Analytical Probability Model Of Structural Frequency Responses And Verification

Signal processing is of vital importance in the fields of structural modal identification,damage detection and finite element model updating.In real applications,structural response measurements are inevitably affected by multiple uncertainties such as inherent randomness of natural excitation,noise contamination and environmental variability,which will distort intrinsic information reflecting the real state of structures,thereby leading to aberrations in real applications.Therefore,it is of great significance to introduce probabilistic model to accommodate these uncertainties and to improve the robustness and accuracy of processing results.However,researches on this topic are still relatively rare.Supported by the project ‘Application of Statistical Properties of Transmissibility function to Modal Identification of Bridge Engineering’(No.51408176)from the Natural Science Foundation of China(NSFC),this study focused on quantifying the uncertainties for fast Fourier transform(FFT)coefficients as well as transmissibility functions which is defined as the ratio of FFT coefficients corresponding to different measurements.Analytical probabilistic model of the FFT coefficients and transmissibility functions are established.Numerical simulations as well as field test data of large-scale engineering structures subject to ambient vibrations are utilized to examine the performance and robustness of the probability model.The main contribution and conclusion of this thesis are outlined as follows:1.The applicability of the complex Gaussian distribution to the probabilistic modeling of FFT coefficients is studied.Furthermore,the joint PDF and the marginal PDF of the real part,the imaginary part,the magnitude and the phase of the FFT coefficients are derived analytically.Theoretical study shows that the real and imaginary parts of FFT coefficients approximately follow the circularly-symmetric Gaussian distribution,while the magnitude is approximately Rayleigh distributed and the phase approximately follows the uniform distribution.2.Modelling the transmissibility function as a ratio random variable,its PDF is obtained by using the the probability density transformation principle and the probabilistic model of FFT coefficients.As a result,the marginal PDFs of the real part,the imaginary part,the magnitude and phase are derived analytically.These formulas are simple,compact,and easy-implemented in real applications.Furthermore,one can figure out that the joint PDF of the real and imaginary parts are bell-shaped.3.On the basis of numerical simulation data and field test data of engineering structures subject to ambient vibrations,the robustness of the probability model of frequency responses is validated by means of probability plot,skewness,kurtosis,Kolmogorov-Smirnov(K-S)test and goodness of fit analysis.These case studies show that the complex Gauss probability model can quantify the uncertainty of FFT coefficients well in most frequencies.Also,the study reveals that the FFT coefficients will deviate from the Gaussian distribution at some frequencies with their PDFs holding the characteristics of sharp-peak and heavy-tail.4.Case studies also show that the distribution curves of the analytical probability model of the transmissibility functions fit well with the histograms obtained from experimental samples.The coefficients of goodness of fit between the theoretical curves and the tested histograms are usually larger than 90%,indicating that the complex Gaussian ratio distribution model proposed in this thesis can quantify the uncertainty of the transmissibility functions effectively.