Model error correction from truncated modal flexibility sensitivity and generic parameters: Part I—simulation

A great advantage of the flexibility based method is the availability of the large amount of measured information from a few lower modes which can be practically measured. But the modal truncated errors would be significant for a structure with high modal density, and this poses a limitation on the usage of existing flexibility methods. This paper presents the truncated modal flexibility sensitivity with respect to the generic parameters, and a model updating method is proposed based on this sensitivity and incomplete measurement. The loss of contribution from the higher modes to the modal flexibility will not be a source of error in the proposed method. The effect of spatial incompleteness and measurement noise is investigated with numerical studies. The proposed approach is found capable of updating both the systematic model error and local stiffness error separately in a single or two stages under noisy environment.     

RESULTS OBTAINED BY MINIMISING NATURAL FREQUENCY AND MAC-VALUE ERRORS OF A BEAM MODEL

This paper reports the procedure followed by the ‘LTAS-Vibrations et Identification des Structures’ research group to generate a low-order finite element (FE) model of the GARTEUR SM-AG19 structure. The model is made of beam elements, local inertia and rigid body elements. The philosophy of the updating method is first based on the correlation of the experimental data with the results of the FE model and on the localisation of errors in the model. Updating parameters are then selected using eigenvalue sensitivity and model error localisation analyses. After updating, the quality of the FE model is assessed in terms of accuracy of the response prediction to structural modifications.     

Comparative study of damped FE model updating methods

Most of finite element (FE) model updating techniques do not employ damping matrices and hence, cannot be used for accurate prediction of complex frequency response functions (FRFs) and complex mode shapes. In this paper, a detailed comparison of two approaches of obtaining damped FE model updating methods are evaluated with the objective that the FRFs obtained from damped updated FE models is able to predict the measured FRFs accurately. In the first method, damped updating FE model is obtained by complex parameter-based updating procedure, which is a single-step procedure. In the second method, damped updated model is obtained by the FE model updating with damping identification, which is a two-step procedure. In the first step, mass and stiffness matrices are updated and in the second step, damping matrix is identified using updated mass and stiffness matrices, which are obtained in the previous step. The effectiveness of both methods is evaluated by numerical examples as well as by actual experimental data. Firstly, a study is performed using a numerical simulation based on fixed–fixed beam structure with non-proportional viscous damping model. The numerical study is followed by a case involving actual measured data for the case of F-shaped test structure. The updated results have shown that the complex parameter-based FE model updating procedure gives better matching of complex FRFs with the experimental data.     

New iterative method for model updating based on model reduction

Model reduction technique is usually employed in model updating process. Here, a new iterative method associating the model updating method with the model reduction technique is investigated. Using the traditional iterative method, the errors resulted from replacing the reduction matrix of the experimental model with that of the finite element (FE) model are not fully considered, which needs more iterations and computing time. In order to reduce the errors produced in the replacement, a new iterative method is proposed based on the traditional method, in which the correction term related to the errors is added. The comparisons between the traditional iterative method and the proposed iterative method are shown by model updating examples of solar panels and both of these two iterative methods combine the cross-model cross-mode (CMCM) method and the succession-level approximate reduction (SAR) technique. The results indicate that the convergence rate and the computing time of the new method are significantly superior to those of the traditional iterative method with or without noise.     

Model updating of damped structures using FRF data

Due to the important contribution of damping on structural vibration, model updating of damped structures becomes significant and remains an issue in most model updating methods developed to date. In this paper, the frequency response function(FRF) method, which is one of the most frequently referenced model updating methods, has been further developed to identify damping matrices of structural systems, as well as mass and stiffness matrices. In order to overcome the problem of complexity of measured FRF and modal data, complex updating formulations using FRF data to identify damping coefficients have been established for the cases of proportional damping and general non-proportional damping. To demonstrate the effectiveness of the proposed complex FRF updating method, numerical simulations based on the GARTEUR structure with structural damping have been presented. The updated results have shown that the complex FRF updating method can be used to derive accurate updated mass and stiffness modelling errors and system damping matrices.     

Interval model updating with irreducible uncertainty using the Kriging predictor

Interval model updating in the presence of irreducible uncertain measured data is defined and solutions are made available for two cases. In the first case, the parameter vertex solution is used but is found to be valid only for particular parameterisation of the finite element model and particular output data. In the second case, a general solution is considered, based on the use of a meta-model which acts as a surrogate for the full finite element mathematical model. Thus, a region of input data is mapped to a region of output data with parameters obtained by regression analysis. The Kriging predictor is chosen as the meta-model in this paper and is found to be capable of predicting the regions of input and output parameter variations with very good accuracy. The interval model updating approach is formulated based on the Kriging predictor and an iterative procedure is developed. The method is validated numerically using a three degree of freedom mass-spring system with both well-separated and close modes. A significant advantage of Kriging interpolation is that it enables the use of updating parameters that are difficult to use by conventional correction of the finite element model. An example of this is demonstrated in an experimental exercise where the positions of two beams in a frame structure are selected as updating parameters.     

Finite element model updating based on eigenvalue and strain energy residuals using multiobjective optimisation technique

The numerical results from a finite element (FE) model often differ from the experimental results of real structures. FE model updating is often required to identify and correct the uncertain parameters of FE model and is usually posed as an optimisation problem. Setting up of an objective function, selecting updating parameters and using robust optimisation algorithm are three crucial steps in FE model updating. In this paper, a multiobjective optimisation technique is used to extremise two objective functions simultaneously which overcomes the difficulty of weighing the individual objective function of more objectives in conventional FE model updating procedure. Eigenfrequency residual and modal strain energy residual are used as two objective functions of the multiobjective optimisation. Only few updating parameters are selected on the basis of the prior knowledge of the dynamic behaviours of the structure and eigenfrequency sensitivity study. The proposed FE model updating procedure is first applied to the simulated simply supported beam. This case study shows that the methodology is robust with an effective detection of assumed damaged elements. The procedure is then successfully applied to the updating of a precast continuous box girder bridge that was tested on field under operational conditions.     

Perturbation methods for the estimation of parameter variability in stochastic model updating

The problem of model updating in the presence of test-structure variability is addressed. Model updating equations are developed using the sensitivity method and presented in a stochastic form with terms that each consist of a deterministic part and a random variable. Two perturbation methods are then developed for the estimation of the first and second statistical moments of randomised updating parameters from measured variability in modal responses (e.g. natural frequencies and mode shapes). A particular aspect of the stochastic model updating problem is the requirement for large amounts of computing time, which may be reduced by making various assumptions and simplifications. It is shown that when the correlation between the updating parameters and the measurements is omitted, then the requirement to calculate the second-order sensitivities is no longer necessary, yet there is no significant deterioration in the estimated parameter distributions. Numerical simulations and a physical experiment are used to illustrate the stochastic model updating procedure.     

DYNAMIC FINITE ELEMENT MODEL UPDATING BASED ON GLOBAL INFORMATION OF STRUCTURES

Finite element model updating method based on global information is proposed. Prior investigation upon design space of structural parameters is performed before updating using statistic analysis, including parameter screening using variance analysis and response surface fitting using regression analysis. The parameter screening method selects the design parameters considering the result of hypothesis testing, which is a kind of global information. Meanwhile, the traditional updating method considers local sensitivity which only gives the information at sole point in the design space. Response surface fitting constructs a close-form multinomial which describes the relationship between concerned structural feature and selected updating parameters. It is an approximation to finite element models(FEM) and used as a substitution in the updating iterations. The presented updating method can be applied without the restriction of linear assumption. In addition, there is no data exchange between the updating prog     

火箭仪器舱试验与计算模态相关性分析

针对火箭仪器舱,研究了计算模态和试验模态的相关性分析问题。首先介绍模态相关性分析的一般方法。针对仪器舱结构复杂和模态密集的特点,引入整体模态和局部模态的概念,并提出一种用模态应变能比值区分整体模态和局部模态的定量的方法。进而研究了模态向量减缩、误差定位和模型修改等问题。在此基础上,得到了较高的试验和计算模态相关性结果,验证了计算模型,并为进一步的仪器舱减振设计提供了依据。     

A method in model updating using Miscorrelation Index sensitivity

This paper presents a new model updating method based on minimization of an index called Miscorrelation Index (MCI), which is introduced to localize the coordinates carrying error in a finite element (FE) model. MCI can be calculated from measured frequency response functions (FRFs) and dynamic stiffness matrix of the FE model for each coordinate as a function of frequency. Nonzero numerical values for MCI of a coordinate indicate errors in one or more elements of the system matrices corresponding to this coordinate. The sensitivity-driven model updating method presented in this study (MCI Sensitivity Method) is based on minimization of MCI. The application of the method is illustrated with four case studies. In the first and second examples a discrete system is considered, and computationally generated and polluted FRFs are used as pseudo-test data. In the third and fourth case studies, real test data is used and the performance of the method in practical applications is demonstrated on the benchmark structure built to simulate the dynamic behavior of an airplane, namely, GARTEUR SM-AG19 test bed. It is concluded that MCI Sensitivity Method yields successful results even when the measured responses of only a few coordinates are used, especially when miscorrelation is due to local errors.     

A two-level updating procedure of the component-mode synthesis model

The component-mode synthesis method is effective for reducing the degrees of freedom of a finite element model for dynamic analysis of a complex structure. However, in the resulting computational model unacceptable errors can exist due to poor modeling. The model can be corrected by applying system identification methods. By localising the parameter errors it is possible to use the updating procedure of the second author. The modal transformation performed within this updating leaves the parameters unchanged. As a result, the corrected component equations and synthesised component-mode model are derived. The general identification procedure contains many parameters to be identified for a complex system. Minimising the objective function with respect to these parameters results in a highly non-linear system of equations. In order to circumvent this, the two-level identification procedure is developed and applied. If the measurement errors of the component quantities are independent of each other, the quadratic objective function can be decomposed on the basis of these components. By introducing model and goal coordination variables, the identified parameters as subsystem units are corrected at the first level and the second level includes the co-ordination units. A parallel iterative algorithm is presented in detail.     

DAMAGE IDENTIFICATION USING CHANGES OF EIGENFREQUENCIES AND MODE SHAPES

In the present paper, we describe an approach to identify the location and the extent of the damage introduced into the steel frame, using a two-step procedure. In the first step, the measured dynamic response of the original undamaged structure was used to generate a reference finite element (FE) model of the structure. The selected parameters were identified by means of a mathematical optimisation algorithm (‘updating procedure'), minimising an objective function containing the test/analyses differences of eigenfrequencies and mode shapes. The uncertain model parameters had to be chosen with care in order to retain the physical significance of the updated model. In the next step, the experimental modal data of the damaged structure were used to identify the extent of the damages. This was based on comparing the changes of stiffness parameters identified from the undamaged and the damaged structure. With the identified parameters, the FE model was able to reproduce the experimental data as close as possible and allowed the identification of the extent of the damage.     

使用应变模态和遗传算法的有限元模型修正方法

提出了采用应变模态置信度为待修正响应特征的有限元模型修正方法。应变模态置信度是评价有限元仿真与试验测试结果相关性的方法,可以为模型修正提供全局的频率误差信息和局部的应变相关性信息。首先,介绍了应变模态和有限元模型修正的相关理论方法;然后,以某航空加筋壁板结构为对象,通过仿真分析和"仿真试验"获得结构的应变模态频率以及对应的应变振型,进一步计算频率误差和应变模态置信度误差;最后,基于两种误差构造模型修正的目标函数,采用遗传算法对目标函数进行优化,修正结构中的待修正参数,并将修正后参数代入模型,验证所提方法的正确性和有效性。结果表明:所采用的方法获得的修正后有限元模型具有复现修正响应特征的能力,并且对于未修正频段内的响应也具有较好的预测能力。     

步兵战车车体结构有限元模型修正

针对某型步兵战车整车刚柔耦合发射动力学中柔性车体有限元模型精度低的问题,基于模态试验数据,应用支持向量机响应面模型修正理论对车体结构有限元模型进行了修正。应用ANSYS有限元分析软件对车体结构进行模态分析,提取前6阶模态的固有频率和振型。为验证模型,设计了模态试验方案,实测了车体结构的模态信息。基于有限元模型数据与实测数据的相对误差,采用支持向量机响应面模型修正方法对车体结构弹性模量和密度进行修正。模型确认结果和动力学模型应用结果表明,修正后的车体有限元模型精度有了大幅度提高,能更加真实地反映车体的结构特征,为射击精度分析提供了准确的模型基础。     

一种基于模型修改的结构多点损伤识别方法

可用于结构损伤识别的方法很多。一般来讲正向方法直接利用结构模态参数的变化，逆向方法则利用模态参数变化反演结构物理参数变化，还有些方法利用了神经网络和模式识别技术。文中利用模型修改的思想，通过逆向方法计算结构单元刚度变化系数来对结构的多点损伤进行识别。以一个七自由度弹簧阻尼质量系统作为研究对象，用数值模拟方法及特征系统实现算法计算系统的模态参数，并用这些模态参数验证所提出方法的可行性，结果表明该方法对多点损伤的识别是简单而可行的。     

Damage identification by multi-model updating in the modal and in the time domain

Computational model updating techniques are used to adjust selected parameters of finite element models in order to make the models compatible with experimental data. This is done by minimizing the differences (residuals) of analytical and experimental data, for example, natural frequencies and mode shapes by numerical optimization procedures. For a long-time updating techniques have also been investigated with regard to their ability to localize and quantify structural damage. The success of such an approach is mainly governed by the quality of the damage model and its ability to describe the structural property changes due to damage in a physical meaningful way. Our experience has shown that due to unavoidable modelling simplifications and measurement errors the changes of the corresponding damage parameters do not always indicate structural modifications introduced by damage alone but indicate also the existence of other modelling uncertainties which may be distributed all over the structure. This means that there are two types of parameters which have to be distinguished: the damage parameters and the other parameters accounting for general modelling and test data uncertainties. Although these general parameters may be physically meaningless they are necessary to achieve a good fit of the test data and it might happen that they cannot be distinguished from the damage parameters. For complex industrial structures it is seldom possible to generate unique structural models covering all possible damage scenarios so that one has to expect, that the parameters introduced for describing the damage will not be fully consistent with the physical reality. Even then the change of such parameters identified from test data taken continuously or temporarily over the time may serve as a feature for structural health monitoring. It is well known that low-frequency modal test data or static response data are not very well suited for detecting and quantifying localized small size damage. Time domain response data from impact tests carry high-frequency information which usually is lost when experimental modal data are utilized for damage identification. Even so only little literature was found addressing the utilization of experimental time histories for model updating in conjunction with damage identification.In the present paper we summarize the methodology of computational model updating and report about our experience with damage identification using two different model updating techniques. The first is based on classical modal residuals (natural frequencies and mode shapes) which is extended to allow for simultaneous updating of two models, one for the initial undamaged structure and the second for the damaged structure using the test data of both states (multi-model updating). The second technique uses residuals composed of measured and analytical time histories. Time histories have the advantage of carrying high-frequency information which is beneficial for the detection of local damage and which usually is lost when modal residuals are used. Both techniques have been applied to the same beam structure consisting of two thin face sheets which were bonded together by an adhesive layer. It was the aim of this application to study the performance of the two techniques to localize and quantify the damage which was introduced locally in the adhesive layer.     

Model updating of locally non-linear systems based on multi-harmonic extended constitutive relation error

Improving the fidelity of numerical simulations using available test data is an important activity in the overall process of model verification and validation. While model updating or calibration of linear elastodynamic behaviors has been extensively studied for both academic and industrial applications over the past three decades, methodologies capable of treating non-linear dynamics remain relatively immature. The authors propose a novel strategy for updating an important subclass of non-linear models characterized by globally linear stiffness and damping behaviors in the presence of local non-linear effects. The approach combines two well-known methods for structural dynamic analysis. The first is the multi-harmonic balance (MHB) method for solving the non-linear equations of motion of a mechanical system under periodic excitation. This approach has the advantage of being much faster than time domain integration procedures while allowing a wide range of non-linear effects to be taken into account. The second method is the extended constitutive relation error (ECRE) that has been used in the past for error localization and updating of linear elastodynamic models. The proposed updating strategy will be illustrated using academic examples.     

Model updating of lattice structures: A substructure energy approach

Model updating is an inverse problem to identify and correct uncertain modeling parameters, which leads to better predictions of the dynamic behavior of a target structure. This study presents a direct physical property adjustment method and the substructure energy approach, which can accurately update the stiffness and mass matrices of lattice structures using incomplete eigenvectors measured from critical substructures. For validation, the proposed method is applied to update models of a mass-spring system, a two-dimensional, and a three-dimensional lattice structure.     

Plane strain extrusion by sequential limit analysis

In this paper, the analysis of plane strain forward extrusion through wedge shaped dies is implemented by using a finite element procedure based on a dual (upper bound) formulation. Since the von Mises yield criterion is employed and expressed in an inequality form, stresses do not appear in this upper bound formulation. As a result, neither complicated stress updating nor rigid zone treatment is needed in this approach. The effect of material strain hardening can be incorporated in the analysis by updating local yield stresses stepwise with deformation history. The computed results are first compared with existing analytical solutions in good agreement. Effects of friction, die semiangle, height reduction ratio and material strain hardening on the extrusion load of aluminum are then investigated.