Uncertainty in vibration modes induced by spatially random material properties

A method is developed to quantify the variability in the natural frequencies and mode shapes of structural components with spatially random mechanical properties, viz. modulus of elasticity and density of the material. The mechanical properties are considered to be Gaussian, truncated Gaussian, or lognormal random fields. The method is developed within the finite element framework, with the vector of density or modulus of elasticity, associated with the different elements, described by a multivariate distribution corresponding to the underlying random field. The method involves Taylor series-based first order uncertainty propagation to find the means and variances of the natural frequencies, as well as the mean vector and covariance matrix of the mode shapes. Jacobi’s formula for the derivative of a determinant is used to find the variance of the natural frequencies, prior to finding the covariance matrix of the mode shapes. This decouples the uncertainty quantification of the natural frequencies from the mode shapes, making the proposed method simpler than existing techniques. The method is illustrated using simply supported Euler-Bernoulli beam and a rectangular Reissner-Mindlin plate, and is validated against results from Monte Carlo simulations. It is shown that the method is capable of accurately quantifying the uncertainty in the first several vibration modes.