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 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.     

基于近似似然的频响函数不确定性模型修正

贝叶斯模型修正框架下,以频响函数作为目标,提出了一种使用近似似然函数的不确定性模型修正方法。相比于模态参数,频响函数包含了结构更加充分的信息,用于结构动力学模型修正时有诸多优点,但现有的不确定性模型修正方法并不能很好地实现将频响函数作为目标进行修正。针对此问题,介绍了频响函数和贝叶斯框架下的不确定性模型修正理论,基于近似贝叶斯计算提出了一种近似似然函数,可适用于频响函数作为目标进行不确定性修正。将提出的似然函数应用到三自由度数值和H型非对称梁的有限元模型修正算例中,并结合DREAM算法对不确定性参数进行识别。研究结果表明：修正后参数的上、下限与目标值相差无几,修正后模型的频响函数与目标值几乎重合,在一定噪声水平下仍具有较好的修正效果,验证了所提方法的有效性。     

Cross-model cross-mode method for model updating

Model updating has become a common method to improve the correlation between finite-element models and measured data. This paper develops a direct, physical property adjustment model updating method, named as cross-model cross-mode (CMCM) method. This new method is capable of updating the mass and stiffness matrices simultaneously based on very limited measured mode shapes and modal frequencies. Two structural models, a shear building model and a three-dimensional frame structural model, are demonstrated in the numerical examples. Numerical results indicate that an excellent updating is achievable by applying the CMCM method.     

COUPLING OF NON-LINEAR SUBSTRUCTURES USING VARIABLE MODAL PARAMETERS

This paper presents a general methodology for the coupling analysis of systems with relatively weak non-linearities by assuming that the response remains harmonic under harmonic excitation. Standard coupling methods and their current shortcomings were discussed first. Two ways of obtaining non-linear modal parameter variations, namely profile constructing and parameter extracting, were presented next. The profile constructing method uses the system's spatial data directly, while the parameter extracting method is based on a non-linear modal analysis of measured response data. Through numerical test cases, it was shown that both methods yielded virtually identical results. An iterative algorithm for the coupling of non-linear subsystems was presented in a form compatible with profile building. A six-degree-of-freedom system with cubic stiffness non-linearity was chosen for a detailed numerical study. Two subsystems, one linear and the other non-linear, were coupled to obtain the modal parameter variations of the coupled system. Using the non-linear modal parameters, the response of the coupled system was predicted at various force levels and the findings were checked via direct simulations using the harmonic balance method. Finally, the methodology was validated by coupling experimentally derived non-linear modal models of two substructures. As for the numerical study, the response of the coupled non-linear structure was predicted at various force levels and the findings were checked against direct measurements. Very good agreement was obtained in all cases studied.     

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.     

USE OF AN UPDATED FINITE ELEMENT MODEL FOR DYNAMIC DESIGN

Model updating techniques are used to update a finite element model of a structure so that an updated model predicts more accurately the dynamics of a structure. The application of such an updated model in dynamic design demands that it also predict the effects of structural modifications with a reasonable accuracy. This paper deals with the updating of a finite element model of a structure using measured modal data and its subsequent use for predicting the effects of structural modifications. An updated model is obtained by employing a method of model updating based on the constrained optimisation. Structural modifications in terms of mass and beam modifications are then introduced to evaluate the updated model for its usefulness in dynamic design.     

Structural response reconstruction based on the modal superposition method in the presence of closely spaced modes

An approach for structural response reconstruction based on the modal superposition method in the presence of closely spaced modes is proposed in this paper. In this method, the entire mode-set of a structure is divided into closely spaced modes and the rest of the modes. The rest of the modal responses whose response is known will be separated into individual modal response by using the empirical mode decomposition method with intermittency criteria. Starting from the mode shapes, derived from these modal responses, the rest of the modal responses at the unavailable locations can be acquired. Furthermore, the contribution of the closely spaced modal responses at the unavailable locations can be obtained based on these responses. This proposed method is valid if there is no periodic excitation. However, in practice it is common that a structure might be excited by transient, stochastic, periodic forces or a combination thereof, i.e. hybrid excitations. A hybrid approach for solving the reconstruction problem of hybrid excitations is also developed, which is based on the proposed method and transmissibility concept in response reconstruction. Numerical studies are conducted and compared with theoretical predictions for validation. Effects of background noise level, high damping ratio and multiple forces are investigated in detail.     

EXTRACTION OF RITZ VECTORS FROM VIBRATION TEST DATA

Modal parameters obtained from modal testing (such as modal vectors, natural frequencies, and damping ratios) have been used extensively in system identification, finite element model updating, and structural health monitoring. As an alternative to modal vectors, load-dependent Ritz vectors have been shown useful in various areas of structural dynamics such as model reduction and damage detection. The applications of Ritz vectors, however, have been mainly limited in analytical and numerical analyses because of the difficulty to identify them from vibration tests. This paper presents a procedure to extract load-dependent Ritz vectors using a complete flexibility matrix constructed from measured vibration test data. The proposed method cannot only construct the Ritz vectors corresponding to the actual load pattern employed in vibration tests, but also generate Ritz vectors from arbitrary load patterns. Experimental data obtained from the vibration test of a grid-type bridge structure are employed to validate and illustrate the proposed extraction procedure.     

使用有效模态质量和遗传算法的有限元模型修正

提出将模态频率和有效模态质量构造的残差作为遗传算法的目标函数进行结构动力学有限元模型修正的方法。有效模态质量不但可以为结构动力学响应分析提供一种判断模态贡献程度的方法,而且能够为有限元模型修正提供更多的信息量。介绍了有效模态质量的概念和基于遗传算法的结构动力学模型修正理论,在此基础上采用仿真算例验证了所提出方法的正确性和有效性。仿真结果显示,模型修正后参数最大误差为-0.062%,不管是在修正频段内还是修正频段外,频率和有效模态质量的均方误差都小于0.025%。研究表明,使用有效模态质量和遗传算法的结构动力学有限元模型修正是有效可行的。     

Localization and quantification of damage in beam-like structures using sensitivities of principal component analysis results

Principal component analysis (PCA) is known as an efficient method for dynamic system identification and diagnosis. This paper addresses a damage diagnosis method based on sensitivities of PCA in the frequency domain for linear-form structures. The aim is not only to detect the presence of damage, but also to localize and to evaluate it. The Frequency response functions measured at different locations on the beam are considered as data for the PCA process. Sensitivities of principal components obtained from PCA to beam parameters are computed and inspected according to the location of sensors; their variation from the healthy state to the damaged state indicates damage locations. The damage can be evaluated next providing that a structural model is available; this evaluation is based on a model updating procedure. It is worth noting that the diagnosis process does not require a modal identification achievement. Both numerical and experimental examples are used for better illustration.     

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。     

Segment-to-segment contact elements for modelling joint interfaces in finite element analysis

This paper presents an efficient approach to model contact interfaces of joints in finite element analysis (FEA) with segment-to-segment contact elements like thin layer or zero thickness elements. These elements originate from geomechanics and have been applied recently in modal analysis as an efficient way to define the contact stiffness of fixed joints for model updating. A big advantage of these elements is that no global contact search algorithm is employed as used in master–slave contacts. Contact search algorithms are not necessary for modelling contact interfaces of fixed joints since the interfaces are always in contact and restricted to small relative movements, which saves much computing time. We first give an introduction into the theory of segment-to-segment contact elements leading to zero thickness and thin layer elements. As a new application of zero thickness elements, we demonstrate the implementation of a structural contact damping model, derived from a Masing model, as non-linear constitutive laws for the contact element. This damping model takes into account the non-linear influence of frictional microslip in the contact interface of fixed joints. With this model we simulate the non-linear response of a bolted structure. This approach constitutes a new way to simulate multi-degree-of-freedom systems with structural joints and predict modal damping properties.     

基于应变模态的模态应变能损伤识别方法

传统模态应变能计算需要完备模态振型信息,而模态振型信息中存在转角自由度难以准确获取的问题,为解决该问题,开展基于应变模态的模态应变能损伤识别研究,实现了结构损伤的定量识别。首先,通过基于应变与位移之间的联系,推导出应变模态与位移模态之间的转换矩阵;其次,利用应变模态代替位移模态计算单元模态应变能,建立基于灵敏度分析的损伤识别方程组;最后,根据奇异值截断法求解该方程组识别结构损伤。以一两端固支梁结构为对象,开展数值仿真和实验研究。结果表明,该方法可以有效识别出结构的损伤位置和损伤程度,相对于基于振型扩充的模态应变能损伤识别方法,具有更好精度和抗噪性能。     

基于灵敏度分析的刀形天线有限元模型修正

准确可靠的有限元模型是结构动态特性分析、设计改进的基础,文中利用模态试验得到的模态参数对某刀形天线进行有限元模型修正.首先建立天线结构的参数化模型,然后通过灵敏度分析选择合适的设计参数作为后续优化对象,利用计算与试验的模态频率之间的相对误差构造加权的优化目标函数,最后应用1阶优化方法修正结构的有限元模型.修正后有限元模型的模态频率最大相对误差降低至10%以内,模态置信度(MAC)均大于0.8.该修正模型可用于后续的动力学分析.     

Damage identification by response surface based model updating using D-optimal design

Statistical tools, as well as mathematical ones, have been widely adopted and their performance has been shown in different engineering problems where randomicity usually exists. In the realm of engineering, merging statistical analysis into structural evaluation and assessment will be a tendency in the future. As a combination of mathematical and statistical techniques, response surface methodology has been successfully applied to design optimization, response prediction and model validation. This methodology provides explicit functions to represent the relationships between the inputs and outputs of a physical system, which is also a desirable advantage in damage identification. However, so far little research has been carried out in applying the response surface methodology to structural damage identification. This paper presents a damage identification method achieved by response surface based model updating using D-optimal designs. Compared with some common designs constructing response surfaces, D-optimal designs generally require a minimum number of numerical samples and this merit is quite desirable when analysts cannot obtain enough samples. In this study, firstly D-optimal designs are used to establish response surface models for screening out non-significant updating parameters and then first-order response surface models are constructed to substitute for finite element models in predicting the dynamic responses of an intact or damaged physical system. Three case studies of a numerical beam, a tested reinforced concrete frame and a tested full-scale bridge have been used to verify the proposed method. Physical properties such as Young’s modulus and section inertias were chosen as the input features and modal frequency was the only response feature. It has been observed that the proposed method gives enough accuracy in damage prediction of not only the numerical but also the real-world structures with single and multiple damage scenarios, and the first-order response surface models based on the D-optimal criterion are adequate for such damage identification purposes.     

基于模态综合技术的结构有限元模型修正

由于结构的动力分析需要大量的计算时间和占用大量的计算机内存,常规的数值迭代计算方法难以实现,提出了基于模态综合技术的模型修正方法。该方法首先得到缩减后结构模型的频率与振型,并将该振型转换为缩减前模型物理坐标下的振型。然后,用缩减后模型的频率和转换后的振型,共同构成模型修正的优化目标函数,进而通过优化求解实现结构的模型修正。该方法既保证了计算精度又提高了模型修正的计算效率,使大型复杂结构的模型修正成为可能。最后,对某吊杆拱桥模型进行了动态测试和模型修正,验证了该算法的有效性。     

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。     

齿轮系统非线性动力学微分方程的渐进法

利用渐进法研究了一类考虑时变啮合刚度和间隙的齿轮系统的非线性动力学微分方程,建立了这类模型的解的统一形式。通过计算表明,渐进法具有良好的通用性,尤其适用于求解大型非线性微分方程组。最后给出了渐进法的计算结果,并且验证能与数值计算结果很好地吻合。     

LOCALISATION OF DAMAGE IN SMART STRUCTURES THROUGH SENSITIVITY ENHANCING FEEDBACK CONTROL

The concept of sensitivity enhancing control (SEC) introduced previously by the authors exploits the relationship between feedback control gains and classic measures of root and system sensitivity in order to increase the magnitude of variation in modal characteristics when damage occurs. This paper further develops the SEC concept, focusing on the ability tolocate damage in smart structures. Two distinct methods are considered for locating damage. Forward methods rely on a priori knowledge of how certain damage scenarios affect modal properties. Using this information, the correlation between measured modal frequency shifts and predicted modal frequency shifts for a given set of damage scenarios is used to identify damage location. Inverse, or model updating methods attempt to update mass and/or stiffness matrices of structural models based on measured modal frequencies and mode shapes. The resulting perturbation matrices indicate damage location. Here, both methods are evaluated through finite-element simulation of controlled structures with local stiffness damage, with the goal of determining whether inclusion of modal data from feedback controlled structures enhances the localisation process. Results show that application of both forward and inverse methods to measured closed-loop modal characteristics increases both the accuracy with which stiffness damage can be located and the ability to tolerate noise in measurement of modal properties used to locate damage.