Further experience with model updating incorporating damping matrices

Most of the 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, response function method (RFM) is extended to deal with the complexity of FRF by updating damping matrices along with mass and stiffness matrices. The effectiveness of the damped FE model updating procedure is demonstrated by actual laboratory experiments of an F-shaped test structure. The updated results have shown that the damped RFM model updating procedure can be used to derive accurate model of the system. This is illustrated by matching of the complex FRFs obtained from the updated model with the experimental data.     

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.     

On the use of damped updated FE model for dynamic design

Model updating techniques are used to update the finite element model of a structure, so that updated model predicts the dynamics of a structure more accurately. The application of such an updated model in dynamic design demands that it also predicts the effects of structural modifications with a reasonable accuracy. Most of the model updating techniques neglect damping and so these updated models cannot be used for predicting amplitudes of vibration at resonance and antiresonance frequencies. This paper deals with updating of the finite element model using the FRF data with damping identification using complex modal data and its subsequent use for predicting the effects of structure modifications. The updated model is obtained in two steps. In the first step, mass and stiffness matrices are updated using FRF-based model updating method. In the second step, damping is identified using updated mass and stiffness matrices, which are obtained in first step. Structural modifications in terms of mass and beam modifications are then introduced to evaluate the updated model for its usefulness in dynamic design.     

Distributed parameter model updating using the Karhunen–Loève expansion

Discrepancies between experimentally measured data and computational predictions are unavoidable for complex engineering dynamical systems. To reduce this gap, model updating methods have been developed over the past three decades. Current methods for model updating often use discrete parameters, such as thickness or joint stiffness, for model updating. However, there are many parameters in a numerical model which are spatially distributed in nature. Such parameters include, but are not limited to, thickness, Poisson's ratio, Young's modulus, density and damping. In this paper a novel approach is proposed which takes account of the distributed nature of the parameters to be updated, by expressing the parameters as spatially correlated random fields. Based on this assumption, the random fields corresponding to the parameters to be updated have been expanded in a spectral decomposition known as the Karhunen–Loève (KL) expansion. Using the KL expansion, the mass and stiffness matrices are expanded in series in terms of discrete parameters. These parameters in turn are obtained using a sensitivity based optimization approach. A numerical example involving a beam with distributed updating parameters is used to illustrate this new idea.     

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.     

Mechanical joint parameter estimation using frequency response functions and component mode synthesis

This paper treats the problem of finite element model updating of structures consisting of substructures connected through mechanical joints whose stiffness and damping properties are unknown. The model is updated by estimating the mechanical joint parameters via the curve fit of measured frequency response functions using a non-linear least-squares method. A damped component mode synthesis formulation is used to calculate the theoretical frequency response functions of the assembled structure given the joint stiffness and damping parameter values. It is shown that a good approximation of the sensitivity matrix may be obtained by finite differences at a smaller computational cost than the closed form expression. Identifiability criteria and estimation errors are addressed. A numerical example consisting of two free-free beams connected through axial springs and dampers illustrates the proposed method.     

基于频响函数相关性的灵敏度分析的有限元模型修正

有限元模型的修正对机械结构的动态特性进行准确而可靠的预测是很重要的。利用试验测试和预测的有限元模型计算得到的频响函数（FRF），引入两种频响函数相关性的判定标准，提出基于频响相关函数的灵敏度分析的修正方程。数值实例研究结果表明，该方法利用少量的测量数据，即使测试数据含附加噪声，也可在很宽的频率范围内得到接近真实结构的有限元模型修正解。本文的方法可适用于大型复杂结构的模型修正。     

IDENTIFICATION OF NON-PROPORTIONAL MODAL DAMPING MATRIX AND REAL NORMAL MODES

Equations of motion for non-proportionally damped structures cannot be decoupled by using the real normal modes. For such structures, the complex normal modes are in common use for this purpose, but for the validation of finite element mass and stiffness matrices where physical damping matrices are not available, the related experimental real normal modes must be known. In previous publications, an identification theory using the real normal modes and the non-diagonal modal damping matrix for the non-proportionally damped system and some applications with the computer code ISSPA were presented. However, the theory cannot assure the symmetry of the identified modal damping matrix, which must be theoretically symmetric. In this paper, a method for identifying the symmetric non-proportional modal damping matrix using undamped modal parameters obtained from ISSPA is presented and the validity of the method is demon-strated through both numerical and experimental examples.     

Finite element model updating of vibrating structures under free-free boundary conditions for modal damping prediction

A method to predict resonance frequencies and modal loss factors of bare and damped samples, using constrained layer damping treatment, under free-free boundary conditions is proposed. In a first phase, measurements of the frequency response functions of these two specimens are performed. In a second phase, a finite element model of the undamped sample is developed. The novelty lies in the consistent modelling of the suspension with spring-damper elements defined with stiffness and damping coefficients with fixed values over the whole considered frequency range. By updating these, the agreement between experiments and simulation is further improved. In a third phase, a finite element model of the damped sample, with constrained layer damping material, is realized. A good agreement with experimental results is obtained thanks to an optimization algorithm used to determine the material parameters of the viscoelastic layer at various frequency. A comparison with experimental results, from a Dynamic Mechanical Analysis, confirms the consistency of the results from the optimization process.     

Model updating of rotors supported on ball bearings and its application in response prediction and balancing

This work attempts experimental studies in finite element model updating of an actual rotor system mounted on ball bearings by using Inverse Eigen Sensitivity Method (IESM). The IESM is applied on state space representation of equations of motion and is used to identify bearing stiffness, damping and shaft material damping parameters. Non-proportional viscous damping model is used to model the bearing and shaft material damping. The experimental identification of viscous coefficient of shaft material damping was not found in the available literature and this work attempts the same as well. The updated model is validated for its accuracy by comparing the predicted frequency response with that obtained from the experiments. Finally, it is shown that the updated finite element model of the rotor system can be efficiently used to predict the unbalance in the rotor.     

An identification method for joint structural parameters using an FRF-based substructuring method and an optimization technique

A new method is proposed to identify the joint structural parameters of complex systems using a frequency response function (FRF)-based substructuring method and an optimization technique. The FRF method is used to estimate the joint parameters indirectly by minimizing the difference between the reference and calculated responses using a gradient-based optimization technique with analytical gradient information. To assess the robustness of the identification method with respect to noisy input data, FRFs contaminated by uniformly distributed random noise were tested in a numerical example. The effects of the random noise and the magnitude of the connection stiffness values on the accuracy of the method were investigated while identifying the joint parameters. When the FRFs were contaminated with random noise, the proposed procedure performed well when used to identify the stiffness values, but the accuracy of identification is deteriorative when used to identify the damping coefficients. The joint parameters of a real bolted structure were also identified by the proposed method. The results show that it can be applied successfully to real structures, and that a hybrid approach using both calculated and measured FRFs in the substructure model can enhance the quality of the identification results.     

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.     

Inverse identification of the mechanical parameters of a pipeline hoop and analysis of the effect of preload

To create a dynamic model of a pipeline system effectively and analyze its vibration characteristics, the mechanical characteristic parameters of the pipeline hoop, such as support stiffness and damping under dynamic load, must be obtained. In this study, an inverse method was developed by utilizing measured vibration data to identify the support stiffness and damping of a hoop. The procedure of identifying such parameters was described based on the measured natural frequencies and amplitudes of the frequency response functions (FRFs) of a pipeline system supported by two hoops. A dynamic model of the pipe-hoop system was built with the finite element method, and the formulas for solving the FRF of the pipeline system were provided. On the premise of selecting initial values reasonably, an inverse identification algorithm based on sensitivity analysis was proposed. A case study was performed, and the mechanical parameters of the hoop were identified using the proposed method. After introducing the identified values into the analysis model, the reliability of the identification results was validated by comparing the predicted and measured FRFs of the pipeline. Then, the developed method was used to identify the support stiffness and damping of the pipeline hoop under different preloads of the bolts. The influence of preload was also discussed. Results indicated that the support stiffness and damping of the hoop exhibited frequency-dependent characteristics. When the preloads of the bolts increased, the support stiffness increased, whereas the support damping decreased.     

A geometrical interpretation of damping for discrete classically damped systems

A geometrical interpretation of complex eigenvalues for discrete classically damped systems is demonstrated. Systems with symmetric mass, stiffness, and damping matrices are considered. Graphical results are presented for mass-proportional, stiffness-proportional and for Rayleigh damping. The basic consequences of proportional damping are underlined.     

Adaptive regularization parameter optimization in output-error-based finite element model updating

In finite element (FE) model updating, regularization methods are required to alter the ill-conditioned system of equations towards a well-conditioned one. The present study addresses the regularization parameter determination when implementing the Tikhonov regularization technique in output-error-based FE model updating. As the output-error-based FE model updating results in a nonlinear least-squares problem which requires iteration for solution, an adaptive strategy that allows varying value of the regularization parameter at different iteration steps is formulated, where the optimal regularization parameter at each iteration step is determined based on the computationally efficient minimum product criterion (MPC). The performance of MPC in output-error-based FE model updating is examined and compared with the commonly used L-curve method (LCM) and the generalized cross validation (GCV) through numerical studies of a truss bridge using noise-free and noise-corrupted modal data. It is shown that MPC is effective and robust in determining the regularization parameter compared with the other two methods, especially when noise-corrupted data are used. The adaptive strategy is more efficient than the fixed strategy that uses a constant value of the regularization parameter throughout the iteration process.     

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.     

螺栓结合部切向动力学行为辨识方法

首先,以螺栓结合部切向动力学特性为研究对象,基于频响函数的子结构综合法为理论主线,建立结合部切向动力学行为辨识基本方程,推导其切向动刚度Za的理论表达式;其次,联合奇异值分解与最小二乘法,优化辨识结合部切向等效刚度ka与阻尼ca,由此建立螺栓结合部切向动力学数值模型;最后,利用该模型计算结构测点频响函数,进而将其与实测值进行对比。结果表明,二者吻合程度较高,且特征峰对应频率误差分别为1.38%,1.51%和0.84%,验证了所提方法能有效辨识螺栓结合部切向动力学行为,所得等效动力学参数精度较高,可为机械系统整机精准建模提供理论参考与数据支撑。     

SYMMETRIC INVERSE EIGENVALUE VIBRATION PROBLEM AND ITS APPLICATION

This paper summarises the authors' previous effort on inverse eigenvalue problem for linear vibrating systems described by a vector differential equation with constant coefficient matrices and non-proportional damping. The inverse problem of interest here is that of determining real symmetric coefficient matrices assumed to represent mass normalised velocity and position coefficient matrices, given a set of specified complex eigenvalues and eigenvectors. There are given two solutions of a symmetric inverse eigenvalue problem presented by Starek and Inman [1, 2].The theory of inverse eigenvalue problem is applied to the model updating problem. The goal of this paper is to recognise that the model updating problem is a subset of the inverse eigenvalue problem. The approach proposed here is to use the results of inverse eigenvalue problem to develop methods for model updating.Comments are made on how their procedure may be used to solve the damage detection problem.     

考虑模型偏差的结构动力学模型修正

针对结构有限元模型修正后仍可能存在模型偏差的问题,提出用待修正参数的不确定性来表征模型偏差的有限元模型修正方法。首先,基于响应面方法识别得到待修正参数的最优值,并通过计算结果与试验结果比较获得模型偏差;然后,基于响应面模型并结合灵敏度分析计算得到模型偏差对待修正参数的影响,从而得到考虑模型偏差后待修正参数的区间;最后,通过一个悬臂梁工程实例的模型修正,验证了笔者所提出方法的可行性。结果表明,考虑模型偏差的修正可以提高模型可靠性。