Time-frequency Representation Of Fractional Fourier Transform And Structural Instantaneous Frequency Identification

The civil engineering structure in complex operation process are essentially timevarying structural systems and nonlinear structural systems.The response of timevarying structures are non-stationary signals,and the structural parameters exhibit some time-varying characters.Identifying the time-varying parameters of the structure is more consistent with the actual situation,which is not only a major demand for engineering practice and structural health monitoring,but also the development trend of engineering structure parameter identification theory.Compared with the parameter identification of time-invariant systems whose structural parameters do not change with time,the instantaneous parameter identification of time-varying structural systems is more challenging.In this dissertation,the theoretical analysis,numerical simulation examples and laboratory experiments are combined to identify the instantaneous frequency parameters of time-varying structural systems.The fractional Fourier transform is projected to the time-frequency plane based on the fractional Fourier transform theory.The analytical expression of the time-frequency representation of fractional Fourier transform is derived theoretically,and the method for identifying the instantaneous frequency of the structure based on the fractional Fourier transform is established.In order to further improve the identification accuracy,the fractional Fourier transform based on Kaiser window is proposed to reduce frequency leakage,and the synchrosqueezing fractional Fourier transform algorithm is proposed to enhance the timefrequency aggregation of frequencies.Several numerical simulation examples and experiments are employed to verify the proposed instantaneous frequency methods.The main research work of this dissertation is as follows:1.As a generalization of the traditional Fourier transform,fractional Fourier transform transforms the signal or function into any intermediate domain(fractional Fourier domain)between time domain and frequency domain.Since the fractional Fourier transform can not be used for time-frequency analysis directly,the time-frequency projection process of fractional Fourier transform is analyzed in detail,and the mechanism of the fractional Fourier transform time-frequency analysis is demonstrated.The analytical expression of time-frequency representation of fractional Fourier transform is deduced theoretically,it is proved that the essence of time-frequency representation of fractional Fourier transform is the window with dilation and translation,which is similar to the time-frequency analysis process of wavelet transform.Then the properties of window function are investigated in detail,and the adaptive dilation and translation characteristics of the window are obtained.The time-frequency analysis function of time-frequency representation of fractional Fourier transform is verified by several numerical simulation examples,and the results are compared with those of short-time Fourier transform and wavelet transform.It is shown that the general expression of time-frequency representation of fractional Fourier transform can be used to process non-stationary signals and instantaneous frequency identification.2.Based on the derived analytical expression of the time-frequency representation of the fractional Fourier transform,the fractional Fourier transform ridge points are defined,and the curve formed by connecting all ridge points is defined as the fractional Fourier transform ridge line.At the same time,the analytical expression of the ridge line is derived by the poles of the fractional Fourier spectrum.A method based on the maximum modulus of the fractional Fourier transform coefficient is proposed to extract the ridge line,and the general expression of the instantaneous frequency is obtained.Finally,the instantaneous frequency identification of the nonstationary signal is realized.The effectiveness of instantaneous frequency identification method based on fractional Fourier transform is verified by several numerical examples and compared with the results of short-time Fourier transform and wavelet transform.Due to the basis function of fractional Fourier transform is trigonometric as traditional Fourier transform,the orthogonal property of trigonometric function shows that fractional Fourier transform algorithm is not affected by the cross term of multi-component signals.And discrete fractional Fourier transform has the same efficient calculation rate as fast Fourier transform.Therefore,the instantaneous frequency identification based on fractional Fourier transform has advantages in both computational efficiency and accuracy.3.In order to realize the instantaneous frequency identification of the time-varying structure system,the definition of instantaneous frequency of the time-varying structure is introduced,and the method of identifying the instantaneous frequency of structure by fractional Fourier transform is established.Based on the extracted fractional Fourier transform ridge line,the instantaneous frequency of time-varying structures can be directly identified.The accuracy and effectiveness of the algorithm are verified by the numerical simulation example with noise and the time-varying cable structure experiment,and the results are compared with those of short-time Fourier transform and wavelet transform,indicating that the algorithm has certain robustness.For time invariant structures,the structural response signal is stationary,and the fractional Fourier transform degenerates into the traditional Fourier transform.The fast Fourier transform method can be directly used to identify the modal frequencies of structures.4.For the spectrum leakage problem of fractional Fourier transform in signal timefrequency analysis process,in order to reduce the spectrum leakage error and improve the accuracy of fractional Fourier transform time-frequency analysis,the dissertation proposes the fractional Fourier transform based on Kaiser window.The theoretical formula representation for Kaiser window based fractional Fourier transform is then derived.The function mechanism of Kaiser window in fractional Fourier transform is discussed,and the time-frequency analysis process of the algorithm is given.Finally,the time-varying structure parameter identification algorithm of non-stationary signal is obtained.The experiment of three-story frame on the shaking table excited by nonstationary inputs with different signal-to-noise ratios is designed.The acceleration response of the frame structure is collected at the same time by outputting linear frequency modulation signal,nonlinear frequency modulation signal superimposed harmonic signal and sinusoidal frequency modulation signal superimposed harmonic signal.The numerical simulation example of arbitrary multi-component nonstationary signal and the non-stationary excitation experiment results of shaking table structural model show that the established fractional Fourier transform based on Kaiser window can effectively improve the spectral leakage of the fractional Fourier transform algorithm and increase the accuracy of parameter identification of timevarying systems.5.As a new time-frequency analysis tool,the time-frequency representation of fractional Fourier transform is affected by Heisenberg uncertainty principle as other timefrequency analysis methods,and the time-frequency distribution of fractional Fourier is blurred.In order to improve the time-frequency analysis accuracy of fractional Fourier transform,the dissertation proposes the synchrosqueezing fractional Fourier transform by using synchrosqueezing transform to rearrange the energy in the timefrequency plane of signal and focus on the real frequency.The theoretical timefrequency expression of synchrosqueezing fractional Fourier transform and its inverse transformation are then derived.The algorithm of synchrosqueezing fractional Fourier transform in analyzing instantaneous frequency of structures is given.The experiment of a moving vehicle-bridge time-varying system is designed.The timevarying structure system is formed by the vehicle running on the simply supported beam and the continuous beam,and the acceleration response signal of the system is collected.The numerical simulation example of the three-degree-of-freedom damped time-varying structure system and the experiment of mobile vehicle-bridge system in the laboratory show that the synchrosqueezing fractional Fourier transform can effectively improve the time-frequency distribution divergence of the fractional Fourier transform and the accuracy of the instantaneous frequency identification,which can be applied to the identification of the instantaneous parameters of the timevarying structural system.