameters of the system. Eventually, the equation modeling the originalnonlinear system can be obtained.In the last part of the paper, the numerical simulations and theexperimental study are carried out to validate the proposed identificationmethod in this paper, and to prove that the method has a good quality ofanti-noise.