A simplified procedure is presented for the determination of the derivatives of eigenvectors of nth order algebraic eigensystems. The method is applicable to symmetric or nonsymmetric systems, and requires knowledge of only one eigenvalue and its associated right and left eigenvectors. In the procedure, the matrix of the original eigensystem of rank (/?-!) is modified to convert it to a matrix of rank /?, which then is solved directly for a vector which, together with the eigenvector, gives the eigenvector derivative to within an arbitrary constant. The norm of the eigenvector is used to determine this constant and complete the calculation. The method is simple, since the modified n rank matrix is formed without matrix multiplication or extensive manipulation. Since the matrix has the same bandedness as the original eigensystems, it can be treated efficiently using the same banded equation solution algorithms that are used to find the eigenvectors.

Simplified calculation of eigenvector derivatives

In a full Bayesian probabilistic framework for "robust" system identification, structural response predictions and performance reliability are updated using structural test data D by considering the predictions of a whole set of possible structural models that are
weighted by their updated probability. This involves integrating h(θ)p(θ|D) over the whole parameter space, where θ is a parameter vector defining each model within the set of possible models of the structure, h(θ) is a model prediction of a response quantity of interest,
and p(θ|D) is the updated probability density for θ, which provides a measure of how plausible each model is given the data D. The evaluation of this integral is difficult because the dimension of the parameter space is usually too large for direct numerical integration and
p(θ|D) is concentrated in a small region in the parameter space and only known up to a scaling constant. An adaptive Markov chain
Monte Carlo simulation approach is proposed to evaluate the desired integral that is based on the Metropolis-Hastings algorithm and a
concept similar to simulated annealing. By carrying out a series of Markov chain simulations with limiting stationary distributions equal
to a sequence of intermediate probability densities that converge on p(θ|D), the region of concentration of p(θ|D) is gradually portrayed.
The Markov chain samples are used to estimate the desired integral by statistical averaging. The method is illustrated using simulated
dynamic test data to update the robust response variance and reliability of a moment-resisting frame for two cases: one where the model
is only locally identifiable based on the data and the other where it is unidentifiable.

Bayesian Updating of Structural Models and Reliability using Markov Chain Monte Carlo Simulation

The sensitivity method in finite element model updating: A tutorial

This article reviews recently developed methods for constitutive parameter identification based on kinematic full-field measurements, namely the finite element model updating method (FEMU), the constitutive equation gap method (CEGM), the virtual fields method (VFM), the equilibrium gap method (EGM) and the reciprocity gap method (RGM) Their formulation and underlying principles are presented and discussed These identification techniques are then applied to full-field experimental data obtained on four different experiments, namely (i) a tensile test, (ii) the Brazilian test, (iii) a shear-flexural test, and (iv) a biaxial test Test (iv) features a non-uniform damage field, and hence non-uniform equivalent elastic properties, while tests (i), (ii) and (iii) deal with the identification of uniform anisotropic elastic properties Tests (ii), (iii) and (iv) involve non-uniform strain fields in the region of interest

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Overview of Identification Methods of Mechanical Parameters Based on Full-field Measurements

A method has been developed which uses measured normal modes and natural frequencies to improve an analytical mass and stiffness matrix model of a structure. The method directly identifies, without iteration, a set of minimum changes in the analytical matrices which force the eigensolutions to agree with the test measurements. An application is presented in which the analytical model had 508 degrees of freedom and 19 modes were measured at 101 locations on the structure. The resulting changes in the model are judged to be small compared to expectations of error in the analysis. Thus, the improved model is accepted as a reasonable model of the structure with improved dynamic response characteristics. In addition, it is shown that the procedure may be a useful tool in identifying apparent measured modes which are not true normal modes of the structure. Nomenclature - analytical matrix = matrix of changes = identity matrix = full improved stiffness and mass matrices (n x n) = full analytical K, M matrices (n x n) = partitions of KA,MA corresponding to test coordinates = partitions of KA,MA corresponding to coupling elements = partitions of KA,MA corresponding to unmea- sured coordinates = number of measured modes = number of degrees of freedom in the model = measures of changes, Eqs. (15-17) = matrix norm, sum of the squares of all the elements = rectangular modal matrix, normalized (n x m) = /th mode, /th column of $ = measured and unmeasured partitions of ,- = diagonal matrix of measured natural frequencies (m xm) = natural frequency of /th mode = 12 17 = sum of the squares of all elements of matrix ( )

Improvement of a Large Analytical Model Using Test Data

A method is formulated for systematically using experimental measurements of the natural frequencies and mode shapes of a structure to modify stiffness and mass characteristics of a finite element model. Throughout the modification process, which does not require complete data, the finite element model remains consistent. An additional feature is that the engineer's confidence in the modeling of the various finite elements is quantified and incorporated into the revision procedure. Examples demonstrate the convergence and versatility of the method.

Statistical Identification of Structures

The treatment of matrix eigenvalue problems with statistical properties is presented in a general form including correlation between matrix elements. Linear equations are formulated for the statistics of the eigenvectors as well as eigenvalues. Results are applied to dynamic systems and verified by an independent Monte Carlo simulation. Although the applications in this study are all structural systems, the theory as developed is applicable to any problem leading to similar eigenvalue equations.

The eigenvalue problem for structural systems with statistical properties.

The Treatment of Randomness in Finite Element Modeling

An analysis procedure is developed for assessing the risk associated with the launch of a missile or space booster. The method considers the various hazardous vehicle failure modes and employs a statistical characterization of the vehicle state vector following a failure; the flight termination process and the effects of the atmosphere on the fragments resulting from vehicle breakup. Based on these statistics, the impact distributions for all impacting vehicle debris are computed and used to define the impact probability and casualty expectation for locations hazarded by the launch. An incremental technique is used in these risk computations whereby fragment impact distributions are computed for discrete vehicle flight times. This allows for a considerably more accurate handling of the various sources of impact uncertainty.

Launch Risk Analysis

Several modal synthesis procedures for the dynamic analysis of largy composite structural systems are surveyed. The matrix formulation of the free-free modal synthesis procedures is presented. Also given are schematic flow charts of the analysis procedure used in three prominent methods. The advantages and disadvantages of several modal synthesis methods for different classes of structural problems are presented in order to enable the engineer to select the best procedure for his particular type of problem.

Jon D. Collins

Papers

Statistical Identification of Structures

The eigenvalue problem for structural systems with statistical properties.

The Treatment of Randomness in Finite Element Modeling

Launch Risk Analysis

A survey of modal synthesis methods.