Approximations for parameter-dependent eigenvalue problems arising in structural vibrations

Abstract

Eigenvalue problems, that depend on a parameter, are frequently encountered in structural engineering. The most common contexts are free vibration and structural stability, where the calculations of natural frequency, or the critical buckling load, for a large number of design parameters is of practical interest. In the present work, approximations suitable for early stages of design exploration are presented, since reanalysis is an expensive process. A novel approach that leads to approximations for several classes of problems is presented. The essence of the proposed approximations is to obtain a trial vector, that is rich in the components of the actual eigenvector, first. This is achieved economically via an interpolation of eigenvectors as proposed here. Following this, Rayleigh quotient approximation with these trial vectors is carried out, which is found to provide excellent approximations for a range of problems economically. The method is then applied to illustrative examples arising in structural vibration. The computational gain is found to be relatively more significant as the size of the problem increases. The computational complexity of the proposed method is assessed.

The proposed approximation requires adaptations depending upon whether the matrices involved are symmetric, skew-symmetric, or general asymmetric, and also if the parameter-dependent eigenvalue problem is standard, or generalised i.e. in terms of a single matrix, or a matrix pencil. There are three primary reasons for separately dealing with different classes of problems associated with different symmetries of the matrices involved: (i) because of different orthogonality relations, the Rayleigh quotient for each case is different, and (ii) the nature of eigensolutions, e.g. real vs pure imaginary vs complex could depend upon the class or problem at hand, and (iii) the practical contexts in which each of these arise are very different: e.g. the presence of gyroscopy, aeroelastic effects, dissipation, follower forces, or a combination of these. Excellent approximations are obtained for various classes of problems considered here, while providing considerable computational economy.

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