Sensitivity Analysis Of Frequency Response Function And Transmissibility Function Based On Direct Algebra Algorithm And Their Applications In Structural Damage Detection

Frequency response function(FRF)and transmissibility function(TF)are two types characterization functions representing the input-output and input-to-output relationships of a dynamic system.FRF and TF have drawn widespread attention in the field of structural damage detection due to their clear physical meanings,containing complete modal information and avoiding the procedure of modal identification.Evaluating their sensitivity to damage parameters is of fundamental importance for their application into structural damage identification.In this regard,this thesis is devoted to studying the sensitivity of FRF/TF and their application in structural damage identification under the support of the National Science Foundation of China entitled “Uncertainty Quantification and Propagation Mechanism for Frequency Response Function-based Structural System Identification”(NO.51778203).The closed-form solutions of the sensitivity of FRF and TF are analytically derived based on a direct algebraic method.The formulae are simple and compact.Moreover,the properties of FRF and TF sensitivity with respect to different damage parameters are analyzed in depth.Based on the algebraic closed-form solutions,the damage equations driven by FRF/TF are established using the Taylor expansion.Furthermore,comparison among several common regularization methods is carried out to evaluate their applicability and performance in solving the damage equations which are prone to ill-conditioned problems.Finally,two numerical case studies,including a continuous beam and a plane frame,are conducted to verify the effectiveness and accuracy of the proposed method.The effects of various parametric factors such as the extent of damage,the frequency range,the noise level,and the number of measured points on the identified results are also investigated.The main contributions and conclusions of this dissertation are outlined as follows:1.The first-order sensitivity formula for FRF as well as for TF is analytically derived by a direct algebraic method.These formulae in matrix form are simple,compact,and convenient for programming and storage.The sensitivity and relative sensitivity of FRF and TF to different damage parameters are analyzed using simulated data from a simply-supported beam and a plate.The results indicate that:(a)The sensitivity of FRF/TF varies over frequencies,and the FRF sensitivity peak occurs around the resonant frequencies,while the peak of TF sensitivity appear around both resonant frequencies and anti-resonance frequencies in FRF formulated by the related outputs;(b)compared with modal frequency and mode shape,FRF and TF are more sensitive to structural damage parameters,indicating good applicability for structural damage detection;(c)the sensitivity of FRF/TF formulated by the outputs measured close to damaged elements are not always larger than by those measured from undamaged elements,demonstrating that it is difficult to detect the location of structural damage solely using the FRF/TF-based damage indices without resorting to a physical model of the structure;(d)the results from FRF/TF-based damage indices are obviously affected by the selection of frequency band,and often only part of damage elements can be detected by selecting a specific frequency band.2.Based on the closed-form solutions of FRF and TF sensitivity,together with Taylor expansion,the structural damage equations are analytically derived.For the illconditioned problems caused by incomplete test and measurement noise in practical engineering applications,regularization methods are introduced to ensure the accuracy of solutions of damage equations.The effectiveness and accuracy of the proposed method are verified by numerical case studies of a continuous beam and a plane frame.The results indicate that the derived FRF/TF-based damage equations can accurately detect damage(location and extent).And FRF-based method can obtain a more accurate damage extent together with less false-positive errors when compared to TF-based method.3.To address the ill-conditioned problems of solving the damage equations,this study further investigates several commonly-used regularization methods.The differences and relationships between these regularization methods are illustrated systematically under the framework of Bayesian statistical inference.Subsequently,different regularization methods are utilized to solve the damage equations driven by the sensitivity of FRF and TF.The results show that the identification results of L1-norm regularization and sparse Bayesian learning are sparse,indicating that these two approaches are more suitable in solving structural damage equations.4.The effects of various parametric factors,including the damage extent,the noise level,and the number of measured points,are discussed in detail to provide a reference for the optimal selection of regularization methods for solving structural damage equations.It is demonstrated that L1-norm regularization and sparse Bayesian learning method are the most robust in solving damage equations,followed by Bayesian regularization method,and truncated singular value decomposition and L2-norm regularization method have poorest robustness.Moreover,L1-norm regularization can obtain more accurate damage extent than those of sparse Bayesian learning.The sparse Bayesian learning algorithm always gives acceptable results for precision with high efficiency and avoid the difficulty in regularization parameter selection involved in the L1-norm regularization algorithm.