Random matrix eigenvalue problems in structural dynamics

Large - Scale Eigenvalue Problems in Structural Dynamics

Some Unusual Matrix Eigenvalue Problems

We survey some unusual eigenvalue problems arising in different applications. We show that all these problems can be cast as problems of estimating quadratic forms. Numerical algorithms based on the well-known Gauss-type quadrature rules and Lanczos process are reviewed for computing these quadratic forms. These algorithms reference the matrix in question only through a matrix-vector product op...

Eigenvalue Separation in Some Random Matrix Models

The eigenvalue density for members of the Gaussian orthogonal and unitary ensembles follows the Wigner semi-circle law. If the Gaussian entries are all shifted by a constant amount c/(2N)1/2, where N is the size of the matrix, in the large N limit a single eigenvalue will separate from the support of the Wigner semi-circle provided c > 1. In this study, using an asymptotic analysis of the secul...

Sign Wave Analysis in Matrix Eigenvalue Problems

Random Eigenvalue Problems for Large Systems

The inherent uncertainties in geometry, material properties, etc. of engineering structures can be represented by stochastic models, where the parameters are described by probabilistic laws. Results from any analysis based on stochastic models inherit probabilistic information as well, which can be used e.g. for reliability analysis. Particularly in linear dynamics of structures the calculation...