A new Gibbs sampling based algorithm for Bayesian model updating with incomplete complex modal data

Correction of stiffness and mass matrices utilizing simulated measured modal data

<div class="htmlview paragraph">With highly complex structures, it is sometimes desirable to derive a dynamic model of the system from experimental vibration data. This paper presents algorithms for synthesizing the mass and stiffness matrices from experimentally derived modal data in a way which preserves the physical significance of the individual mass and stiffness elements. The synthesizing procedures allow for the incorporation of other mass and stiffness data, whether empirical or based on the analyst's insight. The mass and stiffness matrices are derived for a cantilever beam example and are compared with those obtained using earlier techniques.</div>

A two-stage approach to stochastic finite element model updating using FRF data

This research proposed and verified an innovative method to identify and locate structural damage using only the response of operational vibration, that is the displacement acquired by a non-contact optical method. The most efficient and economical way to detect damage within the structure is to monitor its structural health while in operation. However, the uncertainties and the randomness of ambient vibrations due to the operation and environments cause a challenge in conducting the operational analysis. Current technology limits the ability to collect data on the properties of the structure without the interruption of operation. Frequencies and mode shapes have been widely used in structural damage detection, but they are not sensitive enough and cannot provide sufficient information for identifying damage locations and their quantification. Therefore, the goal of this research is to design and verify a method to detect the damage, as well as its location and severity, of structures in operation without any physical contacts for data acquisition (i.e., non-contact based structural health monitoring (SHM)). Three algorithms are integrated into this SHM process. The first algorithm is the determination of structural characteristics (frequencies and mode shapes) of a vibrating structure from output-only data. Stochastic Subspace Identification (SSI) method is applied to measured displacements over time to extract the structural characteristics. The second algorithm is to estimate the scaling factor. The mode shapes obtained from the output-only model analysis are unscaled due to the absence of the information of input excitation forces. Mass Change Modal Scale (McMS) algorithm is used to estimate the modal scaling factors and determine the scaled mode shapes. The third algorithm is to estimate the structural system matrices (i.e., mass and stiffness matrices) and assess the damages. A Finite Element Model Updating (FEMU) is applied and the system matrices are updated from frequencies and scaled mode shapes. The damage within the structure can then be detected by analysing changes in mass and stiffness matrices. All three phases are verified by numerical simulation and laboratory experiments with deflections acquired by non-contact optical methods through video system. At last, to achieve the non-contact based SHM, a modal scaling method based on temperature change is proposed and verified by numerical simulation. Experimental program reveals that the proposed algorithm using McMS method is applicable to detect damage locations and their mass losses. With proposed non-contacted based SHM, the limitations of contact based sensor can be addressed, and the structural damage

On the use of damped updated FE model for dynamic design

In order to present an efficient, practical technique to determine progressive failure mechanism of structures, modelling of member deterioration by using a spring system is outlined. The procedure uses updates of member stiffness and mass matrices as well as the random load vector in incremental forms. In this procedure, the assembly process produces redistributions of the system stiffness and mass matrices, and the load vector. In the calculation of response spectral values, the original forms remain unchanged. Inversion of the stiffness matrix is calculated by using the Neumann expansion solution in which the original stiffness matrix is inverted only once so that a considerable computation time is saved in the whole calculation process. An incremental solution technique is presented for spectral analyses of both static and dynamic sensitive structures. In the case of dynamic analysis, special attention is paid to estimations of modified natural frequencies and mode shapes of deteriorated structures, which may affect response spectral values considerably. The technique, which is presented in the paper, is attractive in practical applications and can be efficiently used in the reliability calculation as well, and also it can be successfully used to determine a progressive failure mechanism of the structure.

Component mode synthesis (CMS) is a widely employed model reduction technique used to reduce the computational cost associated with the dynamic analysis of complex engineering structures. To generate CMS models, specifically the formulation of Craig and Bampton, both normal fixed-interface modes and constraint modes of the component’s structure are calculated. These modes are used in conjunction with the component level mass and stiffness matrices to generate reduced mass and stiffness matrices used in the final analyses. For some component models, the most computationally expensive part of this procedure is calculating the component normal modes information. Several different approaches are utilized to investigate the sensitivity of system level responses to variations in several aspects of the CMS models. One approach evaluates changes due to modifications of the reduced mass and stiffness matrices assuming that the mode shapes do not change. The second approach assumes that the mode shapes change but the reduced mass and stiffness matrices do not change. An example is presented to show the influence of these two approaches.

This paper proposes a method to identify mass and stiffness matrices of structures. Element mass and stiffness matrices have mass and stiffness distributions. Therefore, a ratio between two elements in these matrices has a proper value. Element mass and stiffness matrices are divided into several groups considering this relation. A group is represented by a sparse matrix (called group matrix) with only elements in the group matrix. Mass and stiffness matrices are modified group by group. The objective function to be minimized is the sum of each norm of difference between analytical and identified group matrices. A numerical example is given to show the effectiveness of the method.

We consider damped elastodynamic networks where the damping matrix is assumed to be a non-negative linear combination of the stiffness and mass matrices (also known as Rayleigh or proportional damping). We give here a characterization of the frequency response of such networks. We also answer the synthesis question for such networks, i.e., how to construct a Rayleigh damped elastodynamic network with a given frequency response. Our analysis shows that not all damped elastodynamic networks can be realized when the proportionality constants between the damping matrix and the mass and stiffness matrices are fixed.

This paper presents a new Gibbs sampling based approach for Bayesian model updating of a linear dynamic system based on modal data (natural frequencies and partial mode shapes of some of the dominant modes) obtained from a structure using multiple setups. Modal data from multiple setups pose a problem as mode shapes identified from multiple setups are normalized individually and the scaling factors to form the overall mode shape are not known a priori. For comprehensive quantification of the uncertainties, the proposed approach allows for an efficient update of the probability distribution of the model parameters, overall mode shapes, scaling factors, and prediction error variances. The proposed approach does not require solving the eigenvalue problem of any structural model or matching of model and experimental modes, and is robust to the dimension of the problem. The effectiveness and efficiency of the proposed method are illustrated by simulated numerical examples.

A method for determining mass and stiffness modifications to achieve desired natural frequencies is presented. The given data are modal testing results, which consist of a truncated set of natural frequencies and mode shapes. The difficulty arising from the incompleteness of data is overcome by solving an optimization problem rather than seeking an exact solution. The obtained modifications are optimal in a Rayleigh-Ritz sense. The case where the mass and stiffness matrices are interrelated is also considered. Numerical examples demonstrating the various results and the sensitivity of the problem to perturbations are presented.

In this work, we choose the points and weights of the Gauss–Jacobi, Gauss–Radau–Jacobi and Gauss–Lobatto–Jacobi quadrature rules that optimize the number of operations for the mass and stiffness matrices of the high‐order finite element method. The procedure is particularly applied to the mass and stiffness matrices using the tensor‐based nodal and modal shape functions given in (Int. J. Numer. Meth. Engng 2007; 71(5):529–563). For square and hexahedron elements, we show that it is possible to use tensor product of the 1D mass and stiffness matrices for the Poisson and elasticity problem. For the triangular and tetrahedron elements, an analogous analysis given in (Int. J. Numer. Meth. Engng 2005; 63(2):1530–1558) was considered for the selection of the optimal points and weights for the stiffness matrix coefficients for triangles and mass and stiffness matrices for tetrahedra. Copyright © 2009 John Wiley & Sons, Ltd.

Large deformation analysis for anisotropic and inhomogeneous beams using exact linear static solutions

Damage can be identified using generalized flexibility matrix based methods, by using the first natural frequency and the corresponding mode shape. However, the first mode is not always appropriate to be used in damage detection. The contact interface of rod-fastened-rotor may be partially separated under bending moment which decreases the flexural stiffness of the rotor. The bending moment on the interface varies as rotating speed changes, so that the first- and second-modal parameters obtained are corresponding to different damage scenarios. In this paper, a structural damage detection method requiring single nonfirst mode is proposed. Firstly, the system is updated via restricting the first few mode shapes. The mass matrix, stiffness matrix, and modal parameters of the updated system are derived. Then, the generalized flexibility matrix of the updated system is obtained, and its changes and sensitivity to damage are derived. The changes and sensitivity are used to calculate the location and severity of damage. Finally, this method is tested through numerical means on a cantilever beam and a rod-fastened-rotor with different damage scenarios when only the second mode is available. The results indicate that the proposed method can effectively identify single, double, and multiple damage using single nonfirst mode.

The aim of this paper is to do materially nonlinear free vibration analysis of a beam by using the finite element method. In finite element modeling, two approaches are used. In the first approach, the material matrices of concrete and reinforcement are constructed separately, then superimposed to obtain the element stiffness matrix. In the second approach, the reinforcement is assumed to be uniformly distributed throughout the beam. So, the beam is modeled as a single composite element with increasing the modulus of elasticity of concrete by considering the reinforcement ratio. For these two approaches, elastic-perfectly plastic stress-strain relationship is used in compression. In tension, a stress-strain relationship considering tension softening is used. In order to obtain the stiffness and mass matrices of the beam at initial, yield, and failure, a computer program is coded in MATLAB. After these matrices are obtained, MATLAB is used to obtain the nonlinear frequency parameters and mode shapes. It is concluded that the approaches and the model considered in this study can effectively be used in the materially nonlinear free vibration analysis of reinforced concrete (RC) beams. It is also concluded that the effects of the change in the stiffness matrix from initial to yield on the nonlinear frequency parameters of the beam are always larger than those of the change from yield to failure.