On some stochastic eigenvalue problems for technical applications

In general the randomness of the system properties is by far less significant for the reliability of structures than the stochastic nature of the loading. But there exist some exceptions, for example systems subjected to a given harmonic excitation or the loss of stability of structures caused by given compressive loadings or caused by wind in the context of flutter. The name examples can be described by eigenvalue problems. The stability problem as well as the flutter problem are transcendental eigenvalue problems. It is possible however in reasonable simplification to substitute them by linear matrix eigenvalue problems, which are considered in this paper. A simple approach will be discussed, by which the stochastic properties of the lowest eigenvalues and the corresponding eigenvectors can be approximated. It is related to a Galerkin approach and it works with a description of the random fluctuations of the results by means of a Polynomial Chaos expansion. Typical examples are considered to show the application of the procedure, the accuracy of which is compared for several cases with the results of a Monte Carlo simulation.