Methods for Component Mode Synthesis Model Generation for Uncertainty Quantification

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>

Reliable and efficient vibroacoustic loads prediction is often critical in structural design, yet it remains a challenging task for many applications. Spacecraft structures are characterised by extensive use of composite materials, complex connections between components and various non-trivial geometrical features. Accurate treatment necessitates the construction of highly detailed numerical models, traditionally employing deterministic representations. Simultaneously, the broadband acoustic excitation due to the diffuse sound field experienced during launch requires modelling the fluid domain and solving the resulting elasto-acoustic interaction at = multiple frequencies. To alleviate the computational demand implications for large problem sizes, substructuring and reduction techniques for the structural domain are commonplace, component mode synthesis (CMS) being a framework widely adopted in the aerospace industry. Nevertheless, despite ongoing research, the topic still presents a range of difficulties when a universal, robust method of accounting for model uncertainties is sought. In this study, two CMS based approaches are proposed and evaluated. Firstly, the Craig-Bampton stochastic method (CBSM) is improved via a set of modifications enhancing its efficiency, and subsequently adapted for use in a vibroacoustic setting. Optimal perturbation levels and scope of validity of the technique are established against a probabilistic structural analysis (PSA) simulation for a spacecraft structure. Secondly, a novel stochastic finite element method (FEM) is presented. The underlying mathematical foundation is derived so that uncertainty can naturally be controlled at the subsystem level, in partitions of the corresponding condensed mass and stiffness matrices. This decomposition based approach ensures that realisations of the random matrices have key properties such as positive (semi)definiteness strictly preserved, guaranteeing complete robustness. The method is validated with a spacecraft test case, comparing its predictions against PSA, the improved CBSM and experimental data. A coupling scheme with a hierarchical matrix accelerated boundary element method is formulated, resulting in the construction of a complete stochastic vibroacoustic solver.

Current dynamic substructuring methods as approximations to condensation model reduction

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.

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.

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

<div class="htmlview paragraph">This paper presents a method for computing the statistical variance of eigenvalues and eigenvectors by component mode synthesis. The method relies on a modal summation to obtain eigenvector derivatives where the contributions of individual modes are shown to diminish in importance as their natural frequencies become further separated from that of the eigenvector being differentiated. The convergence of mean eigenvalues and eigenvectors, and their standard deviations, is evaluated as the number of component modes used in the synthesis is increased. It is found that convergence proceeds in that order, with the standard deviations of eigenvectors requiring the largest number of modes for convergence. However, numerical investigations have shown that even the standard deviations of eigenvectors tend to converge for the first several system modes, when only a small fraction of the total number of component modes is taken into account. The dependence of convergence on the distribution of randomness and its spatial correlation is also discussed.</div>

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.

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

Of the two main finite element approaches, the displacement method has progressed more rapidly than the force method in the area of element representation. Many authors have contributed to the displacement method with both static and dynamic stiffness matrices for beam elements, plane stress elements and plate bending elements. The plane stress and bending elements being rectangular, triangular or quadrilateral in form. A dynamic stiffness matrix consists of a static stiffness matrix and a mass matrix. In the force method, .static flexibility matrices have been developed for beam elements and plane stress elements. However, static flexibility matrices for plate bending elements have only recently been published (Kaufman and Hall, Morley, Robinson, Przemieniecki). For the past few years, the author has been investigating the rank force method for structural vibration analysis. In ref. 6 a dynamic flexibility matrix is presented for a beam element. This matrix consists of a static flexibility matrix and an inverse mass matrix. Ref. 5 contains the derivation of a dynamic flexibility matrix for a rectangular plate element in bending, twisting and shear. The author, in collaboration with Petyt, demonstrated how a dynamic stiffness and flexibility matrix can be extended to include material damping.

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

A new method for random vibration analysis of stochastic truss structures

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