The random eigenvalue problem for stochastic systems

The description of real-life engineering structures systems is associated with some amount of uncertainty related to material properties, geometric parameters, boundary conditions and applied loads. In the context of structural dynamics, it is necessary to consider random eigenvalue problems in order to account for these uncertainties. A proposed approach based on the combination of the probabilistic Transformation methods for a random variable and the Rayleigh method in order to evaluate the probability density function of the eigenvalue of stochastic systems. This approach has the advantage of giving directly the whole density function (closed-form) of the eigenvalues, which is very helpful for probabilistic analysis. To show the accuracy of the proposed method, an example of a beam is analyzed for an uncertainty in the material (Young's modulus) and the geometry (beam length). The results are compared with Monte Carlo Simulations.