Regularised finite element model updating using measured incomplete modal data

This paper presents an effective approach for directly updating finite element model from measured incomplete vibration modal data with regularised algorithms. The proposed method is based on the relationship between the perturbation of structural parameters such as stiffness change and the modal data measurements of the tested structure such as measured mode shape readings. In order to adjust structural parameters at detailed locations, structural updating parameters will be selected at critical point level to reflect the modelling errors at the connections of structural elements. These updating parameters are then evaluated by an iterative or a direct solution procedure, which gives optimised solutions in the least squares sense without requiring an optimisation technique. In order to reduce the influence of modal measurement uncertainty, the Tikhonov regularisation method incorporating the L-curve criterion is employed to produce reliable solutions for the chosen updating parameters. Numerical simulation investigations and experimental studies for the laboratory tested space steel frame structure are undertaken to verify the accuracy and effectiveness of the proposed methods for adjusting the stiffness at the joints of structural members. The results demonstrate that the proposed methods provide reliable estimates of finite element model updating using the measured incomplete modal data.     

PREDICTION OF DYNAMIC CHARACTERISTICS USING UPDATED FINITE ELEMENT MODELS

Model updating techniques are used to update a finite element model of a structure so that an updated model predicts the dynamics of a structure more accurately. 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 updating of a finite element model of a structure and its subsequent use for predicting the effects of structural modifications. Updated models have been obtained by a direct model updating method and by an iterative method of model updating based on the frequency response function (FRF) data. The suitability of updated models for predicting the effect of structural modifications is evaluated by some computer and laboratory experiments. First a study is performed using a simulated fixed-fixed beam. Cases of complete, incomplete and noisy data are considered. Updated models are obtained by the direct and the FRF-based method in each of these cases. These models are then used for predicting the changes in the dynamic characteristics brought about due to a mass and a beam modification. The simulated study is followed by a study involving actual measured data for the case of an F-shape test structure. The updated finite element models for this structure are obtained again by the direct and the FRF-based method. Structural modifications in terms of mass and beam modifications are then introduced to evaluate the updated model for its usefulness in dynamic design. It is found that the predictions based on the iterative method based updated model are reasonably accurate and, therefore, this updated model can be used with reasonable accuracy to perform dynamic design. The predictions on the basis of the direct method based updated model are found to be reasonably accurate in the lower portion of the updating frequency range but the predictions are in a significant error in the remaining portion of the updating frequency range. It is concluded that the updated models that are closer to the structure physically are likely to perform better in predicting the effects of structural modification.     

Damage identification in plates using finite element model updating in time domain

A response sensitivity-based approach is presented for identifying the local damages in isotropic plate structures from the measured structural dynamic responses. The local damage is simulated by a reduction in the elemental Young's modulus of the plate. In the forward analysis, the forced vibration responses of the plate under external force are obtained from Newmark direct integration. In the inverse analysis, a response sensitivity-based finite element model updating approach is used to identify local damages of the plate in time domain. The damage identification results are obtained iteratively with the penalty function method with Tikhonov regularization using the measured structural dynamic responses. Two numerical examples are investigated to illustrate the correctness and efficiency of the proposed method. Both single damage and multiple damages cases are studied. The effects of measurement noise and measurement point on the identification results are investigated. Studies in this paper indicate that the proposed method is efficient and robust for both single and multiple damages for plate structures. Good identified results can be obtained from the short time histories of a few number of measurement points.     

Dynamic property analysis of the space-frame structure using the spectral element method

In this paper, the space-frame structure is analyzed using the spectral element method (SEM). It is periodic and composed of many geometrically uniform components. They can be considered as space beam-column elements. The spectral stiffness matrixes of the elements are established by using the variational method and the discrete Fourier transform theory. Then frequency responses of the structure are obtained by solving the spectral equations. In comparison with the results of the finite element method, those in the SEM are more accurate by taking less time, and the vibration bandgaps are observed from the responses of the structure. Moreover, the span, the substructure height, the substructure number, the material combination, the structural damping and the load location have obvious influences on the structural dynamic properties. These findings will provide the engineers with new ideas of the structural and the vibration isolation design.     

Structural damage identification of the highway bridge Z24 by FE model updating

The development of a methodology for accurate and reliable condition assessment of civil structures has become very important. The finite element (FE) model updating method provides an efficient, non-destructive, global damage identification technique, which is based on the fact that the modal parameters (eigenfrequencies and mode shapes) of the structure are affected by structural damage. In the FE model the damage is represented by a reduction of the stiffness properties of the elements and can be identified by tuning the FE model to the measured modal parameters. This paper describes an iterative sensitivity based FE model updating method in which the discrepancies in both the eigenfrequencies and unscaled mode shape data obtained from ambient tests are minimized. Furthermore, the paper proposes the use of damage functions to approximate the stiffness distribution, as an efficient approach to reduce the number of unknowns. Additionally the optimization process is made more robust by using the trust region strategy in the implementation of the Gauss-Newton method, which is another original contribution of this work. The combination of the damage function approach with the trust region strategy is a practical alternative to the pure mathematical regularization techniques such as Tikhonov approach. Afterwards the updating procedure is validated with a real application to a prestressed concrete bridge. The damage in the highway bridge is identified by updating the Young's and the shear modulus, whose distribution over the FE model are approximated by piecewise linear functions.     

Identification and updating of loading in frameworks using dynamic measurements

This paper is concerned with the effect of structural loading on dynamic performance. This topic is recognised as being of importance when validating finite element (FE) models with experimental data. A strategy for including axial load effects in a model updating procedure is developed. The method can be used to identify loading in structural frameworks using measured dynamic data.The effectiveness of the new method is demonstrated by means of case studies involving both simulated and experimental data. The theoretical study allows aspects of the sensitivity of the method to realistic levels of experimental noise to be studied as well as the way in which dynamic load identification can be enhanced with static measurements. The experimental case study proves the practical success of the technique. Updated axial load parameters are compared with static measurements of the same quantities.     

Statistical damage identification of structures with frequency changes

Model updating methods based on structural vibration data have being rapidly developed and applied to detect structural damage in civil engineering. But uncertainties existing in the structural model and measured vibration data might lead to unreliable damage detection. In this paper a statistical damage identification algorithm based on frequency changes is developed to account for the effects of random noise in both the vibration data and finite element model. The structural stiffness parameters in the intact state and damaged state are, respectively, derived with a two-stage model updating process. The statistics of the parameters are estimated by the perturbation method and verified by Monte Carlo technique. The probability of damage existence is then estimated based on the probability density functions of the parameters in the two states. A higher probability statistically implies a more likelihood of damage occurrence. The presented technique is applied to detect damages in a numerical cantilever beam and a laboratory tested steel cantilever plate. The effects of using different number of modal frequencies, noise level and damage level on damage identification results are also discussed.     

Nonlinear finite element model updating of an infilled frame based on identified time-varying modal parameters during an earthquake

A model updating methodology is proposed for calibration of nonlinear finite element (FE) models simulating the behavior of real-world complex civil structures subjected to seismic excitations. In the proposed methodology, parameters of hysteretic material models assigned to elements (or substructures) of a nonlinear FE model are updated by minimizing an objective function. The objective function used in this study is the misfit between the experimentally identified time-varying modal parameters of the structure and those of the FE model at selected time instances along the response time history. The time-varying modal parameters are estimated using the deterministic–stochastic subspace identification method which is an input–output system identification approach. The performance of the proposed updating method is evaluated through numerical and experimental applications on a large-scale three-story reinforced concrete frame with masonry infills. The test structure was subjected to seismic base excitations of increasing amplitude at a large outdoor shake-table. A nonlinear FE model of the test structure has been calibrated to match the time-varying modal parameters of the test structure identified from measured data during a seismic base excitation. The accuracy of the proposed nonlinear FE model updating procedure is quantified in numerical and experimental applications using different error metrics. The calibrated models predict the exact simulated response very accurately in the numerical application, while the updated models match the measured response reasonably well in the experimental application.     

Inverse substructure method for model updating of structures

Traditional model updating of large-scale structures is usually time-consuming because the global structural model needs to be repeatedly re-analyzed as a whole to match global measurements. This paper proposes a new substructural model updating method. The modal data measured on the global structure are disassembled to obtain the independent substructural dynamic flexibility matrices under force and displacement compatibility conditions. The method is extended to the case when the measurement is carried out at partial degrees-of-freedom of the structure. The extracted substructural flexibility matrices are then used as references for updating the corresponding substructural models. An orthogonal projector is employed on both the extracted substructural measurements and the substructural models to remove the rigid body modes of the free–free substructures. Compared with the traditional model updating at the global structure level, only the sub-models at the substructural level are re-analyzed in the proposed substructure-based model updating process, resulting in a rapid convergence of optimization. Moreover, only measurement on the local area corresponding to the concerned substructures is required, and those on other components can be avoided. The effectiveness and efficiency of the proposed substructuring method are verified through applications to a laboratory-tested frame structure and a large-scale 600 m tall Guangzhou New TV Tower. The present technique is referred to as the inverse substructuring model updating method as the measured global modal data are disassembled into the substructure level and then the updating is conducted on the substructures only. This differs from the substructuring model updating method previously proposed by the authors, in which the model updating is still conducted in the global level and the numerical global modal data are assembled from those of substructures. That can be referred to as the forward substructuring model updating method.     

Structural damage detection of controlled building structures using frequency response functions

If a building structure requires both a vibration control system and a health monitoring system, the integration of the two systems will be cost-effective and beneficial. One of the key problems of this integrated system is how to use control devices to enhance system identification and damage detection. This paper presents a new method for system identification and damage detection of controlled building structures equipped with semi-active friction dampers through model updating based on frequency response functions. The two states of the building are first created by adding a known stiffness using semi-active friction dampers. A scheme based on the frequency response functions of the two states of the building is then presented to identify stiffness parameters of structural members in consideration of structural connectivity and transformation information. By applying the proposed model updating scheme to the damaged building, a damage detection scheme is proposed based on the identified stiffness parameters of structural members of both the original and damaged buildings. The feasibility of the proposed schemes is finally demonstrated through a detailed numerical investigation in terms of an example building, in which the effects of measurement noise and excitation conditions are discussed. The numerical results clearly show that the proposed method can locate and quantify damage satisfactorily even though measurement noise is taken into consideration.     

A sensitivity-based structural damage identification method with unknown input excitation using transmissibility concept

Unknown input excitation and local damages universally coexist in a practical situation. Therefore, in this paper a structural damage identification method based on the transmissibility concept in state space domain is proposed without the need for input measurements. On the basis of the transformation matrix which is computed using the system Markov parameters in state space, the relationship between two different sets of acceleration response measurements can be formulated under the same input excitation. A sensitivity-based model updating approach is applied to identify the local damages by minimizing the difference between the measured response and the reconstructed response. The sensitivity of the dynamic acceleration response with respect to the elemental stiffness factors is derived analytically in the state space domain, which accelerates the process of damage identification. A numerical cantilever beam is employed to validate that the variation of structural parameters induced by the local damages can be accurately and effectively identified without the input excitation information by the proposed method even with measurement noise considered. A laboratory test is further carried out to verify the proposed structural damage identification method based on the response reconstruction technique.     

Finite element model updating of a rod-type ultrasonicmotor based on response surface method

The conventional finite element model(FEM) of a rod-type ultrasonic motor is usually simplified by means of continuous composite structure. Because the actual contact characteristics between the parts of the ultrasonic motor is ignored, there is bigger error between the calculated values and experimental results. Aiming at solving problem, a new modeling method of a rod-type ultrasonic motor is presented to obtain a high-accuracy FEM. The bolt pretension and the normal contact stiffness and friction coefficient of the contact surface of ultrasonic motor are all considered in this method, and the significant parameters of working mode of the motor are selected by the response surface method, and the goal of calculating the structural response rapidly is realized by building the response surface model to replace the FEM. The result of finite element model updating shows that the average error of modal frequencies of updated model drops to 0.21% from 1.20%. The accuracy of FEM is obviously improved, which indicates that the FEM updating based on response surface method is of great application value on the design for a rod-type ultrasonic motor.     

A state space force identification method based on Markov parameters precise computation and regularization technique

A time domain force identification approach for linear system is proposed. This approach can found a highly precise force identification model within the scope of general computer precision while it does not cost much computing time. Although the force identification model is accurate, the force identification process, like other inverse methods, is still ill-posed due to the inversion process and the white noise in measured structural responses. The singular value decomposition is used to reveal the intrinsically matter of the ill-posedness of force identification problem and a regularization technique is utilized to deal with this issue. Finally, the proposed method with the aid of regularization technique is successfully applied to identify the input forces in two numerical simulations.     

Nonlinear structural identification by the “Reverse Path” spectral method

When dealing with nonlinear dynamical systems, it is important to have efficient, accurate and reliable tools for estimating both the linear and nonlinear system parameters from measured data. An approach for nonlinear system identification widely studied in recent years is “Reverse Path”. This method is based on broad-band excitation and treats the nonlinear terms as feedback forces acting on an underlying linear system. Parameter estimation is performed in the frequency domain using conventional multiple-input–multiple-output or multiple-input–single-output techniques. This paper presents a generalized approach to apply the method of “Reverse Path” on continuous mechanical systems with multiple nonlinearities. The method requires few spectral calculations and is therefore suitable for use in iterative processes to locate and estimate structural nonlinearities. The proposed method is demonstrated in both simulations and experiments on continuous nonlinear mechanical structures. The results show that the method is effective on both simulated as well as experimental data.     

Effects of internal mass distribution and its isolation on the acoustic characteristics of a submerged hull

The primary aim of machinery isolation in marine vessels is to isolate structural vibration of the onboard machinery from the hull and to reduce far-field radiation of underwater noise. A substantial proportion of the total submarine mass is on flexible mounts that isolate supported masses from the hull at frequencies above the mounting system resonant frequency. This reduces the dynamically effective mass of the hull and affects the signature of the marine vessel due to propeller excitation. A fully coupled finite element/boundary element (FE/BE) model has been developed to investigate the effect of mass distribution and isolation in a submerged hull. The finite element model of the structure includes internal structures to represent the machinery and other flexibly mounted components. Changes in the radiated sound power demonstrate the effect of machinery isolation on the acoustic signature of the submerged hull due to the external propeller forces. Results are also presented to show how the arrangement of flexible mounts for a large internal structure can influence the radiation due to machinery forces.     

A time-varying identification method for mixed response measurements

The proper orthogonal decomposition is a method that may be applied to linear and nonlinear structures for extracting important information from a measured structural response. This method is often applied for model reduction of linear and nonlinear systems and has been applied recently for time-varying system identification. Although methods have previously been developed to identify time-varying models for simple linear and nonlinear structures using the proper orthogonal decomposition of a measured structural response, the application of these methods has been limited to cases where the excitation is either an initial condition or an applied load but not a combination of the two. This paper presents a method for combining previously published proper orthogonal decomposition-based identification techniques for strictly free or strictly forced systems to identify predictive models for a system when only mixed response data are available, i.e. response data resulting from initial conditions and loads that are applied together. This method extends the applicability of the previous proper orthogonal decomposition-based identification techniques to operational data acquired outside of a controlled laboratory setting. The method is applied to response data generated by finite element models of simple linear time-invariant, time-varying, and nonlinear beams and the strengths and weaknesses of the method are discussed.     

TESPDA-SEI: Tensor embedding substructure preserving domain adaptation for specific emitter identification

For the specific emitter identification (SEI) with few or no labels, domain adaptation make the model respond quickly with the help of empirical information. However, the more extreme case is that there are so few labeled samples in the source domain that it is difficult to train an excellent recognition model. In fact, it is more valuable to make full use of these limited label information. This work aims at proposing an unsupervised domain adaptation (UDA)-based method to accommodate the typical case of no labels in the target domain and small samples in the source domain when new devices are first introduced. The basic principle is to learn tensor embedding shared feature space and preserving inter-class substructure, which perform feature space mapping under the joint source and target domain led by mapping error minimize in the source domain. Specifically, this tensor embedding substructure preserving domain adaptation (TESPDA) consist of three parts, tensor invariant subspace learning, substructure preserving feature space mapping and pseudo-label prediction, which are used to learn inter-class substructure after tensor space mapping and identify the predict labels for the target domain. Finally, experiments are conducted on the real-word ADS-B dataset to demonstrate the effectiveness of the TESPDA method.     

A generalized modelling technique for linearized motions of mechanisms with flexible parts

Analysis and control of vibrations of agricultural machines to improve machine performance and vibration comfort of the operator is a major concern of manufacturers these days. In this paper, an analytical method to build the linearized equations of motion of an elastic tree structured mechanism, is presented. The method is based on the principle of virtual work resulting in a set of parameterized linear equations that are functions of the mechanical parameters and the geometry and the interconnection structure of different bodies in the mechanism. The rigid-body motions of the mechanical system are represented by Lagrangian generalized co-ordinates while elastic deformations are described by nodal co-ordinates from a finite element formulation. Explicit expressions for external distributed and concentrated forces and internal concentrated forces acting on the mechanism are given.     

基于电磁力近似算法的驱动电机NVH快速仿真方法

电动汽车驱动电机产生的电磁噪声是汽车NVH关注的重点问题,对其进行全转速段多工况NVH仿真通常需要耗费大量时间和计算资源。该文通过有限单元分析揭示了电机气隙电磁力随转速变化的规律,并根据这一规律提出了基于外特性曲线的电磁力时间缩放及插值的近似算法。文章采用电磁力到结构网格的映射算法对结构振动有限元模型进行激励力加载,使用声场有限元方法计算电机的辐射噪声,最终实现了车用驱动电机的电磁振动及噪声的快速仿真。使用该方法对车用电机进行全转速段振动噪声仿真,可大大压缩多工况电磁场有限元分析所需的计算时间,提升仿真效率。     

A new iterative model updating method using incomplete frequency response function data

A new iterative model updating method is proposed for reduced model using incomplete frequency response function (FRF) data. It uses a modified difference vector between the analytical and experimental FRF data to construct a linear sensitivity updating equation system. To improve the convergence performance of the proposed algorithm, a concept of pseudo master degree-of-freedom (DOF) is put forward and the finite element (FE) model is reduced to the measured and user selected pseudo DOFs. The FRFs at pseudo master DOFs are estimated using the impedance matrix of iteratively modified analytical model and the measured FRFs at master DOFs. They are only used to improve the sensitivity matrix and difference calculation between the analytical and experimental FRF data without introducing additional difference equation. At the end, a 25 truss structure is used to evaluate the performance of the proposed method.