Probabilistic And Non-probabilistic Methods Of Uncertainty Quantification For Complex Ratio Variables And Their Application In Vibration Testing

In the field of dynamics,the dynamic characterization functions including frequency response functions and transmissibility functions defined as the ratios of two frequency-domain responses also fall into the category of complex ratio random variables.As the most prevalent frequency domain tools,they represent input-output relationships and output-to-output relationships which are of fundamental importance in parameter detection,damage detection,model updating,etc.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.In order to consider the influence of uncertain factors on frequency response functions and transmissibility functions,the uncertainty quantification of frequency response functions and transmissibility functions is studied based on probabilistic and non-probabilistic methods.The accuracy and robustness of the proposed method are verified by numerical simulation,experimental beam vibration test and civil engineering structure vibration experiment.The main research contents of this dissertation are as follows:(1)A unified scheme is presented to efficiently calculate the PDF of a complex-valued ratio random variable with FFT coefficients specified by arbitrary complex-valued distributions.With the use of probability density transformation principle in the complex domain,the unified formula is derived for complex ratio distributions by reducing the concerned problem into multi-dimensional integrals.When it is difficult or impossible to discover a closed-form solution,one ought to resort to numerical algorithms.For Gaussian quadrature rule,the number of points to be calculated increases exponentially with the dimension of integrals,leading to uncomfortable computational burden.In this study,a novel sparse-grid quadrature scheme will be employed to address the curse of dimensionality problem and improve the computational efficiency.The problems can be solved which has been revealed that the FFT coefficients may deviate from a Gaussian distribution when the sampling time is short.(2)Based on the monitoring data of civil engineering structures under hammer excitation,the probability model of FFT coefficients are verified by K-S test.The K-S test results assume that the real and imaginary parts of FFT coefficients follow a t distribution and the passing rate of t distribution is generally higher than Gaussian distribution.The accuracy of the probability model of frequency response functions and transmissibility functions is also verified by the examples.When FFT coefficients obey Gaussian distribution or non Gaussian distribution,there is good consistency between the observed histograms,the curves of the analytical values and and the numerical values of the distribution of frequency response functions and transmissibility functions.(3)Based on the general mathematical expression of frequency response functions and transmissibility functions,it is regarded as an arbitrary complex ratio interval function.Due to the multiple operations of the real and imaginary parts of numerators in the functions,it will inevitably lead to interval extension.In this dissertation,the division operation is transformed into the form of product through mapping theory and deformation operation,which avoids multiple operations of the same variable and inhibits interval expansion to a certain extent.Based on numerical simulation and the monitoring data of civil engineering structures under hammer excitation,the intervals of frequency response functions and transmissibility functions are calculated by deformation operation.Compared with samples and traditional algorithms,deformation operation can better quantify the uncertainty of frequency response functions and transmissibility functions.(4)By employing the parallelogram uncertain domain model,the interval function with correlative variables is transformed and normalized into the function with independent variables in the standard rectangular domain,and subsequently the corresponding real valued function is analyzed to determine the optimal solution for arbitrary binary interval functions.The solution process to arbitrary binary interval functions is presented,and the optimal solutions for the interval multiplication function and interval division function is derived.From the example analyses,it is found that the present optimal solution not only inhibits the interval extension caused by not considering or not fully considering the correlation of interval variables,but also eliminates it caused by the repeated operation of interval variables.(5)For the complex ratio interval function,the polar representation and Euler formula are used to express it in the form of mode ratio and phase difference,so that each variable appears only once,avoiding the interval extension caused by multiple operations of the same variable.Considering the correlation between FFT coefficients,the complex domain parallelogram interval model is used quantifying the correlation between modes and phases.Then,the complex ratio interval function is analyzed by regularization algorithm and partition direct solution method,and a simple analytical solution is obtained.(6)Based on numerical simulation and monitoring data of civil engineering structures under hammer excitation,the complex ratio interval model is verified by quantifying uncertainty of frequency response functions and transmissibility functions.The example study shows that the extremum algorithm of parallelogram interval model in complex field proposed in this dissertation inhibits the interval expansion of frequency response functions and transmissibility functions to a great extent,which can provide a reference for practical engineering application.