Operational Modal Analysis For Bridge Engineering Based On Power Spectral Density Transmissibility(PSDT) Functions And Error Analysis

Transmissibility-based operational modal analysis(TOMA)methods have received increasing attention in recent years.TOMA is devoid of the white-noise excitation assumption and can theoretically reduce the risk of extracting non-physical modes caused by a deviation from the excitation assumption.It has broad engineering application prospects due to the potentiality in the presence of colored-noise excitations.As a new transmissibility function,power spectral density transmissibility(PSDT)is defined as the ratio of two response PSDs under the same reference output and has achieved widespread attention.However,the theory and application of PSDT-driven OMA procedure still need to be further studied,especially in terms of its robustness and the applicability to large-scale civil engineering structures.For this reason,with the support from the National Science Foundation of China entitled "Analysis of strain modal parameter identification of bridge structures based on transmissibility measurements"(NO.51778204),this thesis is focused on the methodology of PSDT method and its application to the modal analysis of bridge structures under ambient excitation,and the research is performed from theoretical,numerical and experimental aspects.The main contributions and conclusions of this dissertation are outlined as follows:1)Firstly,the application of the classical TOMA to the modal analysis of bridges under ambient vibration is investigated.To meet the requirement of multiple loading conditions,the original response record is split into several shorter data blocks,each considered as a response corresponding to an unrelated loading condition.Field tests demonstrate that the adopted strategy is efficient.However,it is worth mentioning that since the time duration of acceleration used for modal analysis is the same for both classic TOMA and PSDT,the plot of class TOMA is shown to be less smooth than that of PSDT due to the reduction of averaging times in PSD estimation.2)In order to solve the problem of requiring transmissibility functions corresponding to multiple load conditions involved in the classical TOMA,the PSDT-driven modal analysis is further studied.PSDT at the system poles is independent of the excitation nature and reference output.Based on the unique properties of PSDT,it is possible to locate the system poles from the peak plot of the inverse of PSDT subtraction functions with a number of different reference outputs.Meanwhile,the modal frequencies also can be extracted by taking the SVD for PSDT matrix.The results indicate that compared to the classical TOMA technique,PSDT method doesn't require multiple response records to various loading conditions and is with easy implementation.To make improvements of modal parameter estimation,PSDT is combined with least-squares complex frequency-domain estimator.The system poles are identified by fitting a parametric model,while the stabilization diagram is constructed to separate the stable system poles.3)The applicability of PSDT method to large engineering structures is validated by employing ambient vibration testing data of three large-scale cable-stayed bridges,including a railway bridge,a highway bridge,and a pedestrian bridge.The results show that the peaks of PSDT amplitude do not coincide with the system resonances of the bridge,while PSDTs between different transferring outputs are identical at the resonant peaks.Compared to the conventional PSD-driven peak picking method,PSDT method is more insensitive to harmonic excitation.The natural frequencies of cable-stayed bridges are compared to analyze the discrepancies in the vibration behaviors of the three subject bridges.High-speed railway bridges are often designed with improved vertical and lateral stiffness to satisfy stricter limits on vertical deflection and beam-end rotation.Consequently,both the vertical and torsional frequencies of such bridges are significantly higher than those of highway bridges.The first natural frequency for vertical vibrations of a pedestrian bridge should be above the range of normal pedestrian walking frequencies to avoid serviceability problems induced by human-induced vibrations.4)The modal parameters identified from PSDT method are with high accuracy.However,the error in PSDT induced by PSD estimation is still a worthy issue to investigate.In this thesis,perturbation methodology and statistical moment theory are utilized to derive approximated expectations of the quotient function of random variables.The quotient of random variables is expanded using Taylor series at the mean values of the two variables.Based on the definition of the first two moments,the variance and the mean of this quotient are expressed in terms of expectations(i.e.variance and mean)of the given random variables.Consequently,the approximated variance of PSDT estimation is derived by substituting the variance and the mean of PSD estimation into the above-derived formula.This error formula reveals that the number of averages and reference output are important parameters affecting the accuracy of PSDT estimation.Furthermore,it indicates that the variance of PSDT at the resonant frequencies tends to be zero,and the PSDT around resonant frequencies is more insensitive to noise than PSD is.Moreover,this error formula can provide one measurement to select optimal estimation parameters,such as signal length and reference output.5)The errors in the modal parameters identified from PSDT method are qualified based on the error formula for PSDT measurements.According to the property of PSDT's variance which has local minimum value at the system's poles,the variance of PSDT with different degree of freedoms are combined to construct the maximum likelihood estimation(MLE)function.Then the modal frequencies can be identified based on PSDT variance.Subsequently,the inverse of Hessian matrix of the MLE function is calculated to qualify the error in modal frequencies.Furthermore,the variance and mean of PSDT at the system poles are derived to qualify the error in the amplitude of modal shape and its phase.To verify the correctness of the results of error analysis on modal parameters by using the error formula of PSDT estimation,the Bayesian modal identification is introduced.6)Based on the case studies of a simulated frame and three real bridge structures under ambient excitation,the applicability of the proposed procedure for qualifying the uncertainty in modal parameters is investigated.The results indicate that the variance of PSD has local maximum values around resonant frequencies,while the variance of PSDT and of its subtraction function have local minimum values as approaching resonant frequencies.Selecting the outputs,which have a good coherence with the basic outputs of PSDT,can generate one PSDT estimation with less uncertainty than other cases.Comparison among the uncertainty in modal parameters qualified by error formula of PSDT and Bayesian method is made.The result shows that the results from the two techniques have similar uncertainty which indicates that spectral error is an important factor causing the uncertainty in the modal parameters.