A Stochastic Model Updating Method For Joint-assembled Structures In A Bayesian Framework

A majority of industrial structures are composed of various components that are assembled with joints.Abundant researches indicate that the mechanical characteristics of joints have a significant impact on the dynamic behavior of an assembled structure,and its high-fidelity numerical model is the basis for the dynamic response prediction of an assembled structure.In fact,there are a number of uncertain parameters at the contact interface of a joint.Deterministic model updating methods may lead to over-design.Therefore,stochastic model updating methods considering joint uncertainty have attracted more and more attention from industry and academia.The Bayesian approach based on probability theory is one of the most important theoretical approaches to parameter uncertainty quantification.However,there are not many studies on its application to the stochastic model updating of assembled structures with bolted joints and the relevant techniques are not yet mature.The main application difficulties are: a)It is difficult to characterize the local features of a joint,and to construct a likelihood function with high sensitivity to the features;b)Repeated analyses of the theoretical model of an assembled structure leads to high computational costs in parameter uncertainty quantification and uncertainty propagation analysis;c)Multiple and coupled uncertain parameters in a joint makes the quantification more difficult.Based on the research status and progress,the main work of this thesis is as follows.(1)Aiming at the problem that the local features of joints are difficult to identify,a Bayesian SMUM(stochastic model updating method)for assembled structures under multi-point excitation is proposed.This method introduces multi-point random excitation to fully excite the local mechanical properties of the joints and solves the PSD(power spectral density)of the assembled structure under multi-point random excitation by the pseudo-excitation method.Using the property that the measured PSD obeys a two-degree-of-freedom chi-square distribution and a Bayesian inference framework,a posterior distribution of the joint parameters in the multiple-input-multiple-output mode is established.The numerical examples show that the proposed method can reduce the demanding requirements for measured data and enable the simultaneous identification of joint stiffness and damping in a narrow frequency domain.(2)Aiming at the problem of high computational cost of Bayesian SMUM,an efficient Bayesian SMUM based on semi-analytic sensitivity analysis of PSD is proposed.This method addresses this difficulty from two aspects: a)The component model synthesis is introduced to isolate bolted joints with uncertain parameters from an assembled structure,which can avoid repeated analyses of the other part,and also obtain a reduced-order dynamic model of an assembled structure.b)The first-order and second-order sensitivities of the structure with respect to the joint parameters are calculated by the pseudo-excitation method,and the semi-analytic gradient vector and Hessian matrix of the power spectrum are further derived.Using the property that the measured average PSD matrix obeys the complex Wishart distribution,the posterior distribution of the joint parameters conditional on the measured PSD matrix is established and used to construct the objective function for the maximum posterior estimate of the joint parameters.The semi-analytic gradient vector and Hessian matrix of the objective function are derived from the gradient vector and Hessian matrix of the PSD to accelerate the optimization process.The numerical example results show that the number of iterations of the Bayesian optimization process with semi-analytic sensitivity of the joint parameters is significantly reduced.The component model synthesis can also reduce the computational cost appropriately.(3)Aiming at the calculation accuracy problem caused by the component model synthesis ignoring the influence of truncated high-order modes,a SMUM for assembled structures based on a high-precision modal synthesis method is proposed.This method develops a high-fidelity reduced dynamic model for the assembled structure with elastic boundary conditions based on the low-order modes of the substructure and the truncated generalized constraint modes.Using the concept of prediction error,the reduced model is embedded in a stochastic analysis model.Furthermore,a posteriori distribution for the joint parameters using eigenvalues and eigenvectors of the assembled structure as measured data are developed with a Bayesian inference framework.Sampling from posterior distributions and quantifying uncertainty in joint parameters are accomplished by using an adaptive rejection delay sampling method.The numerical example show that the proposed method can minimize the number of substructural modes while ensuring accuracy of updated results.(4)Aiming at the problem of mixed uncertainties of joints,a method of uncertainty quantification and uncertainty propagation analysis in joint parameters is proposed which considers both the inherent variability of the joints and the measurement error.In this method,hyperparameters and prediction errors are used to model the parameter variability and measurement noise respectively,and establish a hierarchical Bayesian inference framework for the joint parameters,and then derive marginal posterior distributions for the hyperparameters,and deduce posterior distributions for the joint parameters using samples from hyperparameter sampling.In the uncertainty propagation analysis of the joint parameters,in order to solve the computational difficulties caused by calculating response solutions for a large number of joint parameter samples,a polynomial dimensional decomposition method is used to establish an efficient mapping relationship between the joint parameters and the PSD through a membership function with increasing dimensionality of the uncertain parameters.The numerical example show that the proposed method can accurately quantify the uncertainty of the joint parameters and the results of the uncertainty propagation analysis match the statistical results of the measured power spectrum,with a significant advantage in computational efficiency.